NAME______________________________
IB PHYSICS HL
REVIEW PACKET: OPTION “G”—RELATIVITY
1.
This question is about spacetime, gravity and black holes.
(a)
In both the Special and General Theories of Relativity, Einstein introduced the idea of spacetime.
Consider a particle that is a long way from any large mass. The particle is moving with constant
velocity in the x-direction.
Use this example and the axes below to describe what is meant by spacetime.
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(3)
(b)
The Theory of General Relativity suggests, at distances a long way from large masses, spacetime is
flat. The effect of large masses is to warp spacetime. Explain briefly how Einstein used this idea to
describe, for example, the gravitational attraction between the Earth and an orbiting satellite.
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(3)
(c)
Describe what is meant by a black hole.
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(2)
1
(d)
Estimate the radius of the Sun for it to become a black hole. (Mass of the Sun ≈ 2 × 1030 kg)
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(2)
(Total 10 marks)
2.
This question is about the equivalence principle.
(a)
In a famous “thought experiment” relating to general relativity, Einstein considered a “spaceship”
in a gravity-free region of space accelerating uniformly with respect to an inertial observer, in a
direction perpendicular to its “base”. A narrow beam of light is initially directed parallel to the
base. The diagrams below show this situation.
Diagram 1: View with respect to
the inertial observer
Diagram 2: View with respect to the
observer in the spaceship
Acceleration
Initial direction
of light beam
Acceleration
Initial direction
of light beam
Base
(i)
Base
Draw on Diagram 1, the path taken by the light beam as observed by an inertial observer.
(1)
(ii)
Draw on Diagram 2, the path taken by the light beam as observed from within the
spaceship.
(1)
(iii)
Explain the shape of the paths you have drawn.
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(3)
(b)
Explain how this “thought experiment” relates to the equivalence principle.
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(2)
(Total 7 marks)
2
3.
This question is about black holes.
(a)
Describe, by reference to space-time, what is meant by a black hole.
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(b)
(2)
After a particular star has become a supernova, its mass is 2  1031 kg. Determine the radius of the
black hole it subsequently forms.
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(2)
(Total 4 marks)
4.
(a)
Distinguish between rest mass energy and total energy of a particle.
Rest mass energy:
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Total energy:
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(2)
(b)
Estimate the energy released during the annihilation of an electron-positron pair. Explain why your
answer is an estimate.
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(2)
(c)
The graph shows the variation with speed v of the kinetic energy EK of a particle according to
Newtonian mechanics.
EK
0
0
0.5c
1.0c
1.5c
2.0c
speed / v
On the graph above, draw a line to represent the variation with speed v of the kinetic energy
according to relativistic mechanics.
(2)
(Total 6 marks)
3
5.
This question is about relativistic kinematics.
(a)
State what is meant by an inertial frame of reference.
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(2)
(b)
A spacecraft is moving with a speed of 0.80c with respect to observers on Earth. After 6.0 years of
travel, according to the spacecraft clocks, the spacecraft arrives at a distant solar system.
(i)
Calculate the time the journey has taken according to an observer on Earth.
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(3)
(ii)
Calculate the distance between the Earth and the solar system according to an observer on
Earth.
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(2)
(iii)
The spacecraft observers send a signal to Earth to announce that they have arrived at the
solar system. The spacecraft continues to move. Determine how long it will take the signal
to arrive on Earth according to the spacecraft observers.
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(3)
(Total 10 marks)
4
6.
This question is about the Michelson-Morley experiment.
The diagram below shows the essential features of the apparatus used in the Michelson-Morley
experiment.
moveable mirror
A
fixed mirror
light source
observer
A is a half-silvered mirror.
(a)
State the purpose of the experiment.
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(1)
(b)
On the diagram above, draw rays to show the paths of the light from the source that produce the
interference pattern seen by the observer.
(3)
(c)
For part of the experiment, the whole apparatus was rotated though 90°. Explain why.
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(2)
(d)
Explain the function of the moveable mirror.
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(1)
(e)
Describe the results of the experiment and explain how the result supports the Special Theory of
Relativity.
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(2)
(Total 9 marks)
5
7.
A particle is accelerated from rest by a constant force. The graph below shows the variation with time t of
v
the ratio
where v is the speed of the particle and c is the free space speed of light, as calculated using
c
Newtonian mechanics.
1.5
1.0
v
c
0.5
0.0
t / arbitrary units
(a)
On the graph above, draw the variation with time t of the speed v as calculated using relativistic
mechanics.
