A.5 Galilean and special relativity
Practice exam-style IB Physics questions for Galilean and special relativity, aligned with the syllabus and grouped by topic.
Question 1
An event in special relativity is best described as
A.
the complete path followed by a moving object through spacetime.
B.
a coordinate system containing only synchronized clocks.
C.
a single occurrence with specified position and time coordinates in a reference frame.
D.
a physical quantity that is unchanged in all inertial frames.
Question 2
A train moves with constant velocity relative to a platform. A frame fixed to the train is inertial if
A.
the train is not accelerating or rotating.
B.
all objects inside the train are stationary.
C.
the train has a speed much less than the speed of light.
D.
the train is at rest relative to Earth.
Question 3
Frame S′ moves at +12 m s⁻¹ relative to frame S. A cart has velocity +20 m s⁻¹ in S. The Galilean velocity of the cart in S′ is
A.
+32 m s⁻¹
B.
+12 m s⁻¹
C.
−8 m s⁻¹
D.
+8 m s⁻¹
Question 4
In the Galilean transformation between two inertial frames with coincident origins at t = 0, the assumption made about time is that
A.
time depends on position through t′ = t − vx/c².
B.
time is the same in both frames, so t′ = t.
C.
moving clocks always run slow by a factor γ.
D.
simultaneous events must occur at the same position.
Question 5
Einstein’s second postulate of special relativity states that
A.
the speed of light depends on the speed of its source.
B.
all inertial observers measure the same speed for light in vacuum.
C.
Newton’s laws are valid in accelerating reference frames.
D.
all observers measure the same time interval between two events.
Question 6
The proper time interval between two events is measured by a clock that
A.
is moving fastest relative to the events.
B.
is present at both events.
C.
measures the largest possible time interval.
D.
is synchronized with clocks in every inertial frame.
Question 7
On a spacetime diagram with axes x and ct, the world line of a photon moving in the +x direction is
A.
a vertical straight line parallel to the ct axis.
B.
a horizontal straight line parallel to the x axis.
C.
a straight line at 45° to the x and ct axes.
D.
a curved line approaching the x axis.
Question 8
Define an inertial reference frame.
[1]
Define an event in the context of relativity.
[1]
Question 9
The Lorentz factor for a spacecraft moving at 0.60c relative to Earth is
A.
0.80
B.
2.50
C.
1.67
D.
1.25
Question 10
Frame S′ moves at +0.60c relative to S. An event has coordinates x = 9.0 m and ct = 15 m in S. The x′ coordinate is
A.
22.5 m
B.
11.25 m
C.
3.6 m
D.
0 m
Question 11
A particle moves at +0.80c in frame S. Frame S′ moves at +0.50c relative to S. The speed of the particle measured in S′ is
A.
0.93c
B.
0.50c
C.
0.30c
D.
1.30c
Question 12
A light pulse travels in the +x direction in S. Frame S′ moves at +0.70c relative to S. The speed of the light pulse in S′ is
A.
0.30c
B.
0.70c
C.
1.70c
D.
c
Question 13
Two events have cΔt = 5.0 m and Δx = 3.0 m in an inertial frame. Their spacetime separation is
A.
not invariant.
B.
light-like.
C.
time-like.
D.
space-like.
Question 14
A rod has proper length 4.0 m and moves at 0.80c parallel to its length relative to a laboratory. Its length measured in the laboratory is
A.
2.4 m
B.
6.7 m
C.
3.2 m
D.
4.0 m
Question 15
A clock on a spacecraft moving at 0.75c records a proper time interval of 12 s between two ticks. The time interval measured on Earth is approximately
A.
27 s
B.
12 s
C.
18 s
D.
7.9 s
Question 16
The detection of many atmospheric muons at sea level supports special relativity because, in Earth’s frame, the muons
A.
experience no decay while passing through the atmosphere.
B.
have an increased proper lifetime due to their motion.
C.
travel faster than light for a short time.
D.
have a dilated lifetime that allows more of them to reach the detector.
Question 17
Frame S′ moves at +18 m s⁻¹ relative to frame S. At t = 4.0 s, an event occurs at x = 100 m in S.
Calculate x′ using the Galilean transformation.
[1]
State the value of t′.
[1]
Question 18
State the two postulates of special relativity.
[1]
Outline why the Galilean velocity transformation is inconsistent with one of these postulates.
[1]
Question 19
A spacecraft moves at 0.80c relative to Earth.
Calculate the Lorentz factor.
[1]
Explain why relativistic effects are negligible for a car moving at 30 m s⁻¹.
[1]
Question 20
A rod is moving relative to a laboratory.
State what is meant by the proper length of the rod.
[1]
Explain why the positions of the two ends of the rod must be measured simultaneously in the laboratory to determine its laboratory length.
