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Option G — Relativity
G1. This question is about concepts of time and length in Special Relativity.
(a)
Define what is meant by a frame of reference.
[1]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) A car moves along a straight level track with velocity v. A and B are points at each end
of the car and O is an observer in the car at the mid-point between A and B. When O and
C are opposite each other, lightning strikes ends A and B of the car. Observer O receives
the light from A and B at the same instant, as measured on his clock.
O
v
A
(i)
B
C
ground
Discuss whether the lightning strikes appear to be simultaneous to observer O and
to observer C.
Observer O: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observer C: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [4]
(ii) The length of the car, as measured by observer O, is 9.0 m. As measured by C, the
length is 7.2 m. Determine the speed, in terms of the speed c of light, of the car as
measured by observer C.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206-6512
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G2. A radar signal is emitted from a source S. The source is moving with speed 0.80c relative to the
ground in a straight line towards an observer O who is stationary with respect to the ground, as
shown below.
source S
0.80c
observer O
ground
The speed of the radar waves is c relative to the ground.
(a)
Calculate the speed of the radar wave relative to the observer O using
(i)
[1]
the Galilean transformation equation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [3]
(ii) the principles of Special Relativity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2]
(b) Explain how your answer to (a) (ii) relates to Maxwell’s electromagnetic theory.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turn over
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G3. (a)
Distinguish between rest mass energy and total energy of a particle.
Rest mass energy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total energy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Estimate the energy released during the annihilation of an electron-positron pair. Explain
why your answer is an estimate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c)
[2]
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
speed / v
1.5c
2.0c
On the graph above, draw a line to represent the variation with speed v of the kinetic
energy according to relativistic mechanics.
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F3. This question is about Olbers’ paradox.
(a)
Newton proposed a model of the universe that is infinite in extent and in which the stars
are uniformly distributed. Olbers suggested that, if this model were correct, then the sky
would never be dark. Explain how Olbers reached this conclusion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Suggest two reasons how the Big Bang model of the universe accounts for the night sky
being dark.
1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206-6518
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Option G — Relativity
G1. This question is about proper time.
A muon at the top of the atmosphere is moving toward the ground with speed v. In the frame
of reference of a person at rest with respect to the ground, the muon takes a time Tg to reach the
ground. In the frame of reference of the muon, the ground takes a time Tm to reach the muon.
(a)
Explain why the proper time is measured by a clock in the muon frame of reference.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2]
[2]
(b) The time Tg was measured to be 10.2 s. The speed v is 0.98 c. Calculate Tm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turn over
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G2. This question is about simultaneity.
The diagram below shows a railway carriage travelling to the right at constant velocity. A
flashbulb is hanging from a point midway between the ends L and R of the carriage. Each flash
produces single pulses sent in opposite directions.
flashbulb
L
Clare
R
Nino
Clare is at rest at the centre of the carriage. Light pulses from the flashbulb are observed by Clare
to strike the opposite walls L and R of the carriage simultaneously. Nino is at rest on the ground.
He is opposite Clare at the moment when the bulb flashes.
State and explain whether Nino observes the pulses striking L and R simultaneously.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206-6518
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G3. This question is about relative velocities.
(a)
Describe what is meant by a Galilean transformation.
[1]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Two electrons travel along the same straight line towards each other. The speed of each
electron with respect to an observer in the laboratory frame of reference is 0.9800 c.
Calculate the relative speed of the electrons using
(i)
[1]
the Galilean transformation equation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2]
(ii) the relativistic transformation equation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c)
[2]
Comment on your answers in (b).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turn over
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G4. This question is about mass-energy.
(a)
Distinguish between the rest mass-energy of a particle and its total energy.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) The rest mass of a proton is 938 MeV c–2. State the value of its rest mass-energy.
[2]
[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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206-6518
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Option G — Relativity
G1. This question is about Special Relativity.
(a)
Explain what is meant by an inertial frame of reference.
[1]
......................................................................
......................................................................
(b)
State the two postulates of the Special Theory of Relativity.
1.
[2]
.................................................................
.................................................................
2.
.................................................................
.................................................................
An observer in a frame of reference A measures the relativistic mass and the length of an object
that is at rest in his frame of reference. He also measures the time interval between two events
that take place at one point in his reference frame. The relativistic mass and length of the
object, and time interval between the two events, are also measured by a second observer in
reference frame B that is moving at constant velocity relative to the observer in frame A.
(c)
(i)
By crossing out the inappropriate words in the table below, state whether the
observer in frame B will measure the quantities as being larger, the same size or
smaller than when measured by the observer in frame A.
Quantity
(ii)
[3]
Measured by observer in frame B
mass
larger / the same / smaller
length
larger / the same / smaller
time interval
larger / the same / smaller
Use your answers in (c) (i) to suggest how the observer in frame B will consider the
density of the object in frame A to be affected.
[3]
.................................................................
.................................................................
.................................................................
Turn over
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G2. This question is about muon decay.
Muons, created in the upper atmosphere, travel towards the Earth’s surface at a speed of 0.994 c
relative to an observer at rest on the Earth’s surface.
A muon detector at a height above the Earth’s surface of 4150 m, as measured by the
observer, detects 2.80 ×104 muons per hour. A similar detector on the Earth’s surface detects
1.40 ×104 muons per hour, as illustrated below.