(2)
(b)
A particle has rest mass 0.51 MeVc–2 and it is moving at speed 0.90c. Calculate the total energy of
this particle.
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(2)
(Total 4 marks)
8.
The total energy of a particle is always given by E = mc2. Calculate the speed at which a particle is
travelling if its total energy is equal to three times its rest mass energy.
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(Total 3 marks)
6
9.
This question is about space-time, gravitational lensing and black holes.
(a)
State two conditions for the path of a particle to be represented as a straight-line on a space-time
diagram.
1.
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2.
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(2)
(b)
(i)
Explain, with the aid of a diagram, what is meant by gravitational lensing.
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(3)
(ii)
Outline one piece of experimental evidence for gravitational lensing.
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(3)
(c)
Suggest two reasons why the planet Jupiter cannot become a black hole.
1.
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2.
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(2)
(Total 10 marks)
7
10.
This question is about black holes.
(a)
Define the Schwarzchild radius of a black hole.
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(1)
(b)
Suggest why the Schwarzchild radius of a black hole is likely to increase with time.
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(2)
(c)
The diagram shows a black hole with Schwarzchild radius R.
X
Schwarzchild radius
black hole
S
An observer at X sends a signal that is received by the spacecraft S.
(i)
On the diagram, draw a line to indicate a possible path of the radio signal.
(1)
(ii)
Explain the path you have drawn.
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(2)
(Total 6 marks)
8
11.
This question is about mass-energy.
(a)
Distinguish between the rest mass-energy of a particle and its total energy.
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(2)
(b)
The rest mass of a proton is 938 MeV c–2. State the value of its rest mass-energy.
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(1)
(c)
A proton is accelerated from rest through a potential difference V until it reaches a speed of 0.980
c. Determine the potential difference V as measured by an observer at rest in the laboratory frame
of reference.
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(4)
(Total 7 marks)
12.
This question is about General Relativity,
(a)
State the principle of equivalence.
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(2)
A spacecraft is initially at rest on the surface of the Earth. It then accelerates away from Earth into deep
space where it then moves with constant velocity. There is a spring balance supporting a mass from the
ceiling.
0
A
0
B
0
C
The diagrams show the readings on the spring balance at these different stages of the motion.
(b)
Identify and explain, in each case, the motion of the spacecraft that could give rise to the reading
shown
(i)
at rest on the Earth’s surface.
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(2)
9
(ii)
moving away from Earth with acceleration.
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(2)
(iii)
moving at constant velocity in deep space.
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(2)
The spacecraft now accelerates in deep space with an acceleration equal to the acceleration of free fall at
the Earth’s surface.
(c)
State and explain which of the readings A, B or C would now be observed on the spring balance.
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(2)
(Total 10 marks)
13.
This question is about mass-energy.
(a)
Define rest mass.
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(2)
(b)
An electron of rest mass m0 is accelerated through a potential difference V. Explain why, for large
values of V, the formula
1
m0v2 = eV
2
is not appropriate for determining the speed v of the accelerated electron.
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(3)
(c)
An electron is accelerated through a potential difference of 5.0 × 106 V. Determine the mass
equivalence of the change in kinetic energy of the electron.
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(2)
(Total 7 marks)
10
14.
The diagram below illustrates the distortion of space by the gravitational field of a black hole.
(a)
(i)
Describe what is meant by the centre and the surface of a black hole.
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(3)
(ii)
With reference to your answer in (i), define the Schwarzchild radius.
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(1)
(iii)
Calculate the Schwarzchild radius for an object having a mass of 2.0 × 1031 kg (ten solar
masses).
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(2)
Science fiction frequently portrays black holes as objects that “swallow” everything in the Universe.
(iv)
A spacecraft is travelling towards the object in (iii) such that, if it continues in a straight line,
its distance of closest approach would be about 107 m. By reference to the diagram and your
answer in (iii), suggest whether the fate of the spacecraft is consistent with the fate as
portrayed in science fiction.
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(2)
In 1979, Wahl, Carswell and Weymann discovered “two” very distant quasars separated by a small angle.
Spectroscopic examination of the images showed that they were identical.
(b)
Outline how these observations give support to the theory of General Relativity.
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(2)
(Total 10 marks)
11
15.
This question is about relativistic motion.
The radioactive decay of a nucleus of actinium-228 involves the release of a β-particle that has a total
energy of 2.51 MeV as measured in the laboratory frame of reference. This total energy is significantly
larger than the rest mass energy of a β-particle.
