IB Physics HL A.5 Galilean and special relativity Questions

SLSL HLHL

Practise IB Physics HL A.5 by comparing inertial frames, transforming space-time coordinates and interpreting relativistic motion, clocks, lengths and muon-decay evidence.

135 questions in this syllabus scope, with question parts counted together.

Exam points & syllabus

Exam points

Define an inertial frame using coordinates and clocks, adding that it is nonaccelerating or obeys Newton’s laws.
Describe spacetime as an event-coordinate system with three spatial coordinates and one time coordinate.
Under the stated Galilean model, keep length unchanged and subtract signed velocities, as 0.30c − 0.78c = −0.48c.
Distinguish a separation rate in one frame from a speed measured in another: opposite 0.85c motions separate at 1.7c in Earth’s frame.
State both special-relativity postulates: the same physical laws in all inertial frames and invariant vacuum light speed.
Synchronize separated clocks by accounting for light-travel delay: start the receiver clock at d/c when a signal sent at time zero arrives.
Alternatively synchronize clocks using a midpoint flash and simultaneous starts on reception, or negligible-speed transport of synchronized clocks.
For a light ray in a uniformly moving rocket, use a straight path in its inertial frame, with direction consistent with the rocket’s motion.
Name the original and target frames before relativistic velocity addition, and keep the object and boost velocities signed in one original frame.
Use inverse velocity addition when a probe’s given speed belongs to the moving rocket frame; solve for the boost when observer speeds are unknown.
Keep c factors consistent in velocity substitutions and show the working for a given target such as the cargo relative speed 0.946c.
Transform an event using x′ = γ(x − vt) and ct′ = γ(ct − βx), retaining the signs and the length units of ct.
For separated simultaneous events, set Δt = 0 but retain the spatial term in Δt′; use the resulting sign to determine emission order.
Define a proper interval by both events occurring at one position in the chosen frame; an object’s proper length is measured in its rest frame.
Check the proper-time condition before using simple dilation; a one-way photon trip between separated source and detector need not satisfy it.
Explain interval invariance by comparing inertial frames, then calculate (Δs)² from coordinate differences, allowing a negative result.
Verify interval invariance numerically in both frames; use approximate equality when transformed coordinates were rounded.
Multiply a proper lifetime or oscillator period by γ for the dilated interval, and divide a journey time by γ to recover the traveller’s time.
Use the actual clock’s rest frame when recovering a period: a base lamp observed every 3 s at 0.80c flashes every 1.8 s in its own frame.
Infer γ from an interval ratio before solving for speed; a proper-to-dilated time ratio of 0.30 gives γ = 1/0.30.
Sketch the dilation factor versus speed from 1 at zero speed, rising towards a vertical asymptote at c.
Compare traveller ages only after specifying their histories: the returning traveller changes direction and inertial frame, resolving the twin paradox.
Use the relative speed between the object and measuring observer to contract L₀/γ or recover a proper length γL.
Infer speed from a length ratio, and recognize L/L₀ as 1/γ: a 100 m train fitting an 80 m tunnel moves at 0.60c.
Measure both rod endpoints simultaneously in the measuring frame; on a diagram intersect the endpoint worldline with the primed simultaneity line.
On a ct-versus-x plot use v/c = Δx/Δ(ct); for tan θ = v/c, measure θ from the time axis with matching physical scales.
Draw primed axes from the common origin with reciprocal gradients; a simultaneity line parallel to x′ fixes the boost.
Project parallel to the other primed axis to read coordinates, and calibrate primed time units using the invariant-interval curve.
Plot and label events explicitly, read worldline intersections, and divide ct by c to obtain time in seconds.
Draw stationary endpoints as vertical worldlines and moving endpoints as parallel sloping lines; trace light at ±45° on equal-scale axes.
Determine what an observer sees from light-ray intersections with that observer’s worldline, rather than from emission time coordinates.
Explain emission order using light paths to a common reception event; simultaneous local receptions remain coincident in every frame.
Resolve the train–tunnel paradox by distinguishing the spatially separated endpoint events that each frame treats as simultaneous.
Distinguish an arrival coordinate from travel duration; a flash emitted at ct = 2 km and received at ct = 5 km has arrival time 5000/c.
For a photon’s emission and reception events, use its full path Δx = ±cΔt in the Lorentz time transformation, rather than the initial gap.
For a signal chasing a receiver in one frame, solve cΔt = d + vΔt; show full substitution when the requested answer is given.
For a probe or ball chasing a moving front, combine the contracted separation with both velocities in the same frame.
Compare crossing times using consistent frame lengths and speeds, rather than mixing a proper length with a speed from another frame.
For co-moving charged particles, distinguish the rest-frame electric interaction from the additional magnetic interaction in the laboratory frame.
Explain the wire example using contraction of moving proton spacing and increased charge density, while the co-moving charge has no magnetic force.
Use muon observations as a prediction–measurement comparison: classical lifetimes predict fewer arrivals than detectors record.
Support the muon explanation numerically in one frame: compare transit time with dilated lifetime, or atmospheric distance with contracted distance.
When half the laboratory muons survive, use one dilated half-life; the equivalent muon-frame detector travel uses the proper half-life.

