1.
This question is about earthquake waves.
(a)
(i)
Light is emitted from a candle flame. Explain why, in this situation, it is correct to
refer to the “speed of the emitted light”, rather than its velocity.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
By reference to displacement, describe the difference between a longitudinal wave
and a transverse wave.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
The centre of an earthquake produces both longitudinal waves (P waves) and transverse waves
(S waves). The graph below shows the variation with time t of the distance d moved by the two
types of wave.
d / km
S wave
P wave
1200
800
400
0
0
25
50
75
100
125
150
175
200
225
t/s
1
(b)
Use the graph to determine the speed of
(i)
the P waves.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
the S waves.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(1)
The waves from an earthquake close to the Earth’s surface are detected at three laboratories L1,
L2 and L3. The laboratories are at the corners of a triangle so that each is separated from the
others by a distance of 900 km, as shown in the diagram below.
900 km
L1
L2
L3
2
The records of the variation with time of the vibrations produced by the earthquake as detected
at the three laboratories are shown below. All three records were started at the same time.
L1
L2
start of trace
L3
time
On each record, one pulse is made by the S wave and the other by the P wave. The separation of
the two pulses is referred to as the S-P interval.
(c)
(i)
On the trace produced by laboratory L2, identify, by reference to your answers in
(b), the pulse due to the P wave (label the pulse P).
(1)
(ii)
Using evidence from the records of the earthquake, state which laboratory was
closest to the site of the earthquake.
...........................................................................................................................
(1)
(iii)
State three separate pieces of evidence for your statement in (c)(ii).
1
.................................................................................................................
.................................................................................................................
2
.................................................................................................................
.................................................................................................................
3
.................................................................................................................
.................................................................................................................
(3)
3
(iv)
The S-P intervals are 68 s, 42 s and 27 s for laboratories L1, L2 and L3 respectively.
Use the graph, or otherwise, to determine the distance of the earthquake from each
laboratory. Explain your working.
Distance from L1 = ......................km
...........................................................................................................................
Distance from L2 = ......................km
...........................................................................................................................
Distance from L3 = ......................km
...........................................................................................................................
(4)
(v)
Mark on the diagram a possible site of the earthquake.
(1)
There is a tall building near to the site of the earthquake, as illustrated below.
building
ground
direction of vibrations
The base of the building vibrates horizontally due to the earthquake.
(d)
(i)
On the diagram, draw the fundamental mode of vibration of the building caused by
these vibrations.
(1)
4
The building is of height 280 m and the mean speed of waves in the structure of the building is
3.4 × 103 ms–1.
(ii)
Explain quantitatively why earthquake waves of frequency about 6 Hz are likely to
be very destructive.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(Total 21 marks)
2.
This question is about trajectory motion.
Antonia stands at the edge of a vertical cliff and throws a stone upwards at an angle of 60° to
the horizontal.
v = 8.0ms –1
60°
Sea
5
The stone leaves Antonia’s hand with a speed v = 8.0 m s–1. The time between the stone leaving
Antonia’s hand and hitting the sea is 3.0 s.
The acceleration of free fall g is 10 m s–2 and all distance measurements are taken from the
point where the stone leaves Antonia’s hand.
Ignoring air resistance calculate
(a)
the maximum height reached by the stone.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(b)
the horizontal distance travelled by the stone.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 5 marks)
6
3.
This question is about projectile motion.
A stone is thrown horizontally from the top of a vertical cliff of height 33 m as shown below.
18 m s –1
33 m
sea level
The initial horizontal velocity of the stone is 18 m s–1 and air resistance may be assumed to be
negligible.
(a)
State values for the horizontal and for the vertical acceleration of the stone.
Horizontal acceleration: ............................................................................................
Vertical acceleration: ................................................................................................
(2)
(b)
Determine the time taken for the stone to reach sea level.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(c)
Calculate the distance of the stone from the base of the cliff when it reaches sea level.
...................................................................................................................................
...................................................................................................................................
(1)
(Total 5 marks)
7
4.
This question is about projectile motion.
A stone of mass 0.44 kg is thrown horizontally from the top of a cliff with a speed of 22 m s–1
as shown below.
22 m s–1
32 m
cliff
sea level
The cliff is 32 m high.
(a)
Calculate the total kinetic energy of the stone at sea level assuming air resistance is
negligible.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(b)
In practice, air resistance is not negligible. During the motion of the stone from the top of
the cliff to sea level, 34 of the total energy of the stone is transferred due to air
resistance. Determine the speed at which the stone reaches sea level.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 5 marks)
8
5.
This question is about the trajectory of a golf ball.
A golfer hits a golf ball at point A on a golf course. The ball lands at point D as shown on the
diagram. Points A and D are on the same horizontal level.
–1
30m s
–1
20m s
A
D
The initial horizontal component of the velocity of the ball is 20 m s–1 and the initial vertical
component is 30 m s–1. The time of flight of the golf ball between point A and point D is 6.0 s.
Air resistance is negligible and the acceleration of free fall g = 10 m s–2.
Calculate
(a)
the maximum height reached by the golf ball.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(b)
the range of the golf ball.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 5 marks)
9
6.
This question is about projectile motion.
A stone is projected horizontally from the top of a cliff with a speed 15 m s–1.
15 m s–1
70 m
sea
The height of the cliff is 70 m and the acceleration of free fall is 10 m s–2. The stone strikes the
surface of the sea at velocity V.
(a)
Ignoring air resistance, deduce that the stone strikes the sea at a speed of 40 m s–1.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
Use your answer in (a) to calculate the angle that the velocity V makes with the surface of
the sea.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 4 marks)
10
7.
This question is about projectile motion.
A ball is projected from ground level with a speed of 28 m s–1 at an angle of 30 to the
horizontal as shown below.
30
wall
h
16m
There is a wall of height h at a distance of 16 m from the point of projection of the ball. Air
resistance is negligible.
(a)
Calculate the initial magnitudes of
(i)
the horizontal velocity of the ball;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(1)
(ii)
the vertical velocity of the ball.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(1)
11
(b)
The ball just passes over the wall. Determine the maximum height of the wall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(Total 5 marks)
12
8.
This question is about projectile motion.
The barrel of a rifle is held at an angle  to the horizontal. A bullet fired from the rifle leaves the
barrel at time t = 0 with a speed 200 m s–1. The graph below shows the variation with time t of
the vertical height h of the bullet.
600
500
400
300
h/m
200
100
0
0
5
10
15
20
25
t/s
(a)
Using the axes below, draw a sketch graph to show the variation of h with the horizontal
distance x travelled by the bullet. (Note: this is a sketch graph; you do not have to add
any values to the axes.)
h
x
(2)
(b)
State the expression for the initial vertical component of speed Vv in terms of the initial
speed of the bullet and the angle .
...................................................................................................................................
(1)
(c)
Use data from the graph to deduce that the angle  = 30. (The acceleration for free fall
g = 10 m s–2)
13
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(Total 6 marks)
9.
This question is about projectile motion.
A ball is kicked at an angle to the horizontal. The diagram below shows the position of the ball
every 0.50 s.
30
25
20
vertical displacement / m
15
10
5
0
0
30
10
20
horizontal displacement / m
40
The acceleration of free fall is g = 10 m s–2. Air resistance may be neglected.
(a)
Using the diagram determine, for the ball
(i)
the horizontal component of the initial velocity.
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
the vertical component of the initial velocity.
...........................................................................................................................
...........................................................................................................................
14
(2)
(iii)
the magnitude of the displacement after 3.0 s.
...........................................................................................................................
...........................................................................................................................
(2)
(b)
On the diagram above draw a line to indicate a possible path for the ball if air resistance
were not negligible.
(2)
(Total 7 marks)
This question is about projectile motion.
A stone is thrown from the top of a cliff of height 28 m above the sea. The stone is thrown at a
speed of 14 m s–1 at an angle above the horizontal. Air resistance is negligible.
14 m s–1
...........
10.