(a)
Explain the difference between total energy and rest mass energy.
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(2)
(b)
Deduce that the Lorentz factor, as measured in the laboratory reference frame, for the β-particle in
this decay is 4.91.
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(3)
A detector is placed 37 cm from the actinium source, as measured in the laboratory reference frame.
(c)
Calculate, for the laboratory reference frame,
(i)
the speed of the β-particle.
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(2)
(ii)
the time taken for the β-particle to reach the detector.
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(2)
The events described in (c) can be described in the β-particle’s frame of reference.
(d)
For this frame,
(i)
identify the moving object.
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(1)
12
(ii)
state the speed of the moving object.
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(1)
(iii)
calculate the distance travelled by the moving object.
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(2)
(Total 13 marks)
16.
This question is about relativistic momentum.
An electron is accelerated from rest through a potential difference of 2.00  106 V. Determine the
momentum in MeV c–1 of the electron after acceleration. (Rest mass of electron = 0.511 MeV c–2.)
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(Total 4 marks)
17.
An electron of rest mass m0 is accelerated through a potential difference V. The total mass of the electron
is 3.0 m0.
(a)
Determine the momentum of the accelerated electron, as measured in the laboratory frame of
reference.
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(2)
(b)
After acceleration, this electron collides with a stationary electron. Discuss whether it is possible
for an additional electron-positron pair to be created by this collision. You should consider both the
momentum and the energy.
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(3)
(Total 5 marks)
13
18.
One prediction of Einstein’s general theory of relativity is the effect of “gravitational lensing”. This effect
can be predicted from the principle of equivalence.
(a)
State the principle of equivalence.
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(b)
(1)
Use the principle to explain gravitational lensing.
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19.
(4)
(Total 5 marks)
This question is about the postulates of special relativity.
(a)
State the two postulates of the special theory of relativity.
Postulate 1
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Postulate 2
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(b)
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Two identical spacecraft are moving in opposite directions each with a speed of 0.80 c as measured
by an observer at rest relative to the ground. The observer on the ground measures the separation
of the spacecraft as increasing at a rate of 1.60 c.
0.80 c
(2)
0.80 c
ground
(i)
Explain how this observation is consistent with the theory of special relativity.
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(ii)
(1)
Calculate the speed of one spacecraft relative to an observer in the other.
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(3)
(Total 6 marks)
14
20.
This question is about General Relativity and experimental evidence to support it.
On 29 March 1919, an experiment was carried out by Eddington to provide evidence to support
Einstein’s General Theory of Relativity. The diagram below (not to scale) shows the relative position of
the Sun, Earth and a star S on this date.
S
Sun
Earth
orbit path of
Earth about Sun
This particular date was chosen because at the place where the experiment was carried out, there was a
total eclipse of the Sun.
Eddington measured the apparent position of the star and six months later, he again measured the position
of the star from Earth.
(a)
State why it was necessary for there to be a total solar eclipse to carry out the experiment.
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(1)
(b)
Explain why it was necessary to measure the position of the star six months later.
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(1)
(c)
On the diagram, draw the path of a ray of light from S to the Earth as suggested by Einstein’s
theory.
(1)
(d)
Explain how Einstein’s theory accounts for the path of the ray that you have drawn.
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(2)
(e)
On the diagram, label with the letter A, the apparent position of the star as seen from Earth.
(1)
(Total 6 marks)
15
21.
This question is about muons.
(a)
A muon is formed 4500 m above the surface of the Earth, as measured by an observer on Earth.
This muon takes 2.2 s, as measured in its frame of reference, to reach the Earth’s surface.
Describe how these observations support the concept of length contraction.
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(4)
(b)
A muon created in the laboratory is accelerated from rest through a potential difference of 2.1 
108 V. The rest mass of the muon is 105 MeV c–2. Calculate the mass of the accelerated muon, as
measured in the laboratory frame of reference. The charge on a muon is the elementary charge e.
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(3)
(Total 7 marks)
16
22.
This question is about gravitational redshift.
Alex and Elspeth are in a spaceship that is moving with constant speed in the direction shown. Close to
Alex is a light source fixed to the floor of the spaceship. Both Alex and Elspeth measure the same value
for the frequency of the light emitted by the source.
Alex
Elspeth
The spaceship now starts to accelerate.
(a)
Explain why, to Elspeth, the light from the source close to Alex will now be observed to emit light
of a lower frequency.
