Exercise
1B
SKILLS
PROBLEM-SOLVING
In this exercise, i and jare unit vectors
acting in a vertical plane, horizontally
and vertically
respectively.
A particle is projected
from a point on a horizontal
plane with an initial
velocity
of
25
ms!
at an angle of 40° above the horizontal.
a Find the horizontal
b Express the initial
and vertical components of the initial
velocity
2 A particle is projected
velocity.
asa vector in the form (pi + qi)
from a cliff top with an initial
velocity
ms-!.
of l8ms-' at an angle of 20° below
the horizontal.
a Find the horizontal
b Express the initial
and vertical components of the initial
velocity
velocity.
asa vector in the form (pi + qi)
ms!,
3 A particle is projected from a point on level ground with an initial
at an angle a above the horizontal, where tan a =
a Find the horizontal
b Express the initial
and vertical components of the initial
velocity
0 below the horizontal,
a Find the horizontal
b Express the initial
5 A particle is projected with
Find the initial
where tan0 =
velocity.
asa vector in terms of i and j.
initial
velocity
U =(6i + 9)ms-!.
with initial
velocity
U =(4i - Sj) ms-!.
speed of the particle and its angle of projection.
7 Aparticle is projected
with initial
velocity
a Find the angle of projection.
Given the initial
velocity
speed of the particle and its angle of projection.
6 A particle is projected
Find the initial
velocity.
with an initial
and vertical components of the initial
velocity
speed is 313 m s!,
b find the value of k.
of 35 m s-!
asa vector in terms of i and j.
4 A particle is projected from the top of a building
at an angle
velocity
U =(3ki + 2kj) ms-.
of 28 ms!
1c
Exercise
In this exercise,
Whenever
PROBLEM-sOLVING
SKILLS
and j are unit vectors
i
acting
in a vertical plane,
horizontally
A particle is projected with speed 35 ms'at an angle of elevation
1
Find
the time
the particle
2 A ball is projected
to reach its greatest
takes
from a point
vertically respectively.
ball above
of 60°.
height.
5m above horizontal
of 40°, Find the height of the
clevation
and
ofg is required, take g = 9.8 ms? unless otherwise stated.
a numerical value
ground
with speed
18
at an
ms-
angle of
2s after projection.
the ground
3 A stone is projected from a point above horizontal ground with speed 32 ms-', at an angle of
10° below the horizontal.The stone takes 2.5s to reach the ground. Find:
a the height of the point of projection
above the ground
b the distance from the point on the ground
point where
A projectile is launched from a point on horizontal
4
10° above
b the range of the
5A
ground
with speed 150
ms
at an
point above
the
ground
projectile.
PROJECTILES
1
O on a horizontal
from a point
particle is projected
of 45°. The particle moves
elevation
freely
plane with speed 20
under gravity
until
it
CHAPTER
to the
Find:
the horizontal.
a the time the projectile takes to reach its highest
12
below the point of projection
vertically
stone reaches the ground.
the
strikes
m s'at an angle of
the ground
at a point X.
Find:
a
the greatest height above
b the distance
the
plane reached by the
particle
OX.
6 A ball is projected from a point A on level ground with speed 24 m s-, The ball is projected
an angle 0 to the horizontal where
the
ground
at a point
freely
under
at
gravity until it strikes
B. Find:
the time of flight of the ball
b the distance
7A
A to B.
from
reached above the point of projection
the nearest
8A
m s- at an angle of elevation
with speed 21
particle is projected
height
particle
a. Given
that
the greatest
m, find the value of a, giving your answer to
degree.
P is projected
from
the origin
and vertical unit vectors
horizontal
is 15
+24j) ms-, where and j are
The particle moves freely under gravity.
with velocity (12i
respectively.
i
a
=3. The ball moves
sin
Find:
the position vector of P after 3s
b the speed of P after 3s.
a
9 A stone
with speed 30
is thrown
a
of the stone
the angle of projection
from the window
which
is 20
m above horizontal ground.
to the point where the stone hits the ground.
.
=
is 3
m above the ground. Find:
the value of U
a
(6 marks
b the time from the instant
the ball is thrown to the instant
A particle Pis projected from a point A
with position vector 20j m with respect
to a fixed origin
projection
0. The velocity
is (Sui
(5ui
that it strikes
the wall.
+4u) ms!
of
+ 4u) m s-.