28m
sea
The maximum height reached by the stone measured from the point from which it is thrown is
8.0 m.
15
(a)
By considering the energy of the stone, determine the speed with which the stone hits the
sea.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(b)
The stone leaves the cliff at time t = 0. It reaches its maximum height at t = TH. On the
axis below, draw a sketch-graph to show the variation with time t of the magnitude of the
vertical component of velocity of the stone from t = 0 to t = TS, the time just before the
stone strikes the sea.
speed
0
0
t
TH
TS
(4)
(Total 7 marks)
16
11.
This question is about projectile motion.
A projectile is fired horizontally from the top of a vertical cliff of height 40 m.
projectile
cliff
40 m
sea
At any instant of time, the vertical distance fallen by the projectile is d. The graph below shows
the variation with distance d, of the kinetic energy per unit mass E of the projectile.
1400
1300
1200
E / J kg –1
1100
1000
900
800
0
5
10
15
20
d/m
25
30
35
40
17
(a)
Use data from the graph to calculate, for the projectile,
(i)
the initial horizontal speed.
...........................................................................................................................
(1)
(ii)
the speed with which it hits the sea.
...........................................................................................................................
(1)
(b)
Use your answers to (a) to calculate the magnitude of the vertical component of velocity
with which the projectile hits the sea.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 4 marks)
18
12.
This question is about throwing a stone from a cliff.
Antonia stands at the edge of a vertical cliff and throws a stone vertically upwards.
The stone leaves Antonia’s hand with a speed v =8.0 m s–1. Ignore air resistance, the
acceleration of free fall g is 10 m s–2 and all distance measurements are taken from the point
where the stone leaves Antonia’s hand.
(a)
Determine,
(i)
the maximum height reached by the stone.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
the time taken by the stone to reach its maximum height.
.........................................................................................................................
.........................................................................................................................
(1)
19
(b)
The time between the stone leaving Antonia’s hand and hitting the sea is 3.0 s. Determine
the height of the cliff.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(Total 6 marks)
13.
This question is about projectile motion.
A marble is projected horizontally from the edge of a wall 1.8 m high with an initial speed V.
V
1.8 m
ground
20
A series of flash photographs are taken of the marble. The photographs are combined into a
single photograph as shown below. The images of the marble are superimposed on a grid that
shows the horizontal distance x and vertical distance y travelled by the marble.
The time interval between each image of the marble is 0.10 s.
0
0.50
x/m
1.0
1.5
2.0
0
–0.50
y/m
–1.0
–1.5
–2.0
(a)
On the images of the marble at x = 0.50 m and x = 1.0 m, draw arrows to represent the
horizontal velocity VH and vertical velocity VV.
(2)
(b)
On the photograph, draw a suitable line to determine the horizontal distance d from the
base of the wall to the point where the marble hits the ground. Explain your reasoning.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
21
(c)
Use data from the photograph to calculate a value of the acceleration of free fall.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 8 marks)
14.
This question is about linear motion.
A police car P is stationary by the side of a road. A car S, exceeding the speed limit, passes the
police car P at a constant speed of 18 m s–1. The police car P sets off to catch car S just as car S
passes the police car P. Car P accelerates at 4.5 m s–2 for a time of 6.0 s and then continues at
constant speed. Car P takes a time t seconds to draw level with car S.
(a)
(i)
State an expression, in terms of t, for the distance car S travels in t seconds.
...........................................................................................................................
(1)
(ii)
Calculate the distance travelled by the police car P during the first 6.0 seconds of
its motion.
...........................................................................................................................
...........................................................................................................................
(1)
(iii)
Calculate the speed of the police car P after it has completed its acceleration.
...........................................................................................................................
...........................................................................................................................
(1)
(iv)
State an expression, in terms of t, for the distance travelled by the police car P
during the time that it is travelling at constant speed.
...........................................................................................................................
(1)
22
(b)
Using your answers to (a), determine the total time t taken for the police car P to draw
level with car S.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 6 marks)
23
15.
Linear motion
At a sports event, a skier descends a slope AB. At B there is a dip BC of width 12 m. The slope
and dip are shown in the diagram below. The vertical height of the slope is 41 m.
A
(not to scale)
slope
41m
C
B
D
1.8m
dip
12m
The graph below shows the variation with time t of the speed v down the slope of the skier.
25.0
20.0
15.0
v / ms –1
10.0
5.0
0.0
0.0
1.0
2.0
3.0
4.0 5.0
t/s
6.0
7.0
8.0
The skier, of mass 72 kg, takes 8.0 s to ski, from rest, down the length AB of the slope.
24
(a)
Use the graph to
(i)
calculate the kinetic energy EK of the skier at point B.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
determine the length of the slope.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
(b)
(i)
Calculate the change EP in the gravitational potential energy of the skier between
point A and point B.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Use your answers to (a) and (b)(i) to determine the average retarding force on the
skier between point A and point B.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
25
(iii)
Suggest two causes of the retarding force calculated in (ii).
1.
...............................................................................................................
2.
...............................................................................................................
(2)
(c)
At point B of the slope, the skier leaves the ground. He “flies” across the dip and lands on
the lower side at point D. The lower side C of the dip is 1.8 m below the upper side B.
Determine the distance CD of the point D from the edge C of the dip. Air resistance may
be assumed to be negligible.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
(d)
The lower side of the dip is altered so that it is inclined to the horizontal, as shown below.
C
B
D
slope
1.8m
dip
12m
(i)
State the effect of this change on the landing position D.
.........................................................................................................................
.........................................................................................................................
(1)
26
(ii)
Suggest the effect of this change on the impact felt by the skier on landing.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(Total 20 marks)
27
16.
This question is about projectile motion.
A small steel ball is projected horizontally from the edge of a bench. Flash photographs of the
ball are taken at 0.10 s intervals. The resulting images are shown against a scale as in the
diagram below.
0
20
distance / cm
40
60
80
100
0
20
40
60
distance / cm
80
100
120
140
28
(a)
Use the diagram to determine
(i)
the constant horizontal speed of the ball.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
29
(ii)
the acceleration of free fall.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
30
(b)
Mark on the diagram the position of the ball 0.50 s after projection.
In the space below, you should carry out any calculations so that you can accurately
position the ball.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
31
(c)
A second ball is projected from the bench at the same speed as the original ball. The ball
has small mass so that air resistance cannot be neglected. Draw on the diagram the
approximate shape of the path you would expect the ball to take.
(3)
(Total 10 marks)
32
17.
This question is about gravitation.
A space probe is launched from the equator in the direction of the north pole of the Earth.
During the launch, the energy E given to the space probe of mass m is
E=
3GMm
4Re
where G is the Gravitational constant and M and Re are, respectively, the mass and radius of the
Earth. Work done in overcoming frictional forces is not to be considered.
33
(a)
(i)
Explain what is meant by escape speed.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
Deduce that the space probe will not be able to travel into deep space.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
34
The space probe is launched into a circular polar orbit of radius R.
(b)
Derive expressions, in terms of G, M, Re, m and R, for
(i)
the change in gravitational potential energy of the space probe as a result of
travelling from the Earth’s surface to its orbit.
...........................................................................................................................
...........................................................................................................................
(1)
35
(ii)
the kinetic energy of the space probe when in its orbit.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
36
(c)
Using your answers in (b) and the total energy supplied to the space probe as given in (a),
determine the height of the orbit above the Earth’s surface.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
37
A space probe in a low orbit round the Earth will experience friction due to the Earth’s
atmosphere.
(d)
(i)
Describe how friction with the air reduces the energy of the space probe.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
38
(ii)
Suggest why the rate of loss of energy of the space probe depends on the density of
the air and also the speed of the space probe.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
39
(iii)
State what will happen to the height of the space probe above the Earth’s surface
and to its speed as air resistance gradually reduces the total energy of the space
probe.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(Total 18 marks)
40
18.
Kepler’s third law.