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(3)
(b)
Outline how the situation described in (a) leads to the idea of gravitational redshift.
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(2)
(Total 5 marks)
23.
Two electrons are travelling directly towards one another. Each has a speed of 0.80c relative to a
stationary observer. Calculate the relative velocity of approach, as measured in the frame of reference of
one of the electrons.
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(Total 3 marks)
17
MARK SCHEME!
1.
(a)
time
x
(b)
correct labelling of the axes;
any sensible straight line path;
some sensible brief description eg “particles move in both space and time
and so the motion of the particle can be represented by a path in spacetime”;
3
Look for an explanation that contains the following points.
all particles follow the shortest path in spacetime;
if the Earth were not present the satellite would move in a straight line;
the warping of spacetime forces the satellite to follow an orbital path;
3
The description can of course be aided by use of a diagram.
(c)
if an object is dense enough it will cause extreme warping of spacetime;
such that any light leaving the surface will not be able to escape the spacetime
surrounding the object;
OWTTE;
2 max
If the explanation is given in terms of Newtonian gravity then award 1
mark only.
(d)
recognize to use Rsch =
 3000 m;
2GM 2  6.6  10 11  2  10 30
;

c2
9  1016
2
[10]
18
2.
(a)
(i)
(ii)
Diagram 1: a straight line path (with or without showing the
spaceship as having moved);
1 max
Diagram 2: a path clearly curving towards the base;
1 max
Diagram 1: View with respect to the
inertial observer
Diagram 2: View with respect to the
observer in the spaceship
Acceleration
Initial direction
of light beam
Initial direction
of light beam
Base
(iii)
Acceleration
Base
Diagram 1:
for the inertial observer, the light is travelling in a gravity-free region
of space so it would be seen to take a straight-line path / light
travels in a straight line in a gravity-free region of space;
Diagram 2:
for the accelerating observer, the spaceship would be displaced in
the direction of the acceleration in the time the light takes to go across it;
and in this frame of reference, this displacement is attributed to
the beam / OWTTE;
An alternative might be:
in both cases the spaceship would be displaced in the direction of
the acceleration in the time the light takes to go across it;
(both observers would see the light strike the other side of the
spaceship closer to the base);
the observer in the accelerating frame of reference would attribute
this to the light beam “bending ‘downwards’”;
(b)
for the observer in the spaceship the constant acceleration (in a
gravity-free region of space) is indistinguishable from rest / an inertial
frame of reference; in a uniform gravitational field – which is the
equivalence principle / OWTTE;
3 max
2 max
Something like “the observer in the spaceship considers himself to be at
rest and would attribute the bending of light to be a gravitational field”
would receive [2].
[7]
19
3.
(a)
(b)
if object is dense / massive enough it will cause severe warping of space-time;
such that light entering the space-time surrounding the object cannot escape;
Do not accept “light cannot escape”.
2
use of RSCH  2GM
2
c

2  6.67 10 11  2 10 31
310 
8 2
;
= 3  104m;
2
[4]
4.
(a)
rest mass energy: E = m0 c2 where m0 is the rest mass;
total energy:
(b)
(c)
sum of rest mass energy and kinetic energy;
2
energy = 2  0.51 = 1.02MeV;
estimate because only rest-mass energy considered / KE not considered;
2
curved line through origin always “above” given line after about 0.4c;
asymptotic at v = c;
2
[6]
5.
(a)
(b)
observers using rulers and clocks to measure positions and times of events;
these observers are not accelerating;
(i)
realization that 6.0 years is the proper time interval;
calculation of gamma factor γ 
(ii)
(iii)
2
1
1  0.80 2

5
1.67 ;
3
time on Earth   6.0 = 10 yrs;
3
realization that spacecraft has been travelling for 10 years at 0.80c;
so distance is 0.80c  10 = 8.0 ly;
2
let t be the time according to the spacecraft observers, then in this time
Earth will move a distance of 0.80c  t according to spacecraft;
Earth and spacecraft are already separated by 0.80c  6 = 4.8ly
according to spacecraft;
and so ct = (0.80c  t) + 4.8  t 
4.8
 24 years;
0.20
3
[10]
20
6.