[1]
Question 21
A particle has a proper mean lifetime of 2.2 μs and moves with γ = 8.0 relative to a detector.
Calculate its mean lifetime measured by the detector.
[1]
Calculate the mean distance travelled before decay in the detector frame, assuming its speed is 0.992c.
[1]
Question 22
A spaceship has proper length 120 m. It passes Earth at speed 0.60c.
Calculate γ.
[1]
Calculate the length of the spaceship measured from Earth.
[1]
Question 23
Cosmic-ray muons are produced high in the atmosphere and detected at sea level.
State what is meant by the proper lifetime of a muon.
[1]
Explain the observation of sea-level muons in Earth’s frame.
[1]
Question 24
The graph shows the Lorentz factor γ as a function of speed v/c.
Use the graph to estimate γ when v = 0.80c.
[1]
Describe how γ changes as v/c approaches 1.
[1]
Explain why Newtonian mechanics is a good approximation at small v/c.
[1]
Question 25
Two lightning strikes are simultaneous in frame S and occur at positions separated by Δx > 0. Frame S′ moves in the +x direction relative to S. The sign of Δt′ for the two events, using Δt′ = γ(Δt − vΔx/c²), is
A.
positive.
B.
zero.
C.
undefined.
D.
negative.
Question 26
Frame S′ moves at +0.60c relative to S. An event has coordinates x = 4.0 light-seconds and t = 10 s in S.
Calculate γ.
[1]
Calculate x′ in light-seconds.
[1]
State whether the event occurs at the origin of S′.
[1]
Question 27
A probe moves at +0.70c in frame S. Frame S′ moves at +0.40c relative to S.
Calculate the Galilean value of the probe velocity in S′.
[1]
Calculate the relativistic value of the probe velocity in S′.
[1]
Question 28
Two events are separated in an inertial frame by Δx = 9.0 m and Δt = 20 ns.
Calculate cΔt using c = 3.0 × 10⁸ m s⁻¹.
[1]
Determine the sign of (Δs)².
[1]
Classify the interval.
[1]
Question 29
Two lamps at opposite ends of a train carriage flash simultaneously in the carriage frame.
State the condition for simultaneity in a specified frame.
[1]
Explain why the flashes need not be simultaneous in the platform frame.
[1]
Question 30
A particle has a straight world line on a spacetime diagram. The angle between the world line and the ct-axis is 30°.
Determine the speed of the particle in terms of c.
[1]
State what a vertical world line represents.
[1]
State the angle for a photon world line relative to the ct-axis.
[1]
Question 31
A spacetime diagram shows S and S′ axes for two inertial frames.
State what the ct′-axis represents in the S diagram.
[1]
State what the x′-axis represents.
[1]
Explain why the same ruler scale cannot generally be used on the x and x′ axes.
[1]
Question 32
The thickness of the atmosphere between muon production and a detector is 12 km in Earth’s frame. Muons move with γ = 10 relative to Earth.
Calculate the atmospheric thickness in the muon frame.
[1]
Explain why this calculation is consistent with the Earth-frame time-dilation explanation.
[1]
Question 33
A table gives velocities u in S and the corresponding relativistic transformed velocities u′ in S′ for a fixed frame speed v.
| u/c | u′/c |
|---|---|
| 0.20 | −0.33 |
| 0.50 | 0.00 |
| 0.80 | 0.50 |
| 0.90 | — |
| 1.00 | 1.00 |
Identify the value of v/c used for the transformation.
[1]
Calculate one missing u′ value using the relativistic velocity addition equation.
[1]
Explain how the table provides evidence that speeds do not transform according to the Galilean rule at relativistic speeds.
[1]
Question 34
A spacetime diagram shows four labelled pairs of events with separations drawn from a common origin.
Identify one pair of events with a light-like separation.
[1]
Determine whether the interval for pair P is time-like, space-like or light-like.
[1]
Suggest whether event P could causally influence the second event in the pair.
[1]
Question 35
The graph shows the measured length L of a moving rod as a function of speed v/c. The proper length of the rod is indicated.
Read the value of L/L0 at v = 0.60c.
[1]
State the direction in which contraction occurs.
[1]
Explain the shape of the graph using the Lorentz factor.
[1]
Question 36
A spacetime diagram shows events A, B and C and the axes of frames S and S′.
Identify two events that are simultaneous in S.
[1]
Identify two events that are simultaneous in S′.
[1]
Explain how the diagram demonstrates the relativity of simultaneity.
[1]
Question 37
A diagram shows a light clock at rest in a spacecraft and the path of the light pulse as seen from Earth while the spacecraft moves horizontally.
Identify which path of the light pulse corresponds to the spacecraft frame.
[1]
State which frame measures the proper time between two ticks.
[1]
Explain why the Earth frame measures a longer time between ticks.