Covered syllabus points

A.5.1 (HL)—Reference frames A.5.2 (HL)—Galilean relativity A.5.3 (HL)—Galilean transformations A.5.4 (HL)—Galilean velocity addition A.5.5 (HL)—Two postulates of special relativity A.5.6 (HL)—Lorentz transformations A.5.7 (HL)—Relativistic velocity addition A.5.8 (HL)—Space-time interval A.5.9 (HL)—Proper time interval and proper length A.5.10 (HL)—Time dilation A.5.11 (HL)—Length contraction A.5.12 (HL)—Relativity of simultaneity A.5.13 (HL)—Space-time diagrams A.5.14 (HL)—World lines and speed A.5.15 (HL)—Muon decay evidence

Exam points

Define a reference frame using coordinate axes and clocks to locate events and measure positions and times.
Define an inertial frame as a non-accelerating coordinate system, or a frame in which Newton’s first law applies.
Use distance and speed measured in the same frame for travel time; light years divided by a speed in units of c give years, with the unit stated.
For the uniformly moving rocket, draw the given transverse light ray as a straight line sloping toward the rocket’s rear in its own frame.
In a twin-return scenario, identify the traveller’s frame change at turnaround: the histories are asymmetric and the Earth twin ages more.
Explain relative rest by zero relative velocity: an observer moving with the same velocity as the electron has no changing position relative to it.

[Maximum number: 1]

This question is about time dilation and relativistic mass.

Two space stations X and Y are at rest relative to each other. The separation of X and Y as measured in their frame of reference is $1.80 \times 10^{11} \mathrm{~m}$ .

State what is meant by a frame of reference.

Exam points

Under Galilean relativity, a moving object’s measured length is unchanged; retain the station’s 100 m length for the other observer.

[Maximum number: 4]

This question is about the Michelson-Morley experiment.

The diagram shows the basic features of the Michelson-Morley experiment.

During the experiment, Michelson and Morley slowly rotated the apparatus through $90^{\circ}$ . Discuss, with reference to Galilean transformations, the changes that Michelson and Morley expected to observe during the experiment.

(according to a Galilean transformation) speed of light depends on the direction of motion (through the ether);
beam which reflects from mirror $M_{1}$ takes different/less time than the beam reflecting from $\mathrm{M}_{2}$ ;
the time taken to travel each path will change as the apparatus is rotated;
a changing interference pattern is observed;

Exam points

For two rockets approaching in the Earth frame, add their Earth-frame speed magnitudes to find the closing rate, then divide separation by that rate.
For the stated Galilean model, use signed u′ = u − v; a faster observer moving in the same direction gives a negative relative velocity.

[Maximum number: 2]

Two rockets, A and B , are moving towards each other on the same path. From the frame of reference of the Earth, an observer measures the speed of A to be 0.6 c and the speed of B to be 0.4 c. According to the observer on Earth, the distance between A and B is $6.0 \times 10^{8} \mathrm{~m}$ .

Calculate, according to the observer on Earth, the time taken for A and B to meet.

