Question 1
The reference frame that is inertial is
A.
a frame fixed to a platform rotating at constant angular speed
B.
a frame fixed to a train moving along a straight track at constant speed
C.
a frame fixed to a car while it is braking uniformly
D.
a frame fixed to a rocket during launch from Earth
Question 2
A spacecraft moving at constant velocity emits a light pulse in vacuum. An observer in the spacecraft and an observer on Earth both measure the speed of the pulse. The statement from special relativity that accounts for their measurements is
A.
the speed of light is found by adding the speed of the source to
B.
the time coordinate of an event is the same in all inertial frames
C.
Newton's laws have the same form in all reference frames
D.
the speed of light in vacuum is the same for all inertial observers
Question 3
Frame moves at in the positive -direction relative to frame . The origins coincide at . An event has coordinates and in . What are the coordinates of the event in using Galilean transformations?
A.
and
B.
and
C.
and
D.
and
Question 4
The proper length of a rod is the length measured in the frame in which
A.
the rod is moving, with the positions of its ends measured at different times
B.
the rod has its shortest measured length
C.
the rod is at rest, with the positions of its ends measured simultaneously in that frame
D.
the observer receives light from both ends of the rod at the same instant
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Question 5
A clock in a spacecraft records a time interval of between two ticks. The spacecraft moves at relative to Earth. The time interval between the same two ticks measured in the Earth frame is
A.
B.
C.
D.
Question 6
A particle has a straight world line on a space-time diagram with axes and . The angle between the world line and the -axis is , where . The speed of the particle is
A.
B.
C.
D.
Question 7
Muon decay experiments detect more atmospheric muons at sea level than predicted by a non-relativistic calculation. The consistent relativistic descriptions are
A.
an increased speed of the muons above in the Earth frame
B.
time dilation in the Earth frame and length contraction of the atmosphere in the muon frame
C.
absolute simultaneity between the muon frame and the Earth frame
D.
length contraction of the muon in the Earth frame and time contraction in the muon frame
Question 8
A spark occurs at the front of a train as the train passes a mark on a platform. The train is moving with constant velocity relative to the platform.
State what is meant by an event in relativity.
[1]
Explain why the train frame may be treated as an inertial reference frame.
[2]
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Question 9
Two inertial frames and have coincident origins at . Frame moves at in the positive x-direction relative to .
An event has coordinates and in . Determine the coordinates of this event in using the Galilean transformation.
[2]
A cart has velocity in . Determine the velocity of the cart in using Galilean velocity addition.
[1]
Question 10
A spacecraft moves at speed relative to Earth.
State the two postulates of special relativity.
[2]
Calculate the Lorentz factor for the spacecraft.
[1]
Question 11
Two flashes occur simultaneously in frame at positions and . Frame moves at in the positive -direction relative to . The order of the flashes in is
A.
and remain simultaneous
B.
occurs before
C.
The order cannot be determined without the distance to the observer
D.
occurs before
Question 12
Frame moves at in the positive -direction relative to frame . An event in has coordinates and . What is the coordinate of the event?
A.
B.
C.
D.
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Question 13
A spacecraft is moving away from Earth at . It launches a probe backwards at relative to the spacecraft. Taking positive velocity as away from Earth, the velocity of the probe relative to Earth is
A.
B.
C.
D.
Question 14
Two events are separated by and in an inertial frame. Take . The space-time interval between the events is
A.
time-like, so the events can be causally connected by an object moving slower than light
B.
space-like, so the events cannot be causally connected
C.
light-like, so the events can be connected only by light in vacuum
D.
invariant only if measured in the rest frame of one event
Question 15
A rectangular panel is at rest in a spacecraft. Its proper dimensions are parallel to the direction of motion and perpendicular to the direction of motion. The spacecraft moves at relative to Earth. The dimensions of the panel measured in the Earth frame are
A.
by
B.
by
C.
by
D.
by
Question 16
A clock fixed in a spacecraft measures a time interval of between two ticks. The spacecraft moves at relative to Earth.
Explain why the interval measured by the spacecraft clock is the proper time interval.
[1]
Calculate the time interval between the two ticks according to an observer on Earth.
[2]
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Question 17
A rod has proper length . It moves parallel to its length at speed relative to a laboratory.
Calculate the length of the rod measured in the laboratory.
[2]
State the effect, if any, on a dimension of the rod perpendicular to the direction of motion.
[1]
Question 18
The diagram shows the world line of a particle moving with constant velocity in an inertial frame. The angle between the world line and the -axis is .
Determine the speed of the particle in terms of .
[2]
State the angle that the world line of a photon in vacuum makes with the -axis on this diagram.