(a)
to measure the speed of the Earth through the ether / to search for an
absolute frame of reference / OWTTE;
1
(b)
moveable mirror
A
fixed mirror
light source
(c)
(d)
(e)
observer
line and arrows to show reflection from the moveable mirror;
line and arrows to show reflection from the fixed mirror;
ray from A to observer;
3
light from the two mirrors will (should) now take different times to
reach the observer / OWTTE;
hence there will be a shift in the interference pattern;
2
by moving the mirror (backwards or forwards), any shift in the pattern
can be measured / OWTTE;
1
no shift in interference pattern observed;
supports the idea that the speed of light does not depend on the speed of the
source / speed of observer / that there is no absolute reference system;
2
[9]
7.
(a)
(b)
same up to about 0.3;
then curve asymptotic to 1.0;
2
calculation of γ = 2.3;
E = (γm0c2) = 1.2 MeV;
2
[4]
8.
E = γmc2 = 3mc2;
1
1
v2
γ=
=3 2 =1– ;
2
9
c
v
1 2
c
v=
8c
= 0.94 c (= 2.8 × 108 m s–1);
3
3 max
[3]
21
9.
(a)
(b)
far away from any other mass;
constant velocity;
(i)
(ii)
(c)
2
diagram showing large mass and distant light source, light bends
round mass;
mass warps space-time so that it is curved;
shortest path is now curved not straight;
3
describes observed effect when mass between observer and source;
describes observed effect when mass not present;
clear statement that star is the same in both observations;
3
mass too small;
radius too large;
2
[10]
10.
(a)
(b)
(c)
the radius from within which nothing can escape to the outside / the
distance from the black hole where the escape speed is equal to the
speed of light;
1
the black hole is likely to increase in mass due to material falling into it;
2GM
since R  2 the radius is likely to increase / reference to radius being
c
proportional to mass;
2
(i)
(ii)
any curved path from observer to spacecraft that does not cross
the event horizon;
Do not accept paths that start straight and then curve around
event horizon.
1
the black hole curves / “warps” spacetime;
Radio signal follows shortest distance / geodesic of the curved spacetime; 2
[6]
11.
(a)
2
RME: rest mass times c ;
TE: sum of RME + kinetic energy (assuming no potential energy);
2
(b)
938 MeV;
1
(c)
 m0c2 = m0 c2 + Ve;
Ve =  m0 c2  m0 c2
Ve = m0 c2 (  1);
Ve = 938(4.0);
V = 3750 MV;
4
[7]
22
12.
(a)
a frame of reference accelerating far from all masses with acceleration a;
is completely equivalent to a frame of reference at rest in a gravitational
field of field strength equal to a;
2
Accept “the impossibility of distinguishing gravitational from inertial
effects” for full marks.
(b)
(i)
B;
because the scale reads the weight of the mass;
2
C;
because the scale reads a force F where F = mg + ma;
2
A;
because there is no acceleration and no gravitational force on the mass;
2
B;
because by the equivalence principle the accelerating frame is equivalent to a
frame at rest on Earth’s surface;
2
(ii)
(iii)
(c)
Allow ecf for full marks if same answer as (b)(i).
[10]
13.
(a)
(b)
(c)
mass of object in observer’s frame of reference;
or
mass when not moving;
relative to observer;
for large V, calculated value of v would be greater than c;
this is not possible;
mass increases, so mass is not m0 / other comment;
c2m = eV or ∆m =
2 max
3
(1.6  10 19  5.0  10 6 )
;
(3.0  10 8 ) 2
∆m = 8.9 ×10–30 kg;
2
[7]
14.
(a)
(i)
centre is single point to which all mass would collapse;
surface is where the escape speed is equal to c;
within this surface, mass has “disappeared” from the universe;
3
(ii)
distance from point of singularity to the event horizon / OWTTE;
1
(iii)
RSCH =
2GM (2  6.67  10 11  2  10 31 )

c2
(3  10 8 ) 2
= 3.0 × 104 m;
(iv)
at 107 km, space is not warped;
so Newtonian physics applies;
or other good comment;
2
2 max
Award [0] for a statement of “no” without justification.
(b)
theory suggests that light is affected by gravitational fields;
diagrams or “words” to explain formation of two images;
2
[10]
23
15.
(a)
(b)
rest mass energy is the energy that is needed to create the particle at rest /
reference to E0 = m0c2;
total energy is the addition of the rest energy and everything
else (kinetic etc) / reference to mass being greater when in
motion / E = mc2;
realization that betas are electrons;
so me = 0.511 MeV c–2;
2.51
γ=
; (= 4.91)
0.511
2 max
3
Ignore any spurious calculation from Lorentz factor equation here as the
use of this equation is rewarded below.