Exam points

State that the laws of physics are the same for all inertial observers when asked for the postulate other than constant light speed.
For the Maxwell/postulate link, identify light as an electromagnetic wave and state that its vacuum speed is independent of source or observer motion.
For a photon emitted by a source in a moving box, give the same vacuum speed c in the ground frame; do not add the box’s speed.
Distinguish a separation rate from an object’s speed: two ships moving oppositely at 0.85c can separate at 1.7c in the Earth frame without violation.
Sketch the electron speed close to the given classical curve at low voltage, then below it and approaching c asymptotically as voltage increases.
To synchronize from the origin, start A when sending a light pulse and start B at reading d/c on its arrival, allowing for light-travel time.
For equal paths from a midpoint light source, set both clocks to zero and start each clock when its light signal arrives.
Alternatively, synchronize the clocks together first, then move one very slowly so that time-dilation effects during transport are negligible.

[Maximum number: 3]

This question is about the Michelson-Morley experiment.

The diagram shows the basic features of the Michelson-Morley experiment.

(a)

Describe how the observations made by Michelson and Morley differed from what was expected.

[ 1 ]

(b)

Explain the importance of the outcome of the Michelson-Morley experiment in support of the special theory of relativity.

[ 2 ]

Exam points

Use the positive Lorentz factor γ = 1/√(1 − β²); when γ is given, rearrange it for speed, as γ = 1.25 gives v = 0.60c.
Transform an event’s position with x′ = γ(x − vt); for the event at x = 1 m and t = 0, the given boost gives x′ = 1.5 m.
For simultaneous events at separated locations, set Δt = 0 in Δt′ = γ(Δt − vΔx/c²); a nonzero spatial term makes them nonsimultaneous.
Use both event separations from the original frame, including its measured length and current velocity stage, before applying a Lorentz transformation.
In ct coordinates, use ct′ = γ(ct − βx), keep ct′ in length units and retain a negative result when the transformed event precedes the origin.
For creation and decay at the same place in the particle’s frame, set Δx′ = 0 and use the invariant interval or the accepted Δt/γ rest-frame relation.
For a moving observer’s journey time, transform both event separations or include the destination’s motion in a relative-speed calculation.
Distinguish an arrival time coordinate from a signal’s travel duration: the ship-worldline event (x, ct) = (3, 5) km has ct′ = 4 km.
For a photon’s emission and reception events, use its path Δx = cΔt in the source frame when transforming the interval, rather than the initial gap.
On a spacetime diagram, project parallel to the primed axes and state both transformed coordinates explicitly, retaining their signs.

[Maximum number: 1]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Apply a Lorentz transformation to show that the time difference between events B and F according to observer P is $2.5 \times 10^{-7} \mathrm{~s}$ .

$\Delta t^{\prime}=1.25\times\frac{0.6\times100}{3\times10^{8}}=2.5\times10^{-7}\ \mathrm s$

Exam points

Assign signed object and observer velocities in the same initial frame before using u′ = (u − v)/(1 − uv/c²).
For opposite directions, retain the negative velocity in the numerator and the resulting plus term in the denominator.
When a probe’s speed is given in a moving rocket frame, use u = (u′ + v)/(1 + u′v/c²) to obtain its ground-frame speed.
Rearrange velocity addition for an unknown frame speed: a spacecraft measured at +0.50c and −0.50c implies a boost of 0.80c.
Interpret a negative transformed velocity as direction; report its magnitude when the question asks for speed.
Keep velocity units consistent in numerator and denominator, then express the electron’s relative speed as a multiple of c.
Distinguish an Earth-frame separation rate from a spacecraft-frame speed: opposite 0.85c motions give 1.7c and about 0.99c respectively.
For a show-that velocity, display the signed substitution; the cargo-ship result 0.946c earns its mark through correct working.

[Maximum number: 2]

A second train is moving at a velocity of -0.70 c with respect to the ground.

Calculate the speed of the second train relative to observer P .