[1]
Question 19
A laser pulse travels in the positive x-direction with speed in frame . Frame moves at in the positive x-direction relative to .
Use Galilean velocity addition to determine the speed of the pulse in .
[1]
Use relativistic velocity addition to show that the speed of the pulse in is .
[2]
State why the Galilean result is not acceptable for light in vacuum.
[1]
Question 20
Two inertial reference frames, and , have origins that coincide at . Frame moves at constant velocity relative to . A motion sensor records events involving a cart moving along the common x-axis.
Event | / s | in / m |
|---|---|---|
1 | 1.0 | 24.0 |
2 | 3.0 | 84.0 |
3 | 5.0 | 144.0 |
Using the Galilean transformation, calculate the coordinate of the cart for the final recorded event.
[1]
Determine the velocity of the cart in frame from the transformed data.
[2]
State the assumption about time that is being used in this transformation.
[1]
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Question 21
A spacetime diagram shows the world lines of four objects in frame . The vertical axis is labelled and the horizontal axis is labelled ; both axes use the same scale.
Identify the world line that represents an object at rest in frame .
[1]
Calculate the speed of the object whose world line makes an angle of with the photon line.
[2]
Explain why no massive particle world line is shown outside the photon line.
[1]
Question 22
A spaceship moves at in the positive x-direction relative to Earth. A probe is launched from the spaceship in the same direction with speed relative to the spaceship.
State the speed of the probe relative to Earth predicted by Galilean velocity addition.
[1]
Calculate the speed of the probe relative to Earth using relativistic velocity addition.
[2]
Explain why the answer to (b) is consistent with special relativity.
[1]
Question 23
Two events are recorded in a laboratory frame. The time separation is and the spatial separation along the x-axis is .
Determine the spacetime interval squared between the two events.
[2]
Classify the interval and state whether the events could be causally connected.
[1]
Calculate the proper time interval between the two events.
[1]
Question 24
Frame moves at in the positive x-direction relative to frame . The origins coincide at . An event has coordinates and in .
Calculate the Lorentz factor for the transformation from to .
[1]
Calculate the position coordinate of the event.
[1]
Calculate the time coordinate of the event.
[2]
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Question 25
Muons are produced high in the atmosphere and travel towards Earth at speed . The proper mean lifetime of the muons is .
Calculate the mean distance travelled by the muons in the Earth frame before decaying.
[3]
Explain how the same observation is described in the muon frame.
[1]
Question 26
A spacecraft emits pulses of light while moving past two different detector stations. The detector stations are inertial reference frames moving at different speeds relative to the spacecraft along the same straight line.
Describe the trend in the measured speed of the light pulse as the relative velocity of the detector frame changes.
[1]
Explain why the Galilean velocity addition model is inconsistent with the measurements.
[2]
Identify the postulate of special relativity directly supported by these measurements.
[1]
Question 27
A probe moves along the x-axis in frame . Frame is a spacecraft moving in the positive x-direction relative to . The table gives speeds as fractions of .
Trial | Probe speed / c | Spacecraft speed / c |
|---|---|---|
1 | 0.20 | 0.05 |
2 | 0.55 | 0.10 |
★ 3 | 0.75 | 0.25 |
4 | 0.90 | 0.35 |
For the highlighted trial, calculate the probe velocity in frame using relativistic velocity addition.
[2]
Calculate the value predicted by Galilean velocity addition for the same trial.
[1]
Explain why the relativistic result is the appropriate model for these speeds.
[1]
Question 28
Three pairs of events are recorded in the same inertial reference frame. The separation of each pair is along one space dimension.
Event pair | / s | / m |
|---|---|---|
A | 1.00 | 250 |
B | 0.80 | 210 |
C | 0.50 | 200 |
For event pair A, calculate .
[2]
Classify event pair A as time-like, space-like or light-like.
[1]
Suggest whether event pair C could be causally connected by a signal travelling below the speed of light.
[1]
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Question 29
A train moves in the positive x-direction at relative to a platform. In the train frame, two lamps at the rear and front of the train flash simultaneously. The separation between the lamps in the train frame is . Take to be front minus rear.
State why simultaneity in this situation is not absolute.
[1]
Calculate the time difference between the flashes according to the platform frame.
[2]
Identify which flash occurs first according to the platform frame.
[1]
Question 30
Two inertial frames and have coincident origins at . Frame moves at speed in the positive x-direction relative to . An event is recorded in .
Use the graph to determine the Lorentz factor for the relative motion.
[1]
Calculate the position coordinate of the event in frame .
[2]
Calculate the time coordinate of the event in frame .
[2]
Question 31
An unstable particle travels through a laboratory at speed . The particle decays after a proper time interval measured in its own rest frame. The laboratory measures the distance between two fixed detectors along the direction of motion.