[1]
Question 38
A spacecraft emits a pulse of light in the +x direction while moving at +0.50c relative to Earth.
State the speed predicted for the pulse in Earth’s frame by Galilean velocity addition if the pulse speed is c in the spacecraft frame.
[1]
State the speed predicted by special relativity.
[1]
Compare the assumptions about time and light speed that lead to these different predictions.
[1]
Question 39
A spacetime diagram shows the world lines of three particles X, Y and Z.
Identify the particle at rest in the frame of the diagram.
[1]
Determine the speed of particle Y in terms of c using the angle shown.
[1]
Explain why particle Z cannot represent a material particle if its world line lies outside the photon line.
[1]
Question 40
A table gives muon count rate measured at several altitudes. A Newtonian prediction and a relativistic prediction are also shown.
| Altitude / m | Measured rate / min^-1 | Newtonian rate / min^-1 | Relativistic rate / min^-1 |
|---|---|---|---|
| 2000 | 563 | 563 | 563 |
| 1500 | 517 | 291 | 516 |
| 1000 | 466 | 150 | 473 |
| 500 | 429 | 77 | 433 |
| 0 | 402 | 40 | 396 |
State how the measured count rate changes as altitude decreases.
[1]
Compare the measured sea-level count rate with the Newtonian and relativistic predictions.
[1]
Evaluate whether the data support time dilation.
[1]
Question 41
A table gives coordinates of the same two events in frame S. Frame S′ moves at a known speed relative to S.
| Event | x in S / m | t in S / s | v of S′ / c |
|---|---|---|---|
| 1 | 1.0 × 10^8 | 1.50 | +0.60 |
| 2 | 3.0 × 10^8 | 1.50 | +0.60 |
Calculate Δx and Δt in S.
[1]
Use the Lorentz time transformation to calculate Δt′.
[1]
Use your result to decide whether the two events are simultaneous in S′.
[1]
State whether simultaneity of these two events is invariant.
[1]
Question 42
A laboratory cart moves at speed u in frame S. A second frame S′ moves at speed v relative to S along the same line.
State the Galilean transformations for position and time, and derive the Galilean velocity addition equation.
[1]
Compare and contrast Galilean relativity and special relativity in their treatment of light and time.
[1]
Question 43
A spacetime diagram is used to represent two inertial frames S and S′.
State how to represent a stationary object, a moving massive particle and a photon on an x–ct diagram.
[1]
Explain how the axes of S′ are interpreted on the S diagram, including the issue of scale.
[1]
Question 44
Measurements of the separation between two events are made in two inertial frames and displayed in a table.
| Frame | Δx / m | cΔt / m |
|---|---|---|
| S | 300 | 500 |
| S′ | 420 | 580 |
Calculate (Δs)² for frame S.
[1]
Calculate (Δs)² for frame S′.
[1]
Compare the two calculated values.
[1]
Evaluate whether the data are consistent with Lorentz transformations.
[1]
Question 45
Two inertial frames S and S′ move relative to each other along the x-axis.
State Einstein’s two postulates of special relativity and define the Lorentz factor γ.
[1]
Explain how the Lorentz transformations differ conceptually from Galilean transformations, including the low-speed limit.
[1]
Question 46
Two events have different coordinates in two inertial frames.
Define the spacetime interval and distinguish time-like, space-like and light-like intervals.
[1]
Evaluate the importance of the spacetime interval for deciding whether two events can be causally connected.
[1]
Question 47
A spacecraft travels from Earth to a star at a constant speed of 0.80c. The distance to the star in Earth’s frame is 12 light-years.
Calculate γ and the travel time measured in Earth’s frame.
[1]
Explain the journey in the spacecraft frame, including proper time and length contraction.
[1]
Question 48
A flash lamp is at the midpoint of a train carriage moving at constant velocity past a platform. The light reaches the front and rear of the carriage.
Describe the order of these two arrival events in the train frame.
[1]
Discuss how the platform frame describes the same events and why this is not a consequence of light travel time to an observer’s eye.
[1]
Question 49
Muons are produced in the upper atmosphere with proper mean lifetime 2.2 μs. A group of muons travels at 0.995c toward Earth through an atmospheric thickness of 15 km as measured in Earth’s frame.
Calculate the mean lifetime of the muons in Earth’s frame and the mean distance travelled before decay in that frame.
[1]
Evaluate how the observation of muons at ground level provides evidence for both time dilation and length contraction.
[1]
Question 50
Two identical spacecraft move directly towards each other. In the frame of a space station, spacecraft A has velocity +0.70c and spacecraft B has velocity −0.70c.
Calculate the velocity of B in the frame of A using the relativistic velocity addition equation.
[1]
Evaluate why the Galilean answer is not acceptable at these speeds and how the relativistic result is consistent with the postulates of special relativity.
[1]
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