(a)
Kepler’s third law states that the period T of the orbit of a planet about the Sun is related
to the average orbital radius R of the planet by the relationship
T2 = KR3
where K is a constant.
(i)
Suggest why the law specifies the average orbital radius.
.........................................................................................................................
.........................................................................................................................
(1)
(ii)
State the name of the force that causes the acceleration of the planets orbiting the
Sun.
.........................................................................................................................
(1)
(iii)
State an expression for the magnitude F of the force in (ii) in terms of the mass MS,
of the Sun, the mass m of the planet, the radius R of the orbit and the universal
gravitational constant G.
.........................................................................................................................
.........................................................................................................................
(1)
(iv)
Hence deduce, explaining your working, that the constant K is given by the
expression
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
(b)
Ganymede is one of the moons of Jupiter and the following data are available.
41
Average orbital radius of Ganymede = 1.1  109 m
(i)
Orbital period of Ganymede
= 6.2  105 s
Universal gravitational constant G
= 6.7  10−11 N m2 kg−2
Deduce that the gravitational field strength of Jupiter at the surface of Ganymede is
approximately 0.1 N kg−1.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Estimate the mass of Jupiter.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(Total 12 marks)
42
19.
Gravitation
The diagram below illustrates the planet Saturn.
Saturn
A ring
2.72 10 8m
Saturn has several rings, each of which consists of many small particles that orbit the planet.
Saturn may be considered to be a sphere with its mass M concentrated at its centre.
(a)
Deduce that, for a particle in one ring moving in a circular orbit of radius R, the linear
speed v of the particle in its orbit is given by the expression
GM = Rv2.
Explain your reasoning.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
One ring, the A ring, has an outer diameter of 2.72  108 m. The mass of Saturn is 5.69 
1026 kg. A particle orbits on the outer edge of this ring. Determine the time for the
particle to complete one orbit of Saturn.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
43
(c)
Another particle of mass m is orbiting at a distance r from the centre of Saturn.
(i)
State a formula, in terms of G, M, m and r for the gravitational potential energy EP
of the particle.
.........................................................................................................................
(1)
(ii)
The gravitational potential energy of this particle decreases. Suggest and explain
the change, if any, in the linear speed of the particle.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(d)
Explain the concept of escape speed.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(e)
A planet has radius R and the acceleration of free fall at its surface is g. The planet may
be considered to be a sphere with its mass concentrated at its centre.
Deduce that the escape speed ves is given by the expression
ves =
(2 gR).
Explain your working and state one assumption that is made in the derivation.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
44
(f)
Calculate the escape speed for a spherical planet of radius 1.7  103 km having an
acceleration of free fall at its surface of 1.6 m s–2.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(g)
The mean kinetic energy EK, in joule, of helium-4 atoms at thermodynamic temperature T
is given by the expression
EK = 2.1  10–23 T.
Determine the surface temperature of the planet such that helium-4 atoms on the surface
of the planet have the escape speed calculated in (f).
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(h)
Suggest one reason why, at temperatures below that calculated in (g), helium will escape
from the planet.
...................................................................................................................................
...................................................................................................................................
(1)
(Total 19 marks)
20.
Motion of a satellite
(a)
Define gravitational potential.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
A satellite of mass m is in a circular orbit around the Earth at height R from the Earth’s
surface. The mass of the Earth may be considered to be a point mass concentrated at the
Earth’s centre. The Earth has mass M and radius R.
45
orbit
satellite mass m
Earth mass M
R
(i)
R
Deduce that the kinetic energy EK of the satellite when in orbit of height R is
EK =
GMm
.
4R
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
46
(ii)
The kinetic energy of the satellite in this orbit is 1.5  1010 J. Calculate the total
energy of the satellite.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(iii)
Explain how your answer to (b)(ii) indicates that the satellite will not escape the
Earth’s gravitational field and state the minimum amount of energy that must be
provided to this satellite so that it does escape.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(Total 11 marks)
21.
This question is about the gravitational field associated with a neutron star.
(a)
Define gravitational field strength.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
47
(b)
Neutron stars are very dense stars of small radius. They are formed as part of the
evolutionary process of stars that are much more massive than the Sun.
A particular neutron star has radius R of 1.6  104 m. The gravitational field strength at its
surface is g0. The escape speed ve from the surface of the star is 3.6  107 m s–1.
(i)
The gravitational potential V at the surface of the star is equal to – g0R. Deduce,
explaining your reasoning, that the escape speed from the surface of the star is
given by the expression
ve = 2 g0 R.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(ii)
Calculate gravitational field strength at the surface of the neutron star.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
48
(c)
The period T of rotation of the neutron star is 0.02 s. Use your answer to (b)(ii) to deduce
that matter is not lost from the surface of the star as a result of its high speed of rotation.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(Total 10 marks)
22.
This question is about gravitation.
(a)
State Newton’s universal law of gravitation.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(b)
The average distance of Earth from the Sun is 1.5  1011 m. The gravitational field
strength due to the Sun at the Earth is 6.0×10–3N kg–1.
Estimate the mass of the Sun.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
49
(c)
Deduce that the orbital period T of a planet about the Sun is given by the expression
T2 = KR3
where R is the radius of the orbit and K is a constant.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(5)
(Total 11 marks)
23.
This question is about gravitation.
A spherical planet has radius R and mass M. A satellite of mass m orbits the planet with constant
linear speed v at a height h above the planet’s surface, as shown below (not to scale).
planet mass M
v
R
satellite mass m
h
50
(a)
Outline why
(i)
although the satellite is moving at constant speed, it is not in equilibrium.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
an object in the satellite appears to be weightless.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(b)
For the satellite in its orbit,
(i)
state an expression, in terms of M, m, R and h, for its potential energy.
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
derive an expression, using the same terms as in (b)(i), for its kinetic energy.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
51
(c)
The total energy of the satellite is reduced. Use your expressions in (b) to outline what
change, if any, occurs in the radius of the orbit and the speed of the satellite.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
(d)
The force of friction between the satellite and the atmospheric air increases as the speed
of the satellite increases. By reference to your answer in (c), suggest why small satellites
will “burn up” as they re-enter the Earth’s atmosphere.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
(Total 17 marks)
24.
Gravitational potential
(a)
Define gravitational potential at a point in a gravitational field.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
52
(b)
The graph below shows the variation with distance R from the centre of a planet of the
gravitational potential V. The radius R0 of the planet = 5.0  106 m. Values of V are not
shown for R  R0.
Use the graph to determine the magnitude of the gravitational field strength at the surface
of the planet.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
53
(c)
A satellite of mass 3.2  103 kg is launched from the surface of the planet. Use the graph
to determine the minimum launch speed that the satellite must have in order to reach a
height of 2.0  107 m above the surface of the planet. (You may assume that it reaches its
maximum speed immediately after launch.)
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
(Total 10 marks)
25.
This question is about the kinematics of an elevator (lift).
(a)
Explain the difference between the gravitational mass and the inertial mass of an object.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
54
An elevator (lift) starts from rest on the ground floor and comes to rest at a higher floor. Its
motion is controlled by an electric motor. A simplified graph of the variation of the elevator’s
velocity with time is shown below.
velocity / m s –1
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0 12.0
time / s
The elevator is supported by a cable. The diagram below is a free-body force diagram for when
the elevator is moving upwards during the first 0.50 s.
tension
weight
(b)
In the space below, draw free-body force diagrams for the elevator during the following
time intervals.
(i)
0.50 to 11.50 s
(ii)
11.50 to 12.00 s
55
(3)
A person is standing on weighing scales in the elevator. Before the elevator rises, the reading on
the scales is W.
(c)
On the axes below, sketch a graph to show how the reading on the scales varies during
the whole 12.00 s upward journey of the elevator. (Note that this is a sketch graph – you
do not need to add any values.)
reading on scales
W
0.00
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0 12.0
time / s
(3)
(d)
The elevator now returns to the ground floor where it comes to rest. Describe and explain
the energy changes that take place during the whole up and down journey.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
(Total 13 marks)
56
26.