The particle moves
freely under gravity,
passing through a point B, which has
position
vector (ki
+
12j)
m, where k is
constant, before reaching the point C
a
on the x-axis, as shown
in the diagram.
20 m
m
11
which
A ball is thrown from a point O on horizontal ground with speed Ums- at an angle of
of 0, where tan
The ball strikes a vertical wall which is 20 m from O at a point
elevation
E/P)
from a window
2
10
s-
3.5 s later. Find:
b the horizontal distance
E/P)
m
The stone hits the ground
(2 marks)
angle of
The particle takes 4s to move from A to B. Find:
a
the value of u
(4 marks)
b the value of k
(2 marks)
as
reaches C.
it
c the angle the velocity of P makes with the x-axis
Watch out
use
g
of
When
=9.8 m s
(6 marks)
finding a square
root involving
to work out an answer,
an exact
surd (irrationalnumber) answer is not acceptable.
12
A stone is thrown from point A with speed
30 m s- at an angle of 15° below the horizontal.
The point A is 14 m above horizontal ground.
a
The stone strikes the ground
as shown
at the point
30 ms
B,
diagram. Find:
in the
14m
a the time the stone takes to travel
A to B
from
b the distance
(6 marks)
AB.
(2 marks)
E/P)
13
from a point on level ground with speed
The maximum
reached by the particle
ground
Find
E/P
14
Ums'and angle of elevation
A particle is projected
height
is 42
a.
B
m above the ground and the particle hits the
m from its point of projection.
196
the value of oa
and the value of U.
(9 marks)
= 10 ms².
In this question use g
Ums!
Ums
An object is projected
with speed
from
A at the top ofa vertical building.
a point
The point 4 is 25 m above the ground.
The object is projected at an angle a above
the horizontal, where tan a
The object
=.
hits the ground
horizontal
at the point
of the building,
The object
is at a
m from the foot
as shown
in the diagram.
is modelled
freely under
B, which
of 42
distance
25 m
as a particle moving
R
gravity.
Find:
42 m
a the value of
from A to B
c the speed of the object when
it is 12.4
m above the ground,
your answer
Find the
b
Find:
i
the
value
a
fixed origin
through
passes
O with velocity (4i +5j) ms-'. The particle moves
P with position vector k(i -j)m, where k is a
the point
of k.
(6 marks)
speed of the object
A basketball
at the instant
of motion of the object
the direction
3.05
(5 marks)
constant.
a
ii
16
from
under gravity and
positive
is standing on
player
when
the floor 10
P
through
it passes
when
at the instant
passes
through P.
(7 marks
m from the basket. The height of the basket is
m, and he shoots the ball from a height of 2m, at an angle of 40° above the horizontal.
The basketball
can be modelled as a particle moving
in
a
vertical plane.
Given
that the ball
passes through the basket,
a
find the speed
b
State
with which the basketball
two factors
that
(6 marks)
is thrown.
can be ignored by modelling
the basketball
Challenge
m
ms
A vertical tower is 85
high. A stone isprojected at a speed of 20
from the top of a tower at an angle of a below the horizontal. At the
m
from a window
s
12
Given
that the
in
the tower
horizontally at a speed
between the moment
of
45m above the ground.
twO stones move freely under
plane, and that they collide
is 2.5
projected
is
same time, a second stone
s.
E/P)
2significant figures.
PROJECTILES
An object is projected
freely
(2 marks)
it
15
to
1
in
E/P)
CHAPTER
(6 marks)
to travel
giving
14
U
b the time taken by the object
gravity in the same
vertical
mid-air, show that the time that elapses
they are projected
and the moment
they collide
as a particle.
(2 marks)
REASONING/ARGUMENTATION
Whenever
a numerical value of
P
g
required,
A particle is launched from a point on
1
of elevation
a. The particle moves
The greatest
height
of the particle
2 A particle is projected
point of projection
is
at an
angle
the plane.
strikes
ms'at an angle of elevation a and moves
moved
particle has
a horizontal
distance
xm, its height above the
ym.