2.80 ×104 muons per hour
4150 m
1.40 ×104 muons per hour
Earth’s surface
The half-life of muons as measured in a reference frame in which the muons are at rest is
1.52 µ s.
(a)
Calculate the half-life of the muons, as observed by the observer on the Earth’s surface.
[2]
......................................................................
......................................................................
......................................................................
......................................................................
(b)
Calculate, as measured in the reference frame in which the muons are at rest,
(i)
[1]
the distance between the detectors.
.................................................................
.................................................................
(ii)
the time it takes for the detectors to pass an undecayed muon.
[1]
.................................................................
.................................................................
(This question continues on the following page)
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(Question G2 continued)
(c)
Use your answers to (a) and (b) to explain the concepts of
(i)
[2]
time dilation.
.................................................................
.................................................................
.................................................................
(ii)
[2]
length contraction.
.................................................................
.................................................................
.................................................................
G3. Two electrons are travelling directly towards one another. Each has a speed of 0.80 c relative
to a stationary observer. Calculate the relative velocity of approach, as measured in the frame
of reference of one of the electrons.
[3]
...........................................................................
...........................................................................
...........................................................................
...........................................................................
Turn over
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Option G — Relativity
G1. This question is about the postulates of special relativity.
(a)
State the two postulates of the special theory of relativity.
[2]
Postulate 1
......................................................................
......................................................................
Postulate 2
......................................................................
......................................................................
(i)
Explain how this observation is consistent with the theory of special relativity.
[1]
.................................................................
.................................................................
.................................................................
(ii)
[3]
Calculate the speed of one spacecraft relative to an observer in the other.
.................................................................
.................................................................
.................................................................
.................................................................
Turn over
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G2. This question is about relativistic kinematics.
(a)
Muons are unstable particles that have an average lifetime of 2.2 ×10−6 s as measured in
a reference frame in which they are at rest. Muons that are created at a height of 3.0 km
above the Earth’s surface move vertically downward with a speed of 0.98 c as measured
by an observer at rest on the Earth’s surface.
(i)
Calculate the average lifetime of a muon as measured by the observer on Earth.
[2]
.................................................................
.................................................................
.................................................................
.................................................................
(ii)
Calculate the distance travelled by a muon during a time equal to the average lifetime
of the muon according to the observer at rest relative to the Earth’s surface.
[2]
.................................................................
.................................................................
.................................................................
.................................................................
(iii) Calculate the distance travelled by the Earth during a time equal to the average
lifetime of the muon according to an observer at rest relative to the muon.
[2]
.................................................................
.................................................................
.................................................................
.................................................................
(This question continues on the following page)
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(Question G2 continued)
(iv) Muons created at a height of 3.0 km above the Earth’s surface are in fact detected on
the surface of the Earth. Use your answers to (ii) and (iii) together with any other
relevant calculations to explain this observation according to
1.
[2]
the observer at rest on the surface of the Earth.
............................................................
............................................................
............................................................
2.
[3]
the observer at rest relative to the muon.
............................................................
............................................................
............................................................
............................................................
(b)
The rest mass of the muon is 106
106 MeVc−2. Calculate the potential difference through
which a muon at rest in the lab must be accelerated in order to have a speed of 0.98 c.
(The electric charge of the muon is identical to that of the electron.)
[3]
......................................................................
......................................................................
......................................................................
......................................................................
Turn over
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Option G — Relativity
G1. This question is about frames of reference.
(a)
Explain what is meant by a reference frame.
[2]
......................................................................
......................................................................
......................................................................
......................................................................
In the diagram below, Jasper regards his reference frame to be at rest and Morgan’s reference
frame to be moving away from him with constant speed v in the x-direction.
v
Jasper
Morgan
light source
x
x
Morgan carries out an experiment to measure the speed of light from a source which is at rest
in her reference frame. The value of the speed that she obtains is c.
(b)
Applying a Galilean transformation to the situation, state the value that Jasper would be
expected to obtain for the speed of light from the source.
[1]
......................................................................
(c)
State the value that Jasper would be expected to obtain for the speed of light from the
source based on Maxwell’s theory of electromagnetic radiation.
[1]
......................................................................
(This question continues on the following page)
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(Question G1 continued)
(d)
Deduce, using the relativistic equation for the addition of velocities, that Jasper will in
fact obtain a value for the velocity of light from the source consistent with that predicted
by the Maxwell theory.
[3]
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In Morgan’s experiment to measure the speed of light she uses a spark as the light source.
According to her, the spark lasts for a time interval of 1.5 µ s. In this particular situation, the
time duration of the spark as measured by Morgan is known in the Special Theory of Relativity
as the proper time.
(e)
(i)
Explain what is meant by proper time.
[1]
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(ii)
According to Jasper, the spark lasts for a time interval of 3.0 µ s. Calculate the
relative velocity between Jasper and Morgan.
[3]
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G2. This question is about the Michelson Morley experiment.
The diagram below shows the essential features of the apparatus used in the Michelson-Morley
experiment.
movable mirror
A
fixed mirror
light source
observer
A is a half-silvered mirror.
(a)
[1]
State the purpose of the experiment.
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(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 90o . Explain why.
[2]
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(Question G2 continued)
(d)
[1]
Explain the function of the moveable mirror.
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(e)
Describe the results of the experiment and explain how the result supports the Special
Theory of Relativity.
[2]
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