This question is about gravitation.
(a)
(i)
Define gravitational potential at a point in a gravitational field.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Explain why values of gravitational potential have negative values.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
The Earth and the Moon may be considered to be two isolated point masses. The masses of the
Earth and the Moon are 5.98  1024kg and 7.35  1022 kg respectively and their separation is
3.84  108 m, as shown below. The diagram is not to scale.
Earth
mass = 5.98 1024kg
Moon
mass = 7.35 1022kg
P
3.84 108m
(b)
(i)
Deduce that, at point P, 3.46  108m from Earth, the gravitational field strength is
approximately zero.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
57
(ii)
The gravitational potential at P is −1.28  106 J kg–1. Calculate the minimum speed
of a space probe at P so that it can escape from the attraction of the Earth and the
Moon.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(Total 10 marks)
27.
This question is about gravitation.
A binary star consists of two stars that each follow circular orbits about a fixed point P as shown
below.
star mass
M1
P
R1
star mass
M2
R2
The stars have the same orbital period T. Each star may be considered to act as a point mass
with its mass concentrated at its centre. The stars, of masses M1 and M2, orbit at distances R1
and R2 respectively from point P.
(a)
State the name of the force that provides the centripetal force for the motion of the stars.
...................................................................................................................................
(1)
(b)
By considering the force acting on one of the stars, deduce that the orbital period T is
given by the expression
58
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(c)
The star of mass M1 is closer to the point P than the star of mass M2. Using the answer in
(b), state and explain which star has the larger mass.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 6 marks)
59
28.
This question is about a spacecraft.
A spacecraft above Earth’s atmosphere is moving away from the Earth. The diagram below
shows two positions of the spacecraft. Position A and position B are well above Earth’s
atmosphere.
A
Earth
B
At position A, the rocket engine is switched off and the spacecraft begins coasting freely. At
position A, the speed of the spacecraft is 5.37  103 m s–1 and at position B, 5.10  103 m s–1.
The time to travel from position A to position B is 6.00  102 s.
(a)
(i)
Explain why the speed is changing between positions A and B.
.........................................................................................................................
.........................................................................................................................
(1)
(ii)
Calculate the average acceleration of the spacecraft between positions A and B.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(iii)
Estimate the average gravitational field strength between positions A and B.
Explain your working.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
60
(b)
The diagram below shows the variation with distance from Earth of the kinetic energy Ek
of the spacecraft. The radius of Earth is R.
energy
Ek
0
R
0
distance
On the diagram above, draw the variation with distance from the surface of Earth of the
gravitational potential energy Ep of the spacecraft.
(2)
(Total 8 marks)
29.
This question is about gravitational potential.
(a)
Define gravitational potential at a point in a gravitational field.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
61
(b)
A planet has mass M and radius R0. The magnitude g0 of the gravitational field strength at
the surface of a planet is
g0 = G
M
R0
2
where G is the gravitational constant.
Use this expression to deduce that the gravitational potential V0 at the surface of the
planet is given by
V0 = –g0R0.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
62
(c)
The graph below shows the variation with distance R from the centre of the planet of the
gravitational potential V. The radius R0 of the planet = 5.0  106 m. Values of V are not
shown for R  R0.
R / 10 7 m
0.5
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
–0.5
–1.0
V / 10 7 Jkg–1
–1.5
–2.0
–2.5
–3.0
–3.5
–4.0
–4.5
–5.0
Use the graph to determine the magnitude of the gravitational field strength at the surface
of the planet.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
63
(d)
A satellite of mass 3.2  103 kg is launched from the surface of the planet. Use the graph
to estimate the minimum launch speed that the satellite must have in order to reach a
height of 2.0  107 m above the surface of the planet. (You may assume that it reaches its
maximum speed immediately after launch.)
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
(Total 12 marks)
30.
This question is about the energy of orbiting satellites.
(a)
Define the term gravitational potential at a point in a gravitational field.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
A satellite is in orbit about Earth at a distance R from the centre of Earth. The Earth may
be regarded as a point mass situated at its centre.
Deduce that the kinetic energy of the satellite is numerically equal to half the potential
energy of the satellite.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(c)
The distance between the centre of the Moon and the centre of Earth is about 4.0  108 m.
The Moon may also be regarded as a point mass situated at its centre. The orbital period
of the Moon about the Earth is 2.4  106 s.
64
(i)
Calculate the orbital speed of the Moon.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Use your answer in (b) and (c)(i) to calculate a value for the gravitational potential
due to Earth at a distance of 4.0  108 m from its centre.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(Total 9 marks)
31.
This question is about orbital motion.
(a)
State Kepler’s third law (the law of periods).
.....................................................................................................................................
.....................................................................................................................................
(1)
65
(b)
A satellite of mass m is in orbit of radius r about Earth. The mass of Earth is ME and the
orbital period of the satellite is T.
State, for the satellite,
(i)
the name of the force that provides the centripetal force.
...........................................................................................................................
(1)
(ii)
the orbital speed in terms of T and r.
...........................................................................................................................
(1)
(c)
Kepler’s third law may be applied to the satellite orbiting the Earth. Use your answers to
(b) to deduce that in Kepler’s third law there is a constant K given by
K=
4π 2
.
GM E
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(d)
State an expression for the gravitational field strength g at the surface of the Earth in
terms of ME and the radius of Earth RE.
.....................................................................................................................................
(1)
66
(e)
For the Earth, the gravitational field strength, g is 10Nkg–1 and the radius RE is 6.4×106m.
Using your answers to (c) and (d), deduce that the orbital period of a satellite that is at a
height RE above the surface of Earth is 1.4×104s.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 10 marks)
32.
This question is about gravitational fields.
(a)
Define gravitational field strength.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
The gravitational field strength at the surface of Jupiter is 25 N kg–1 and the radius of
Jupiter is 7.1  107 m.
(i)
Derive an expression for the gravitational field strength at the surface of a planet in
terms of its mass M, its radius R and the gravitational constant G.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
67
(ii)
Use your expression in (b)(i) above to estimate the mass of Jupiter.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(Total 6 marks)
33.
This question is about gravitational field strength near the surface of a planet.
(a)
(i)
Define gravitational field strength.
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
State why gravitational field strength at a point is numerically equal to the
acceleration of free fall at that point.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(1)
68
(b)
A certain planet is a uniform sphere of mass M and radius R of 5.1  106 m.
(i)
State an expression, in terms of M and R, for the gravitational field strength at the
surface of the planet. State the name of any other symbol you may use.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(1)
(ii)
A mountain on the surface of the planet has a height of 2000 m. Suggest why the
value of the gravitational field strength at the base of the mountain and at the top of
the mountain are almost equal.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
69
(c)
A small sphere is projected horizontally near the surface of the planet in (b). Photographs
of the sphere are taken at time intervals of 0.20 s. The images of the sphere are placed on
a grid and the result is shown below.
point of
release
1.00 cm represents 1.00 m
The first photograph is taken at time t = 0. Each 1.00 cm on the grid represents a distance
of 1.00 m in both the horizontal and the vertical directions.
Use the diagram to
(i)
explain why air resistance on the planet is negligible;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
calculate a value for the acceleration of free fall at the surface of the planet.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
70
(d)
Use your answer to (c)(ii) and data from (b) to calculate the mass of the planet.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 13 marks)
34.
This question is about gravitational potential.
(a)
Define gravitational potential at a point.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
A meteorite moves towards the Moon from a long distance away.
(i)
On the axes below, sketch a graph to show the variation with distance from the
centre of the Moon of the gravitational potential of the meteorite as it approaches
the Moon. The radius of the Moon is r.
gravitational
potential
+ve
0
r
distance from centre of Moon
–ve
(2)
71
(ii)
The radius r of the Moon is 1.7  106 m and its mass is 7.3  1022 kg.
Estimate the impact speed with which the meteorite hits the surface of the Moon.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(iii)
Suggest one factor that will make the impact speed greater than your estimate.