=x tan a
Show that y
a
until
hm.
is
from a point with speed 2l
under gravity. When the
freely
Ums-
plane with initial velocity
horizontal
freely under gravity
sin' a
2g
U
Show that h=
a
g=9.8 m s² unless otherwise stated.
take
it
1D)
is
SKILLS
Exercise
a
90 cos²
b Given that y = 8.1 when x= 36, find the value of tan a.
particle is launched
3
from a point on
of elevation a. The particle moves
Rm.
particle
a
plane with initial speed
horizontal
freely under gravity
until
Ums-
strikes the plane.
at an
angle
The range of the
is
it
O A
a
2Usin
a
Show
c Deduce
Given
that, for a fixed
possible
with a speed of vms!,When
is launched vertically
each with speed 2vm s!.
E/P)
5
the
from a point
and moves
horizontal
land a distance
height,
in opposite directions,
m apart.
g = 10ms.
A particle is projected
O with speed U at an angle of elevation
When the particle has moved
freely under gravity.
O is y.
=x tan
a above the
a horizontal
x,
distance
above
that
y
(4 marks)
a
its height
Show
at which
its maximum
reaches
horizontally
4 y2
the two parts of the firework
that
In this question use
a
=45°.
PROJECTILES
firework explodes into two parts, which are projected
Show
o
of the angle of elevation
have been launched.
1
A firework
the
range is when
values
it
CHAPTER
u, the greatest
R= 2U2
find the two possible
5g
that
particle could
18
seconds.
that R= USin 2a
b Show
d
the time of flight of the particle is
that
20'cos
a
A boy throws a stone from a point P at the end of a pier. The point P is 15 m above sea level.
The stone is projected
with a speed of 8 ms-
stone as a particle moving
find the horizontal
6 A particle is projected
and moves
a
of projection
P when
the stone is
2m above sea level. (5 marks)
from a point with speed Uat an angle of elevation
When it has moved a horizontal
freely under gravity.
a above the horizontal
distance x, its height above
Show that y = x tan a - 2U2l+tan²
(5 marks)
An athlete throws a javelin from a point Pat a height of 2m above horizontal
The javelin is projected
at an angle of elevation
By modelling the javelin
is thrown
1.5
m
in a vertical
hits the ground.
(2 marks)
m
above the ground. The ball moves
plane which is perpendicular to
the net.
The ball just passes over the top of
the net,
which is 2.4
2.4m
m above the ground,
1.5
1.5m above
as a particle
the ground
at an
0.9
it
for the horizontal
was hit, show
=9tan a
0,
angle a to the horizontal.
a By writing down expressions
seconds after
-9
projected
Ums- from point
with initial speed
m
in the diagram.
ball is modelled
I
P
of the javelin from
on horizontal ground
away from a
volleyball
hits the ball towards the net 9
The
distance
(5 marl
to the point when
EP) 7 A girl playing
as shown
freely under gravity,
2significant figures, the time elapsed from the point the javelin
find, to
point
ground.
m s-,
m
c
moving
of 45° with a speed of 30
hits the ground
it
when
as a particle
figures, the horizontal
find, to 3 significant
it
b
the
y.
is
point
of the stone from
distance
of 40°. By modelling the
a).
b
E/F
at an angle of elevation
under gravity,
freely
that
when
and
vertical distances
the ball passes
from
O to the ball,
over the net:
81g
2U2 cos²a
(6 marks)
a = 30°,
that
b
the speed of the ball as
find
(6 marks)
passes over the net.
it
Given
PROJECTILES
CHAPTER1
8 In
this question,
i
E/P
and j are unit
An object is projected
where k is a positive
vectors
in
a
the object
is (xi
vertical direction
ground with velocity
horizontal
The object moves
constant.
and upward
horizontal
from a fixed point A on
where it immediately comes to rest. Relative
1
freely under gravity
to 0, the position
(ki
respectively.
+ 2k) ms-,
until it strikes the ground
vector
at B,
of a point on the path of
+yj) m.
gx?
Show
a
Given
that
y
=2x - 2k2
AB = Rm and the maximum
that
b using the
result in part
of the object above the ground
vertical height
a, or otherwise,
find, in
is
Hm,
terms of k andg.
H
ii
R
i
(5 marks)
(6 marks)
Challenge
2v2 U2
Chapter
A ball
is
1
a
the
Um
1
review
Whenever
-m down
distance
a
show that the stone lands
hill.
projected at an angle of 45° above the horizontal with speed
is
Given that
downwards at an angle of 45°, and that the stone
s-t,
point on a straight sloping
a
hill.
projected from
slopes
hill
the
is
A stone
g is required, take g =9.8 ms unless otherwise stated.
numerical value of
thrown
vertically downwards
from the top of a tower with speed
6 ms!.