$u^{\prime}=\frac{-0.7-0.6}{1+0.7\times0.6}\,c=-0.92\,c$

Exam points

Explain interval invariance as the same value in every inertial reference frame; saying only that it does not change is insufficient.
Read coordinate differences between the two events, then calculate (Δs)² = (cΔt)² − (Δx)²; P(2, 2) and Q(5, 3) give −8 m².
Convert a time separation to cΔt before squaring: Δt = 9.0 × 10⁻⁸ s and Δx = 15 m give an interval squared of 504 m².
Use invariance in the co-moving lamp frame: with Δx′ = 0 and cΔt′ = 1 m, the other frame also has interval squared 1 m².
To show invariance numerically, calculate the interval squared in both frames and compare values, allowing for rounded coordinates.

[Maximum number: 3]

The space-time diagram shows the coordinate axes (x, c t) of an inertial reference frame S. Two events, P and Q, are plotted.

(a)

Calculate the space-time interval $(\Delta s)^{2}$ between P and Q .

[ 2 ]

Correct readoffs $\Delta x=3 \mathrm{~m}, c \Delta t=1 \mathrm{~m}$ $(\Delta s)^{2}=1^{2}-3^{2}=-8 \mathrm{~m}^{2}$

Marking guidance:

Ignore units as they are not required for the answer.

Do not award MP2 if the answer is positive

(b)

$(\Delta s)^{2}$ is an invariant quantity. Outline what is meant by an invariant quantity.

[ 1 ]

Exam points

Define proper length as an object’s length measured in its rest frame, or by an observer at rest relative to that object.
Identify the proper-length observer for a bridge: the observer at rest with the bridge measures its proper length.
For collinear frame comparisons, identify the common-rest-frame star separation as the maximum measured distance: here, 4.8 light years.
Identify proper time by checking that both events occur at the same position in the named observer’s frame.
Reject an Earth-frame journey interval as proper when departure and arrival occur at different Earth-frame positions.
Before using simple time dilation, check that one interval is proper; one-way light emission and detection need not satisfy this condition.

[Maximum number: 1]

State which of the two time intervals is a proper time.

$\Delta t_{\mathrm P}$ : observer sitting in the train

Exam points

Identify which interval is proper before choosing Δt = γΔt₀; a clock co-moving with the particle measures its shorter creation–decay interval.
Divide a station- or Earth-frame journey time by γ to find the traveller’s time, as 50 years at 0.60c becomes 40 years.
Multiply a supplied rest-frame lifetime or period by γ to obtain the moving observer’s longer interval.
Recover a stationary lamp’s flash period by dividing the moving observer’s interval by γ; 3.0 s at 0.80c gives 1.8 s.
Infer speed from a clock ratio: Δt₀/Δt = 0.30 means γ = 1/0.30, giving v ≈ 0.95c.
Combine a dilated journey time with speed in the same frame to find distance; convert minutes to seconds before multiplying by v.
Sketch Δt/Δt₀ against speed with value 1 at v = 0, increasing curvature and a vertical asymptote at v = c.
Use the dilated muon lifetime in Earth-frame distance = vγt₀; using vt₀ alone omits time dilation.
Describe the twin paradox as the apparent reciprocal prediction that each twin’s moving clock runs slower.
Explain unequal ages after reunion by the traveller changing direction and inertial frame, while the stay-at-home observer does not.
State the Lorentz-transformation consequence that elapsed time is frame-dependent rather than absolute.

[Maximum number: 2]

Calculate the speed v of the train for the ratio $\frac{\Delta t_{P}}{\Delta t_{Q}}=0.30$ .

$\gamma=\frac{\Delta t_Q}{\Delta t_P}=\frac{1}{0.30}=3.3$ v=0.95 c

Exam points

Identify the object’s proper length and divide by γ for the observer moving along it; 450 m at 0.80c contracts to 270 m.
Recover a proper separation by multiplying the moving-frame distance by γ; an 8-year ship journey gives Earth–star distance about 13 ly.
Use the relative speed of the two spacecraft to choose γ before calculating contraction, rather than either spacecraft’s Earth-frame speed.
Infer γ from a length ratio, then solve for speed: a 100 m proper train fitting an 80 m tunnel has γ = 1.25 and v = 0.60c.
Calculate a length ratio directly as L/L₀ = 1/γ; at 0.80c the contracted-to-proper ratio is 3/5.
To demonstrate contraction for a decaying pion, calculate distances using each frame’s lifetime and identify the Earth-frame distance as proper.
On a spacetime diagram, locate both rod ends at the same time in the measuring frame, using the end’s worldline and the primed space axis.
Read the contracted length using the calibrated primed axis; the rod’s endpoint intersection gives about 0.7 m.
For a ball chasing a train’s front, combine the contracted train length with the ball–front closing speed in the platform frame.
Explain the co-moving charge’s electric force through increased proton density from wire contraction; the stationary charge has no magnetic force.
Use the accepted rounded-speed route for the electron separation: γ ≈ 5 gives 0.800/γ ≈ 0.159 m.