.........................................................................................................................
.........................................................................................................................
(1)
(c)
A similar meteorite moves towards the Earth from a long distance away.
Suggest how the total energy of the meteorite varies with distance when the meteorite is
(i)
outside the Earth’s atmosphere;
.........................................................................................................................
.........................................................................................................................
(1)
(ii)
inside the Earth’s atmosphere.
.........................................................................................................................
.........................................................................................................................
(1)
(Total 10 marks)
72
35.
This question is about gravitational field strength and gravitational potential.
(a)
Define gravitational field strength.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
State an expression for the magnitude of the gravitational field strength gh at a point
height h above a planet in terms of the mass of the planet M, its radius R and the
gravitational constant G.
...................................................................................................................................
...................................................................................................................................
(1)
(c)
The radius of Mars is 3.4  106 m and the magnitude of the gravitational field strength at
a height of 1.2  106 m above its surface is 1.8 N kg–1. Use your answer to (b) to deduce
that the magnitude of the gravitational potential at height of 1.2  106 m above the
surface of Mars is 8.3  106 J kg–1.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
73
(d)
A lump of rock is moving with speed v, towards Mars. Its closest distance of approach to
Mars is at distance 1.8  106 m above the surface of Mars. Deduce that the lump of rock
will go into circular orbit about Mars if the speed v is less than 3.0  103 m s–1.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(Total 8 marks)
74
36.
Interference of waves
Travelling waves
(a)
The graph below shows the variation with time t of the displacement xA of wave A as it
passes through a point P.
Wave A
3.0
2.0
1.0
xA / mm
0.0
0.0
2.0
4.0
6.0
8.0
10.0 t / ms
–1.0
–2.0
–3.0
The graph below shows the variation with time t of the displacement xB of wave B as it
passes through point P.
Wave B
2.0
1.0
xB / mm
0.0
0.0
2.0
4.0
6.0
8.0
10.0 t / ms
–1.0
–2.0
(i)
Calculate the frequency of the waves.
.........................................................................................................................
.........................................................................................................................
(1)
75
(ii)
The waves pass simultaneously through point P. Use the graphs to determine the
resultant displacement at point P of the two waves at time t = 1.0 ms and at time t =
8.0 ms.
At t = 1.0 ms:
...............................................................................................
...............................................................................................
At t = 8.0 ms:
...............................................................................................
...............................................................................................
(3)
(b)
Monochromatic light passes through a double-slit arrangement. The diagram below
shows the variation with distance of the intensity of the fringes of the interference pattern
as observed on a screen.
intensity
bright fringe
dark fringe
0
distance
The intensity of the monochromatic light passing through one of the slits of the doubleslit arrangement is reduced. State, and explain, the effect of this change on the appearance
of the bright fringes and of the dark fringes.
bright fringes: .........................................................................................................
.........................................................................................................
dark fringes:
.........................................................................................................
.........................................................................................................
(2)
76
Standing (stationary) waves
(c)
A tube is filled with water and a tuning fork is sounded above the tube, as shown in
diagram 1.
Diagram 1
Diagram 2
Diagram 3
tuning fork,
frequency 310 Hz
water
56 cm
Water is allowed to run out of the tube and, at the position shown in diagram 2, a loud
sound is heard for the first time. Water continues to run out of the tube and a loud sound
is next heard at the position shown in diagram 3.
(i)
A loud sound indicates that a standing (stationary) wave has been produced in the
tube. Outline how the standing wave is formed.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
On diagram 3, draw lines to represent the standing wave produced in the tube.
Also, identify, with the letter N, the positions of the nodes of the standing wave.
(2)
77
(iii)
The change in height of the water surface between the positions shown in diagram
2 and diagram 3 is 56 cm. The frequency of the tuning fork is 310 Hz. Calculate
the speed of sound in the tube.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
78
Doppler effect
(d)
(i)
State what is meant by the Doppler effect.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
A source of sound has frequency f and is moving with constant speed v directly
towards a stationary observer. The speed of sound in still air is c. Derive an
expression for the frequency fo of the sound heard by the observer. Explain your
working.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(iii)
A moving police car produces sound from its engine and from its siren. The car
passes a stationary person. The person notices a Doppler shift as the car passes.
When the police car, travelling at the same speed, next passes the person, its siren
is not sounding. The Doppler shift is not as noticeable. Suggest one reason for this
observation.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(Total 20 marks)
79
37.
This question is about a satellite orbiting the Earth.
A satellite S is in orbit round the Earth, a distance R = 4.2 × 107 m from the centre of the Earth.
S
Earth
R = 4.2 × 107 m
(a)
On the diagram above, for the satellite in the position shown, draw arrow(s) to represent
the force(s) acting on the satellite.
(1)
(b)
Deduce that the velocity v of the satellite is given by the expression
v2 =
GM
R
where M is the mass of the Earth.
.....................................................................................................................................
.....................................................................................................................................
(1)
(c)
Hence deduce that the period of orbit T of the satellite is given by the following
expression.
(3)
(d)
Use the following information to determine that the orbital period of the satellite is about
24 hours.
80
Acceleration due to gravity at the surface of the Earth g =
GM
= 10 ms−2, where M is
2
RE
the mass of the Earth and RE is the radius of the Earth = 6.4 × 106 m.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(e)
The satellite is moved into an orbit that is closer to the Earth. State what happens to its
(i)
potential energy.
...........................................................................................................................
(1)
(ii)
kinetic energy.
...........................................................................................................................
(1)
(Total 9 marks)
38.
This question is about escape speed and Kepler’s third law.
Jupiter and Earth are two planets that orbit the Sun.
The Earth has mass Me and diameter De. The escape speed from Earth is 11.2 km s–1.
Data for Jupiter are given below.
Mass:
1.90 × 1027 kg (318 Me)
Mean diameter: 1.38 × 105 km (10.8 De)
81
(a)
(i)
State what is meant by escape speed.
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
Escape speed v is given by the expression
v=
.
Determine the escape speed from Jupiter.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(b)
(i)
State Kepler’s third law.
(1)
...........................................................................................................................
...........................................................................................................................
(ii)
In 1610, the moon Ganymede was discovered orbiting Jupiter. Its orbit was found
to have a radius of 15.0 R and period 7.15 days, where R is the radius of Jupiter.
Another moon of Jupiter, Lysithea, was discovered in 1938 and its orbit was found
to have a radius of 164 R and a period of 260 days. Show that these data are
consistent with Kepler’s third law.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(Total 6 marks)
82
39.
This question is about units and momentum.
(a)
Distinguish between fundamental units and derived units.
.....................................................................................................................................
.....................................................................................................................................
(1)
(b)
The rate of change of momentum R of an object moving at speed v in a stationary fluid of
constant density is given by the expression
R = kv2
where k is a constant.
(i)
State the derived units of speed v.
...........................................................................................................................
(1)
(ii)
Determine the derived units of R.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(iii)
Use the expression and your answers in (b)(i) and (b)(ii) to determine the derived
units of k.
...........................................................................................................................
...........................................................................................................................
(1)
83
(c)
Define
(i)
linear momentum.
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
impulse.
...........................................................................................................................
...........................................................................................................................
(1)
(d)
In a ride in a pleasure park, a carriage of mass 450 kg is travelling horizontally at a speed
of 18 m s–1. It passes through a shallow tank containing stationary water. The tank is of
length 9.3 m. The carriage leaves the tank at a speed of 13 m s–1.
18 m s–1
water-tank
13 m s –1
carriage, mass 450 kg
9.3m
As the carriage passes through the tank, the carriage loses momentum and causes some
water to be pushed forwards with a speed of 19 m s–1 in the direction of motion of the
carriage.
(i)
For the carriage passing through the water-tank, deduce that the magnitude of its
total change in momentum is 2250N s.
...........................................................................................................................
...........................................................................................................................