(c)
(i)
(ii)
(d)
correct substitution into Lorentz factor equation;
to give v = 0.979c = 2.94 × 10 m s–1;
correct substitution into speed =
2
distance
;
time
to give time = 1.26 ns;
2
(i)
the detector / the laboratory / OWTTE;
1
(ii)
same answer as (c) (i) = 2.94 × 108 m s–1;
1
(iii)
realization that length contraction applies;
37
distance =
= 7.5 cm;

2
[13]
16.
use of E2 = p2c2 + m02c4
E = mc2 = Ve + m0c2 = 2.511 (MeV);
(2.511)2 (MeV)2 = p2c2 + (0.511)2 (MeV)2;
p2c2 = 6.04(MeV)2;
p = 2.46 (MeVc1);
To award [4] intermediate and / or final units are not required.
or
calculation of  from (  1) m0c2 = Ve;
 = 4.91;
use of  to calculate
v = 0.979c;
p = m0v = (4.91  0.511  0.979) = 2.46(MeVc1);
To award [4] intermediate and / or final units are not required.
[4]
24
17.
(a)
(b)
correct substitution into E2 = p2c2 + m02c4;
p2c2 = (1.533)2 – (0.511)2 = 2.089(MeV2)
so p = 1.45 MeV c–1 (= 7.71 × 10–22 Ns);
2
realization that energy is sufficient to create electron / positron pair (at rest);
momentum must be conserved so some particles must have KE so
not all the 1.533 is available for particle creation / OWTTE;
so it is not possible;
3
Award [0] for a “bald” statement without any attempt at justification.
[5]
18.
(a)
an observer cannot tell the difference between the effect of acceleration
(in one direction) and a gravitational field (in the opposite direction);
1
Accept “It is impossible to distinguish between inertial or gravitational
forces” or “there is no way in which gravitational effects can be
distinguished from inertial effects” / OWTTE.
(b)
any correct argument to show that light would be expected to be bent in
an accelerating frame (eg observer in lift / rocket etc);
application of principle of equivalence to show that light must also
be bent in a gravitational field;
gravitational lensing is the bending of light around a massive astronomical
object; to produce multiple images or magnified images of a region of space
that is further away / OWTTE;
4
The final [2] marks can be awarded for a clearly drawn and fully
labelled diagram.
[5]
19.
(a)
(b)
the speed of light in vacuum is the same for all inertial observers;
the laws of Physics are the same in all inertial frames of reference;
(i)
(ii)
2
this faster than light speed is not the speed of any physical object /
inertial observer and so is not in violation of the theory of SR;
1
uv
with v = –0.80 c and u = 0.80 c so that
uv
1 2
c
0.80c  0.80c
u =
;
0.80c  0.80c
1
c2
1.60c
u=
;
1.64
u = 0.98 c;
3
u =
[6]
25
20.
(a)
in order that the star could be seen;
1
(b)
in order that the degree / amount of bending of the light by the Sun can
be measured / OWTTE;
1
path showing bending by the Sun;
1
(c)
Note that a correct diagram may also include rays from the other side of
the Sun.
Only accept rays from the star that at the point of closest approach to the
Sun are no more than about 1 cm from the Sun’s surface.
A
S
Sun
Earth
orbit path of
Earth about Sun
(d)
the theory predicts that space-time is curved / warped by the presence
of matter;
the light ray takes the shortest path between the star and Earth in the
curved / warped space;
2 max
To award [2], space-time must be mentioned. An answer such as “gravity
bends light” would only receive [1].
(e)
see diagram;
Position must be consistent with bent ray.
1
[6]
21.
(a)
(b)
if the muon measures4500min its reference frame; recognizes the idea of
two frames of reference
the muon / Earth would have to travel at 2.0  109ms1 / faster than the speed
of light; which is not possible;
distance travelled, as measured in muon’s reference frame must be less /
contracted;
4
mc 2  Ve  m0 c 2 ;
= 210MeV + 105MeV
= 315MeV;
m = 315MeVmc2 or 3m0;
3
[7]
26
22.
(a)
(b)
since the speed of light is independent of the speed of the source;
Alex’s source will appear to be moving away from Elspeth;
so according to the Doppler effect the light will appear to be redshifted;
3
because of the principle of equivalence;
the situation is the same as if Elspeth were observing light emitted
from the surface of a planet / OWTTE;
2
[5]
23.
ux ' 
u x  v 
 ux v 
1  2 
c 

identifies ux as 0.8c;
identifies v as  0.8c;
to give answer of 0.98c;
[3]
27