The ball strikes the ground with speed 25 ms-!. Find the time the ball takes to move
from the top of the tower to the ground.
82
is
2 A child drops a ball from a point at the top of a cliff which
m above the sea.
The ball is initially at rest. Find:
the
c
State
time taken for the ball to reach the sea
one physical
3 The velocity-time
graph represents
of a particle moving
from
a
motion
Velocity+
Use the graph to show that:
iy=u+at
s=("
ii
b
the
the ball hits the sea.
calculation.
in a straight line, accelerating
u at time 0 to velocity v at time
velocity
b the speed with which
which has been ignored in making your
factor
t.
a
Hence show
that:
Time
4
A particle
projected
i s=v-sar
s= ut +sa?
ii
vertically
upwards with a speed of 30
m s-! from a point
A.
E/P
y²=+ 2as
is
i
The point B is h metres above A. The particle moves freely under gravity and is
above
CHAPTER
O 5
motion
(5 marks)
of h.
PROJECTILES
The diagram is a velocity-time
the
the value
1
graph representing
v(ms)4
of a cyclist along a straight road.
Attime t= 0s, the cyclist is moving with velocity
ums-. The velocity is maintained until time
t= 15s, when she slows down with constant
deceleration,
coming
to rest when
t
=23
15
s.
20
B for 2.4 s. Calculate
The total distance she travels in 23 s is 152 m.
23
Find the value of u.
6 A particle P is projected
angle of elevation
the plane. Find:
from a point
45°. After
O on a horizontal
projection,
the particle
a the greatest height above the plane reached by P
b the time of flight of P.
plane with speed 42 ms- and with
moves
freely under gravity
until it strikes
Ts)
7A
is thrown
stone
horizontally
with
speed 21 ms-
of
ball
of h.
the value
thrown from a window
is
8A
distance
Q from the cliff is 56 m.
Calculate
and the angle of elevation
lawn. The velocity of projection
above a horizontal
=.The
is a, where tan a
4
ball takes
a the horizontal distance between the point of projection and
where the ball hits the lawn
is 15 ms-!
sto reach the lawn. Find:
the point
(3 marks)
b the vertical height above the lawn from which the ball was thrown.
projectile is fired with velocity
from a point
gravity
A on horizontal
the ground
until it reaches
at an angle
of elevation
projectile
at the point
A projectile P is projected from a point on a horizontal
U at an angle
speed
!
(5 marks)
of elevation
is 15 m
above the ground.
(5 marks)
plane with
0.
10
of 30°
freely under
B. Find:
b the speed of the projectile at the first instant when
U'sin 20
a Show
b
Hence
that the range of the projectile is
find, as varies, the maximum
(6 marks)
range
of
(2 marks)
the projectile.
2U2
c Given that the range of the projectile isfind the two possible
value
of 0.
(3marks)
Give your answers to the nearest 0.1°.
PROJECTILES
CHAPTER
1
11
B
40 ms!
15.1
m
A golf ball isdriven from a point A with a speed of 40 ms-'at an angle of elevation of 30°.
On its downward
the
diagram
flight, the ball hits a tree at a height
a the time taken by the ball to reach
E/P
12
15.1
m above the level of 4, as shown
in
Find:
above.
its greatest
(3 marks)
above A
height
b the time taken by the ball to travel from A to B
(6 marks)
c the speed
(5 marks)
with
which
the ball hits the tree.
A particle P is projected from a fixed origin O with velocity (12i + 5j) ms-. The particle moves
freely under gravity
a positive
and passes through the point A with position
vector
)m, where A
A(2i-
is
constant.
a Find the value of A.
(6 marks)
b Find:
the speed
of P at the instant when
E/P)
13
passes
through A
of motion of P at the instant when
it
i
ii the direction
it
E/P)
moves
(3 marks)
AB
the distance
a
40 ms-
ground. The
it
9A
!
P on the edge of a cliff
from a point
h metres above sea level. The stone lands in the sea at a point 0, where the horizontal
passes through A.
useg= 10ms-2.