(1)
84
(ii)
Use the answer in (d)(i) to deduce that the mass of water moved in the direction of
motion of the carriage is approximately 120 kg.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(iii)
Calculate the mean value of the magnitude of the acceleration of the carriage in the
water.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(e)
For the carriage in (d) passing through the water-tank, determine
(i)
its total loss in kinetic energy.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(ii)
the gain in kinetic energy of the water that is moved in the direction of motion of
the carriage.
...........................................................................................................................
...........................................................................................................................
(1)
85
(f)
By reference to the principles of conservation of momentum and of energy, explain your
answers in (e).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 20 marks)
40.
This question is about linear and circular motion.
A car moves along a straight road. At time t = 0 the car starts to move from rest and oil begins
to drip from the engine of the car. One drop of oil is produced every 0.80 s. Oil drops are left on
the road. The position of the oil drops are drawn to scale on the grid below such that 1.0 cm
represents 4.0 m. The grid starts at time t = 0.
direction of motion
1.0cm
(a)
(i)
State the feature of the diagram above which indicates that, initially, the car is
accelerating.
...........................................................................................................................
(1)
(ii)
On the grid above, draw further dots to show where oil would have dripped if the
drops had been produced from the time when the car had started to move.
(2)
(iii)
Determine the distance moved by the car during the first 5.6 s of its motion.
...........................................................................................................................
...........................................................................................................................
(1)
(b)
Using information from the grid above, determine for the car,
(i)
the final constant speed.
86
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
the initial acceleration.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(c)
The car then turns a corner at constant speed. Passengers in the car who were sitting
upright feel as if their upper bodies are being “thrown outwards”.
(i)
Identify the force acting on the car, and its line of action, that enables the car to
turn the corner.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
Explain why the passengers feel as if they are being “thrown outwards”.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(Total 13 marks)
87
41.
This question is about Newton’s laws of motion, the dynamics of a model helicopter and the
engine that powers it.
(a)
Explain how Newton’s third law leads to the concept of conservation of momentum in the
collision between two objects in an isolated system.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
(b)
The diagram illustrates a model helicopter that is hovering in a stationary position.
0.70 m
0.70 m
rotating
blades
downward motion of air
The rotating blades of the helicopter force a column of air to move downwards. Explain
how this may enable the helicopter to remain stationary.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(c)
The length of each blade of the helicopter in (b) is 0.70 m. Deduce that the area that the
blades sweep out as they rotate is 1.5 m2. (Area of a circle = r2)
88
...................................................................................................................................
...................................................................................................................................
(1)
(d)
For the hovering helicopter in (b), it is assumed that all the air beneath the blades is
pushed vertically downwards with the same speed of 4.0 m s–1. No other air is disturbed.
The density of the air is 1.2 kg m–3.
Calculate, for the air moved downwards by the rotating blades,
(i)
the mass per second;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
the rate of change of momentum.
.........................................................................................................................
.........................................................................................................................
(1)
(e)
State the magnitude of the force that the air beneath the blades exerts on the blades.
...................................................................................................................................
(1)
(f)
Calculate the mass of the helicopter and its load.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(g)
In order to move forward, the helicopter blades are made to incline at an angle  to the
horizontal as shown schematically below.
89
While moving forward, the helicopter does not move vertically up or down. In the space
provided below draw a free body force diagram that shows the forces acting on the
helicopter blades at the moment that the helicopter starts to move forward. On your
diagram, label the angle.
(4)
90
(h)
Use your diagram in (g) to explain why a forward force F now acts on the helicopter and
deduce that the initial acceleration a of the helicopter is given by
a = g tan
where g is the acceleration of free fall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(5)
(i)
The helicopter is driven by an engine that has a useful power output of 9.0  102 W. The
engine makes 300 revolutions per second. Deduce that the work done in one cycle is 3.0
J.
...................................................................................................................................
...................................................................................................................................
(1)
91
(j)
The diagram below shows the relation between the pressure and the volume of the air in
the engine for one cycle of operation of the engine.
pressure
B
C
D
A
volume
(i)
State the name given to the type of process represented by DA.
.........................................................................................................................
(1)
(ii)
During one cycle of the engine, the gas absorbs Q1 units of thermal energy and Q2
units of thermal energy are transferred from the gas. On the diagram above, draw
labelled arrows to show these energy transfers.
(2)
(iii)
The efficiency of the engine is 60. Using your answer to question (i), calculate
the values of Q1 and Q2.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(Total 30 marks)
92
42.
This question is about satellite motion.
A satellite of mass m orbits a planet of mass M and radius R as shown below. (The diagram is
not to scale.)
planet mass M
x
R
satellite mass m
The radius of the circular orbit of the satellite is x. The planet may be assumed to behave as a
point mass with its mass concentrated at its centre.
(a)
Deduce that the linear speed v of the satellite in its orbit is given by the expression
v=
GM
,
x
where G is the gravitational constant.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
93
(b)
(i)
Derive expressions, in terms of m, G, M and x, for the kinetic energy of the satellite
and for the gravitational potential energy of the satellite.
Kinetic energy:
...........................................................................................................................
...........................................................................................................................
Gravitational potential energy:
...........................................................................................................................
(2)
(ii)
Deduce an expression for the total energy of the satellite.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
The satellite is moved into an orbit closer to the planet where there is friction with the planet’s
atmosphere.
(c)
(i)
State the effect of these frictional forces on the total energy of the satellite.
...........................................................................................................................
(1)
(ii)
Apply your equation in (b)(ii) to deduce that, as a result of this friction, the radius
of the orbit will change continuously.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(iii)
Describe the effect of this change in orbital radius on the speed of the satellite.
...........................................................................................................................
(1)
(iv)
The frictional forces will change as the orbit of the satellite changes. Suggest and
explain the effect on the motion of the satellite of these changing frictional forces.
94
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(Total 13 marks)
43.
Collisions
A large metal ball is hung from a crane by means of a cable of length 5.8 m as shown below.
cable
crane
5.8 m
wall
metal ball
95
In order to knock down a wall, the metal ball of mass 350 kg is pulled away from the wall and
then released. The crane does not move. The graph below shows the variation with time t of the
speed v of the ball after release.
3.0
v / m s–1
2.0
1.0
0.0
0.0
0.2
0.4
0.8
0.6
1.0
1.2
1.4
t/s
The ball makes contact with the wall when the cable from the crane is vertical.
(a)
For the ball just before it hits the wall,
(i)
state why the tension in the cable is not equal to the weight of the ball;
.........................................................................................................................
.........................................................................................................................
(1)
96
(ii)
by reference to the graph, estimate the tension in the cable. The acceleration of free
fall is 9.8 m s–2.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
(b)
Use the graph to determine the distance moved by the ball after coming into contact with
the wall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(c)
For the collision between the ball and the wall, calculate
(i)
the total change in momentum of the ball;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
the average force exerted by the ball on the wall.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
97
(d)
(i)
State the law of conservation of momentum.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
The metal ball has lost momentum. Discuss whether the law applies to this
situation.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(e)
During the impact of the ball with the wall, 12 of the total kinetic energy of the ball is
converted into thermal energy in the ball. The metal of the ball has specific heat capacity
450 J kg–1 K–1. Determine the average rise in temperature of the ball as a result of
colliding with the wall.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(4)
(Total 18 marks)
98
44.
This question is about momentum and energy.
(a)
Define impulse of a force and state the relation between impulse and momentum.
definition
.....................................................................................................................................
.....................................................................................................................................
relation
.....................................................................................................................................
.....................................................................................................................................
(2)
(b)
By applying Newton’s laws of motion to the collision of two particles, deduce that
momentum is conserved in the collision.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(5)
99
(c)
In an experiment to measure the speed of a bullet, the bullet is fired into a piece of
plasticine suspended from a rigid support by a light thread.
24cm
bullet
speed V
plasticine
The speed of the bullet on impact with the plasticine is V. As a result of the impact, the
bullet embeds itself in the plasticine and the plasticine is displaced vertically through a
height of 24 cm. The mass of the bullet is 5.2×10–3 kg and the mass of the plasticine is
0.38 kg.