Quantity | Value | Frame / note |
|---|---|---|
Particle speed | laboratory frame | |
Proper decay time | particle rest frame | |
Detector separation | 800 m | fixed in laboratory |
Calculate the Lorentz factor for the particle.
[1]
Calculate the mean lifetime of the particle in the laboratory frame.
[2]
In the particle rest frame, determine the separation of the two detectors.
[1]
Explain why the laboratory separation is the proper length for the detector separation.
[1]
Question 32
Two flashes occur at the front and rear of a train. In the platform frame , the flashes are simultaneous. The train frame moves at speed in the positive x-direction relative to the platform. The front flash is ahead of the rear flash in .
Use the Lorentz time transformation for intervals to determine the time difference between the front flash and the rear flash in the train frame.
[2]
State which flash occurs first in the train frame.
[1]
Explain why this result is not caused by the time taken for light to reach an observer.
[1]
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Question 33
Atmospheric muons are produced at high altitude and travel towards a detector near sea level. The proper mean lifetime of the muons is . A sample of muons has speed .
Detector | Altitude / km | Measured count rate / min | Non-relativistic model / min | Relativistic model / min |
|---|---|---|---|---|
Upper detector | 10.0 | 100 | 100 | 100 |
Lower detector | 0.0 | 22.0 | 21.8 |
Calculate the Lorentz factor for the muons.
[1]
Calculate the mean lifetime of the muons in the Earth frame.
[1]
Use the data to determine whether the non-relativistic or relativistic model better matches the lower-altitude count rate.
[1]
Explain the same survival of muons in the muon rest frame.
[2]
Question 34
Two events are recorded in frame and in frame . Frame moves at constant speed relative to along the x-axis. The table gives the coordinate separations between the same two events in both frames.
Frame | / m | / s |
|---|---|---|
S | 90 | |
S' | 180 |
Calculate using the measurements in frame .
[1]
Calculate using the measurements in frame .
[1]
State the significance of obtaining the same value in both calculations.
[1]
Determine whether the two events can occur at the same position in some inertial frame.
[1]
Question 35
A small research pod moves at a constant speed of in the positive -direction relative to an inertial laboratory frame . The frame of the pod is . The origins of and coincide when . A flash occurs at position and time in .
The flash is treated as an event in the laboratory frame.
I.
Explain what is meant by an event and an inertial reference frame.
[2]
II.
Using Galilean transformations, determine the coordinates and of the flash in the pod frame.
[2]
Determine the coordinates of the same flash in using the Lorentz transformations, and compare the result with the Galilean result.
[3]
Question 36
A probe moves in the positive -direction at relative to a space station. A spacecraft moves in the same direction at relative to the station. A pulse of light also travels in the positive -direction relative to the station.
First consider the Galilean transformation between the station frame and the spacecraft frame.
I.
Calculate the velocity of the probe relative to the spacecraft using Galilean velocity addition.
[2]
II.
State the speed of the light pulse predicted by Galilean velocity addition in the spacecraft frame.
[1]
Discuss why the result for the light pulse in (a)(ii) is not consistent with special relativity.
[3]
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Question 37
Two spacecraft, A and B, move along the same straight line as measured in the frame of a planet. Spacecraft A has velocity and spacecraft B has velocity .
The pilot of A measures the velocity of B.
I.
Calculate the speed of B relative to A predicted by Galilean velocity addition.
[2]
II.
Calculate the speed of B relative to A using relativistic velocity addition.
[2]
Evaluate which calculation is physically acceptable according to special relativity.
[2]
Question 38
A spacecraft carries a light clock. In the rest frame of the spacecraft the time interval between two ticks is . The spacecraft moves at relative to Earth.
The two ticks are events occurring at the same position in the spacecraft frame.
I.
State which time interval is the proper time interval.
[1]
II.
Calculate the time interval between ticks measured in the Earth frame.
[2]
III.
Calculate the distance travelled by the spacecraft in the Earth frame between the two ticks.
[2]
Explain, using the light-clock model, why the Earth-frame time interval is greater than the spacecraft-frame time interval.
[3]
Question 39
A rectangular space probe has proper length in the direction of travel and proper width perpendicular to the direction of travel. It passes Earth at .
An Earth-based observer measures the dimensions of the moving probe.
I.
Calculate the length of the probe measured by the Earth-based observer.
[2]
II.
Explain the width measured by the Earth-based observer.
[2]
A student claims that a high-speed photograph would simply show the probe as a uniformly shortened rectangle of length . Evaluate this claim in terms of measurement in special relativity.