A boy plays a game at a fairground. He needs to throw a ball through a hole in a ve
In this question
to win a prize. The motion of the
gravity. The ball moves
ball is modelled as that
The boy throws the ball horizontally
It hits the target at a point
a Find the horizontal
of a particle moving
in a vertical plane which is perpendicular
at the same
height
freely
to the plane of the
ta
as the hole with a speed of 10
ms
20 cm below the hole.
distance
from the point where the ball was thrown to the
target.
(4 marks)
The boy throws the ball again with the same speed and at the same
b Work
out the
passes
possible angles
through the
above the horizontal
the boy could throw the ball so that it
(6 marks)
20 ms
O on the ground.
The stone is thrown from P with speed
above the point
20 m s- at an angle of a below the
horizontal,
from the target.
hole.
EP) 14 In this question use g = 10 ms²,
A stone is thrown from a point P at a target,
which is on horizontal ground. The point P
is 10m
distance
where tan a
=
10m
g= 10ms².
14 In this question use
stone is thrown from
E/P)
A
which
is 10
point
a
P at
a
target,
ground. The point P
is on horizontal
20 ms
m above the point O on the ground.
The stone is thrown from P with speed
20 m s- at an angle of a below the
horizontal,
where tan a
10m
=
9m
CHAPTER
PROJECTILES
1
The stone
The stone
is modelled as
as shown
in the diagram.
a
and the target
particle
P
to0
(5 marks)
vertically
above T.
A.
(5 marks)
A vertical mast is 32 m high. Two balls P and O are projected
horizontally
from the bottom
of the mast with speed 30
By considering the horizontal
cos
prove that
b Find
a
P is
time. Ball
m s'. Ball Q is projected
m s-! at an angle a above the horizontal.
in the same
vertical plane and
collide
mid-air.
in
motion of each ball,
=
(4 marks)
time which elapses
the
at the same
from the top of the mast with speed 18
The balls move freely under gravity
a
line,
(4 marks)
c Find the speed of the stone at
projected
9 m.
TQ.
The point A is on the path of the stone
15
0T
distance
Find:
a the time taken by the stone to travel from
E/P)
T. The
the target and hits the ground at the point Q, where OTQ is a straight
misses
b the distance
as a point
is
22
between
the instant
when
the balls are projected
and
the instant when they collide.
(4
Challenge
marks)
Problem-solving
You need to calculate the
in a straight
speed of the ball relative
to the golfer.This
the speed
the ball would appear to
the bow (front).
is
(back) of the cruise ship and hits a golf balltowards
initial
line at a constant rate of 1.5 ms-². A golfer stands at the stern
Given that the golfer hits the golf ball at an angle of elevation of 60,
be travelling at
and that the ball lands directly on the bow of the cruise ship,find
you were
on the ship.
standing
the speed, y,with which the golfer hits the ball.
if
A cruise ship is 250 m long, and isaccelerating forwards
The force of gravity causes all objects
of air resistance, this acceleration
towards the Earth.
to accelerate
is constant.
It
1
If
Summary of key points
you ignore the effects
does not depend on the mass
of the
object.
2 An object moving vertically freely under gravity can be modelled as particle with a constant
downward acceleration of g =9.8 m s-.
3 The horizontal motion of projectile is modelled as having constant velocity (a= 0).
You can use the formula s = v.
a
a
The vertical motion
a
with initialvelocity U, at an angle a above the horizontal:
particle is projected
The horizontal component
• The vertical
6
as having constant acceleration due to
component
of the initialvelocity
Ucos
a.
5 When
of a projectile is modelled
is
(a=
gravity
g).
4
of the initialvelocity is Usin a.
A projectile reaches its point of greatest
height
when the vertical component
of itsvelocity is
PROJECTILES
CHAPTER
7 For a particle which is projected
from a point on a horizontal plane with an initialvelocity
•
an angle a above the horizontal, and that moves
Time of flight =
2Usin
1
equal to 0.
a
Usin a
freely
under gravity:
U at
7 For a particle which is projected from a point on a horizontal plane with an
initial
velocity
U at
an angle a above the horizontal, and that moves freely under gravity.
.
Time of flight
=2Usin a
• Time to reach greatest height =
Usin a
•Range on horizontal plane = U sin 2a
• Equation of trajectory: y =x tan a
a)
-gx(1+tan²
2U2
where y is the vertical height of the particle, x is the horizontal distance from the point of
projection, and
g is the acceleration due to gravity.