(i)
Ignoring the mass of the bullet, calculate the speed of the plasticine immediately
after the impact.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
Deduce that the speed V with which the bullet strikes the plasticine is about
160 m s–1.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
100
(iii)
Estimate the kinetic energy lost in the impact.
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
(3)
(d)
Another bullet is fired from a different gun into a large block of wood. The block remains
stationary after impact and the bullet melts completely. The temperature rise of the block
is negligible. Use the data to estimate the minimum impact speed of the bullet.
mass of bullet
specific heat capacity of the material of the bullet
latent heat of fusion of the material of the bullet
melting point of the material of the bullet
initial temperature of bullet
= 5.3×10–3 kg
= 130 J kg–1 K–1
= 870 J kg–1
= 330°C
= 30°C
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(5)
(Total 19 marks)
101
45.
This question is about driving a metal bar into the ground and the engine used in the process.
Large metal bars can be driven into the ground using a heavy falling object.
object mass = 2.0×103 kg
bar mass = 400kg
In the situation shown, the object has a mass 2.0 × 103 kg and the metal bar has a mass of 400
kg.
The object strikes the bar at a speed of 6.0 m s–1 It comes to rest on the bar without bouncing.
As a result of the collision, the bar is driven into the ground to a depth of 0.75 m.
(a)
Determine the speed of the bar immediately after the object strikes it.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
102
(b)
Determine the average frictional force exerted by the ground on the bar.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(c)
The object is raised by a diesel engine that has a useful power output of 7.2 kW.
In order that the falling object strikes the bar at a speed of 6.0 m s–1, it must be raised to a
certain height above the bar. Assuming that there are no energy losses due to friction,
calculate how long it takes the engine to raise the object to this height.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(4)
103
The diagram below shows the relation between the pressure and the volume of the air in the
diesel engine for one cycle of operation of the engine. During the cycle there are two adiabatic
processes, an isochoric process and an isobaric process.
pressure
thermal energy
B
C
D
A
volume
(d)
Explain what is meant by
(i)
an adiabatic process;
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
an isochoric process;
...........................................................................................................................
...........................................................................................................................
(1)
(iii)
an isobaric process.
...........................................................................................................................
...........................................................................................................................
(1)
(e)
Identify, from the diagram, the following processes.
(i)
Adiabatic processes
104
...........................................................................................................................
(1)
(ii)
Isochoric process
...........................................................................................................................
(1)
(iii)
Isobaric process
...........................................................................................................................
(1)
During the process B  C thermal energy is absorbed.
The diesel engine has a total power output of 8.4 kW and an efficiency of 40%. The cycle of
operation is repeated 40 times every second.
(f)
State what quantity is represented on the diagram by the area ABCD.
.....................................................................................................................................
(1)
(g)
Determine the value of the quantity that is represented by the area ABCD.
.....................................................................................................................................
.....................................................................................................................................
(1)
105
(h)
Determine the thermal energy absorbed during the process B  C.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 22 marks)
46.
Temperature, specific heat and latent heat
(a)
Outline how a temperature scale is constructed.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b)
Discuss why even an accurate thermometer may affect the reliability of a temperature
reading.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
106
(c)
(i)
Define specific heat capacity.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
The table below gives data for water and ice.
specific heat capacity of water
4.2 kJ kg–1 K–1
specific latent heat of fusion of ice
330 kJ kg–1
A beaker contains 450 g of water at a temperature of 24C. The thermal (heat)
capacity of the beaker is negligible and no heat is gained by, or lost to, the
atmosphere. Calculate the mass of ice, initially at 0C, that must be mixed with the
water so that the final temperature of the contents of the beaker is 8.0C.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
107
Ideal gases and heat engines
(d)
An ideal gas is contained in a cylinder by means of a piston as shown below.
cylinder
ideal gas
piston
The piston is pushed quickly into the cylinder.
For the resulting change of state of the gas,
(i)
state, and explain, whether the change is isochoric, isobaric or adiabatic.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
use the molecular model of an ideal gas to explain why the temperature of the gas
changes.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(3)
108
(e)
A heat engine operates between a high-temperature source and a sink at a lower
temperature as shown below.
source
engine
W
680J
sink
The overall efficiency of the engine is 15. The engine transfers 680 J of energy to the
sink.
(i)
Determine the work W done by the engine.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
There is a gain in entropy as a result of the engine doing work W. Identify two
further entropy changes and, by reference to the second law of thermodynamics,
state how the three changes are related.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
(Total 21 marks)
109
47.
This question is about power and an ideal gas.
(a)
Define power.
.....................................................................................................................................
.....................................................................................................................................
(1)
(b)
A constant force of magnitude F moves an object at constant speed v in the direction of
the force. Deduce that the power P required to maintain constant speed is given by the
expression
P = Fv
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(c)
Sand falls vertically on to a horizontal conveyor belt at a rate of 60 kg s–1.
sand
60 kg s -1
2.0 m s -1
The conveyor belt that is driven by an engine, moves with speed 2.0 m s–1.
When the sand hits the conveyor belt, its horizontal speed is zero.
(i)
Identify the force F that accelerates the sand to the speed of the conveyor belt.
...........................................................................................................................
(1)
110
(ii)
Determine the magnitude of the force F.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(iii)
Calculate the power P required to move the conveyor belt at constant speed.
...........................................................................................................................
...........................................................................................................................
(1)
(iv)
Determine the rate of change of kinetic energy K of the sand.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(v)
Explain why P and K are not equal.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
111
(d)
The engine that drives the conveyor belt in (c) operates in a cycle. In part of this cycle, air
is compressed in a cylinder of the engine such that the pressure and the temperature of the
air increases. Assuming that the air in the cylinder behaves as an ideal gas, outline how
the kinetic model of an ideal gas accounts for this increase in temperature and pressure.
temperature:
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
pressure:
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(7)
(Total 18 marks)
112
48.
Mechanical power
(a)
A car drives up a straight incline that is 4.8 km long. The total height of the incline is 0.30
km.
The car moves up the incline at a steady speed of 16 m s–1. During the climb, the average
friction force acting on the car is 5.0  102 N. The total weight of the car and the driver is
1.2  104 N.
(i)
Determine the time it takes the car to travel from the bottom to the top of the
incline.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Detemine the work done against the gravitational force in travelling from the
bottom to the top of the incline.
.........................................................................................................................
(1)
(iii)
Using your answers to (a)(i) and (a)(ii), calculate a value for the minimum power
output of the car engine needed to move the car from the bottom to the top of the
incline.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
113
(b)
From the top of the incline, the road continues downwards in a straight line. At the point
where the road starts to go downwards, the driver of the car in (a), stops the car to look at
the view. In continuing his journey, the driver decides to save fuel. He switches off the
engine and allows the car to move freely down the hill. The car descends a height of 0.30
km in a distance of 6.4 km before levelling out.
The average resistive force acting on the car is 5.0  102 N.
Estimate
(i)
the acceleration of the car down the incline.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(5)
(ii)
the speed of the car at the bottom of the incline.
.........................................................................................................................
.........................................................................................................................
(2)
(c)
In fact, for the last few hundred metres of its journey down the hill, the car travels at
constant speed. State the value of the frictional force acting on the car whilst it is moving
at constant speed.
...................................................................................................................................
(1)
(Total 15 marks)
49.
This question is about power output of an outboard motor.
A small boat is powered by an outboard motor of variable power P. The graph below shows the
114
variation with speed v of P when the boat is carrying different loads.
5.0
4.5
350 kg
4.0
3.5
3.0
300 kg
P / kW 2.5
2.0
1.5
250 kg
1.0
200 kg
0.5
0.0
0.0
0.5
1.0
1.5
2.0 2.5
v / ms–1
3.0 3.5
4.0
The masses shown are the total mass of the boat plus passengers,
(a)
For the boat having a steady speed of 2.0 m s–1 and with a total mass of 350 kg
(i)
use the graph to determine the power of the engine.