[2]
Question 40
A calibrated spacetime diagram shows axes for frame and for a frame moving in the positive x-direction. Invariant hyperbolas are included on the diagram.
State what the -axis represents physically.
[1]
State what is represented by a line parallel to the -axis.
[1]
Explain why the same ruler scale cannot be used directly on both the -axis and the tilted -axis unless the diagram is calibrated.
[2]
Suggest why the and axes tilt towards the photon lines as the relative speed increases.
[1]
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Question 41
A student analyses data from a trolley carrying a lamp and two clocks. The trolley first moves at constant velocity and then accelerates for a short time. The student attempts to apply one Lorentz transformation to all recorded events.
Flash | / s | / m |
|---|---|---|
1 | 0.0 | 0.00 |
2 | 1.0 | 2.00 |
3 | 2.0 | 4.00 |
4 | 3.0 | 6.00 |
5 | 4.0 | 8.00 |
6 | 4.5 | 9.25 |
7 | 5.0 | 11.00 |
8 | 5.5 | 13.25 |
9 | 6.0 | 16.00 |
Identify the time interval during which the trolley frame can be treated as inertial.
[1]
Explain why a single Lorentz transformation is not valid for transforming all the events in the table.
[2]
For two events in the constant-velocity section, state the information required to specify each event in one-dimensional spacetime.
[1]
Suggest one improvement to the analysis so that the special relativity equations are applied appropriately.
[1]
Question 42
Two detector clicks are recorded in an inertial frame . Event A occurs at , . Event B occurs at , .
Analyse the spacetime separation of the two events in frame .
I.
Calculate for the two events.
[3]
II.
Calculate the proper time interval between A and B.
[2]
Discuss whether one event could cause the other.
[2]
Question 43
A straight rod has proper length . It moves at along its length past two stationary sensors in a laboratory frame. The sensors are used to determine the length of the moving rod.
Consider the length measurement made in the laboratory frame.
I.
Calculate the length of the rod measured in the laboratory frame.
[2]
II.
Explain why the positions of the two ends of the rod must be measured simultaneously in the laboratory frame.
[2]
In the laboratory frame the events used for the length measurement are simultaneous and separated by . Determine the time separation of these two events in the rest frame of the rod and comment on the result.
[3]
Question 44
A carriage of proper length moves at relative to a platform. A lamp at the centre of the carriage emits a flash. In the carriage frame, the light reaches the front and rear ends at the same time.
Consider the two events: light reaching the front end and light reaching the rear end.
I.
Define simultaneity in a specified reference frame.
[1]
II.
Calculate the length of the carriage in the platform frame.
[2]
III.
State which end of the carriage receives the light first in the platform frame.
[1]
Use the Lorentz transformation to determine the time difference between the two light-arrival events in the platform frame, taking front minus rear. In the carriage frame, the events are separated by and have .
[2]
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Question 45
A spacecraft frame moves at in the positive -direction relative to an Earth frame . The origins coincide at zero time. A beacon flash has coordinates and in .
Transform the coordinates of the beacon flash into the spacecraft frame.
I.
Calculate the Lorentz factor for the transformation.
[1]
II.
Calculate for the beacon flash.
[2]
III.
Calculate for the beacon flash.
[2]
Compare the Lorentz transformation with the Galilean transformation for ordinary speeds.
[2]
Question 46
The diagram shows a spacetime diagram for an inertial frame . The vertical axis is and the horizontal axis is . A particle P has a straight world line making an angle of with the -axis.
Use the spacetime diagram to analyse the motion of particle P.
I.
Calculate the speed of P in terms of .
[2]
II.
State why the world line of P is straight.
[1]
On the diagram, add the world line of a second particle Q that is at when and moves in the negative -direction at .
[2]
Discuss why the axes of another inertial frame moving relative to cannot be drawn by an ordinary Euclidean rotation of the and axes.
[2]
Question 47
Muons are produced at an altitude of above Earth and travel vertically downward at . The proper mean lifetime of the muons is .
Analyse the motion in the Earth frame.
I.
Calculate the Lorentz factor for the muons.
[1]
II.
Calculate the mean lifetime of the muons measured in the Earth frame.
[1]
III.
Calculate the time taken for the muons to travel in the Earth frame and express this as a number of dilated mean lifetimes.
[2]
Evaluate the same situation in the muon frame and explain why muon observations support special relativity.
[4]
Question 48
Two detector clicks occur in an inertial laboratory frame. The second click is to the right of the first click and occurs later.
Use the invariant spacetime interval to classify the separation between the clicks.
I.
Calculate for the two clicks.
[3]
II.
State the classification of the interval.
[1]
Discuss whether the first click could have caused the second click.
[3]
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