...........................................................................................................................
(1)
(ii)
calculate the frictional (resistive) force acting on the boat.
...........................................................................................................................
...........................................................................................................................
(2)
Consider the case of the boat moving with a speed of 2.5 ms–1.
(b)
(i)
Use the axes below to construct a graph to show the variation of power P with the
total mass W.
115
200
250
300
350
400
450
W / kg
(6)
(ii)
Use data from the graph that you have drawn to determine the power of the motor
for a total mass of 330 kg.
...........................................................................................................................
(1)
116
The relationship between power P and speed v is of the form
P = kvn
where n is an integer and k is a constant.
The graph below shows the variation of lg v (log10 v) with lg P (log10 P) for the situation when
the total mass is 350 kg. P is measured in kW and v is measured in m s–1.
0.7
1g (P / kW)
0.6
0.5
0.4
0.3
0.2
0.1
–0.4
0.0
–0.3 –0.2 –0.1 0.0
–0.1
0.1
0.2
0.3
0.4
0.5
0.6
1g (v / ms –1)
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
–1.0
117
(c)
Use the graph to deduce the value of n and explain how you obtained your answer.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 13 marks)
50.
This question is about estimating the energy changes for an escalator (moving staircase).
The diagram below represents an escalator. People step on to it at point A and step off at
point B.
B
30 m
A
(a)
40°
The escalator is 30 m long and makes an angle of 40° with the horizontal. At full
capacity, 48 people step on at point A and step off at point B every minute.
(i)
Calculate the potential energy gained by a person of weight 700 N in moving from
A to B.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
118
(ii)
Estimate the energy supplied by the escalator motor to the people every minute
when the escalator is working at full capacity.
...........................................................................................................................
...........................................................................................................................
(1)
(iii)
State one assumption that you have made to obtain your answer to (ii).
...........................................................................................................................
...........................................................................................................................
(1)
The escalator is driven by an electric motor that has an efficiency of 70%.
(b)
(i)
Using your answer to (a)(ii), calculate the minimum input power required by the
motor to drive the escalator.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(ii)
Explain why it is not necessary to take into account the weight of the escalator
when calculating the input power.
...........................................................................................................................
...........................................................................................................................
(1)
119
(c)
Explain why in practice, the power of the motor will need to be greater than that
calculated in (b)(i).
.....................................................................................................................................
.....................................................................................................................................
(1)
(Total 9 marks)
51.
This question is about waves and wave properties.
(a)
(i)
Describe what is meant by a continuous travelling wave.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
With reference to your answer in (a)(i), state what is meant by the speed of a
travelling wave.
...........................................................................................................................
...........................................................................................................................
(1)
(b)
Define, for a wave,
(i)
frequency;
...........................................................................................................................
...........................................................................................................................
(1)
(ii)
wavelength.
...........................................................................................................................
...........................................................................................................................
(1)
A tube that is open at both ends is placed in a deep tank of water, as shown below.
120
tuning fork, frequency 256 Hz
tube
tank of water
A tuning fork of frequency 256 Hz is sounded continuously above the tube. The tube is slowly
raised out of the water and, at one position of the tube, a maximum loudness of sound is heard.
(c)
(i)
Explain the formation of a standing wave in the tube.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
121
(ii)
The tube is raised a further small distance. Explain, by reference to resonance, why
the loudness of the sound changes.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(4)
(iii)
The tube is gradually raised from a position of maximum loudness until the next
position of maximum loudness is reached. The length of the tube above the water
surface is increased by 65.0 cm. Calculate the speed of sound in the tube.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2)
122
A sound wave is incident on the ear of a person. The pressure variation of the sound wave
causes a force F to be exerted on a moveable part of the ear called the eardrum. The variation of
the displacement x of the eardrum caused by the force F is shown below.
F/×10–5 N
8
4
–2.0
–1.0
0 0
1.0
2.0
–2
x/×10 mm
–4
–8
(d)
The eardrum has an area of 30 mm2. Calculate the pressure, in pascal, exerted on the
eardrum for a displacement x of 1.0 × 10–2 mm.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(e)
(i)
Calculate the energy required to cause the displacement to change from x = 0 to
x = +1.5 × 10–2 mm.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
123
(3)
The sound wave causing a maximum displacement of the eardrum of 1.5 × 10–2 mm has
frequency 1000 Hz.
(ii)
Deduce that the energy causing the displacement in (e)(i) is delivered in a time of
0.25 ms. Also, determine the mean power of the sound wave to cause this
displacement.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(4)
(iii)
Suggest the form of energy into which the energy of the sound wave has been
transformed at the eardrum.
...........................................................................................................................
(1)
In an experiment to measure the speed of sound, two coherent sources S1 and S2 produce sound
waves of frequency 1700 Hz. A sound detector is moved along a line AB, parallel to S1S2 as
shown below.
B
X
S1
P
S2
A
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When the detector is at P, such that S1P = S2P, maximum loudness of sound is detected. As the
detector is moved along AB, regions of minimum and maximum loudness are detected. Point X
is the third position of minimum loudness from P. The distance (S2X – S1X) is 0.50 m.
(f)
(i)
Determine the speed of the sound.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(ii)
At X, no sound is detected. The loudness of the sound produced by S1 alone is then
reduced. State and explain the effect of this change on the loudness of sound heard
at X and at P.
at X:
...........................................................................................................
...........................................................................................................
...........................................................................................................
at P:
...........................................................................................................
...........................................................................................................
...........................................................................................................
(4)
(Total 30 marks)
52.
This question is about mechanical power and heat engines.
Mechanical power
(a)
Define power.
...................................................................................................................................
...................................................................................................................................
(1)
125
(b)
A car is travelling with constant speed v along a horizontal straight road. There is a total
resistive force F acting on the car.
Deduce that the power P to overcome the force F is
P = Fv.
...................................................................................................................................
...................................................................................................................................
(2)
126
(c)
A car drives up a straight incline that is 4.80 km long. The total height of the incline is
0.30 km.
4.80 km
0.30 km
The car moves up the incline at a steady speed of 16 m s−1. During the climb, the average
resistive force acting on the car is 5.0  102 N. The total weight of the car and the driver
is 1.2  104 N.
(i)
Determine the time it takes the car to travel from the bottom to the top of the
incline.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
Determine the work done against the gravitational force in travelling from the
bottom to the top of the incline.
.........................................................................................................................
(1)
(iii)
Using your answers to (i) and (ii), calculate a value for the minimum power output
of the car engine needed to move the car from the bottom to the top of the incline.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(4)
127
(d)
From the top of the incline, the road continues downwards in a straight-line. At the point
where the incline starts to go downwards, the driver of the car in (c) stops the car to look
at the view. In continuing his journey, the driver decides to save fuel. He switches off the
engine and allows the car to move freely down the incline. The car descends a height of
0.30 km in a distance of 6.40 km before levelling out.
6.40 km
0.30 km
The average resistive force acting on the car is 5.0  102 N.
Estimate
(i)
the acceleration of the car down the incline;
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(5)
(ii)
the speed of the car at the bottom of the incline.
.........................................................................................................................
.........................................................................................................................
(2)
(e)
In fact, for the last few hundred metres of its journey down the incline, the car travels at
constant speed. State the value of the frictional force acting on the car whilst it is moving
at constant speed.
...................................................................................................................................
(1)
The heat engine
128
(f)
The diagram below shows the idealised pressure-volume (P-V) diagram for one cycle of
the gases in an engine similar to that used in the car.
pressure P
B
C
D
A
volume V
The changes A  B and C  D are adiabatic changes.
(i)
Explain what is meant by an adiabatic change.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
(ii)
State the name given to the change B  C.
.........................................................................................................................
(1)
129
(g)
The useful power output of the engine is 20 kW and the overall efficiency of the engine is
32. The car engine completes 50 cycles every second. Deduce that QH = 1.3 kJ.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(3)
(Total 24 marks)
130