[Maximum number: 2]

Later the train is travelling at a speed of 0.60 c . Observer P measures the length of the train to be 125 m . The train enters a tunnel of length 100 m according to observer Q .

Show that the length of the train according to observer Q is 100 m .

$\gamma=1.25$

length of train according to Q $=125/1.25=100\ \mathrm m$

Exam points

Read simultaneity in the unprimed frame from equal ct coordinates: events A and D on a horizontal line are simultaneous for P.
In a moving frame, events joined by a line parallel to x′ have equal time coordinates; the mirror strikes are simultaneous for the passenger.
Project each event parallel to x′ onto ct′ to establish its order; include the construction when the question asks for diagram evidence.
Use the sign of the transformed time difference with a defined event order to identify which separated lamp flashes first.
Explain nonsimultaneous clock setting using equal light speed and the motion of each receiver towards or away from the signal.
Distinguish flash emission order from what an observer sees; compare light paths to the common receiver whose arrivals coincide.
Resolve the train–tunnel paradox by showing that spatially separated end coincidences simultaneous in one frame occur at different times in the other.

[Maximum number: 3]

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Deduce, using the spacetime diagram, which light was turned on first according to observer P.

lines parallel to the $x^{\prime}$ axis and passing through B and F;
intersections on the $ct^{\prime}$ axis at $\mathrm B^{\prime}$ and $\mathrm F^{\prime}$ ;
light at the front of the train must have been turned on first

Exam points

Construct primed axes through the common origin with reciprocal gradients β and 1/β; they lie symmetrically about the light line.
When two plotted events must be simultaneous in S′, draw x′ through the origin parallel to their joining line, then draw a consistent ct′ axis.
Project an event parallel to ct′ to read x′; use the primed-axis calibration to place another event with a specified x′ coordinate.
Use a constant-interval curve to calibrate the primed time unit, and a line parallel to x′ to identify simultaneous events.
Read event positions from worldline intersections and convert the ct reading to time by dividing by c, including any kilometre-to-metre conversion.
Label event coordinates explicitly and in the correct order: the event at (x, ct) = (3, 1) must be marked E.
Draw stationary endpoints as vertical worldlines and the two ends of a uniformly moving train as parallel sloping worldlines.
With equal x and ct scales, draw light paths at ±45° from the emission event and label their intersections with receiver worldlines.
For a reflected radio signal, draw the inward ray to Earth first, then the outward ray to the moving ship; mark both reception events.
Determine what an observer sees by tracing light rays to that observer’s worldline; this arrival order can differ from emission order.
Distinguish an arrival time coordinate from travel duration: a flash emitted at ct = 2 km arrives at ct = 5 km, so its arrival time is 5000/c.
To justify a negative primed event time, project parallel to x′ onto ct′ and show that the intercept lies before the shared origin.
Explain why a co-moving lamp interval ct′ = 1 m has a larger ct coordinate in the other frame: ct = γct′ is time dilation.

[Maximum number: 1]

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Draw the time $c t^{\prime}$ and space $x^{\prime}$ axes for observer P's reference frame on the spacetime diagram.

axes drawn with correct gradients of $\frac{5}{3}$ for $ct^{\prime}$ and 0.6 for $x^{\prime}$

Exam points

On a ct-versus-x worldline plot, calculate v/c = Δx/Δ(ct); the usual vertical-over-horizontal gradient is c/v.
Measure the worldline angle from the time axis when using tan θ = v/c; a speed of 0.80c gives θ ≈ 39°.
Choose a worldline using both its speed magnitude and direction; a line through (x, ct) = (−2, 2.5) has velocity −0.80c.
Infer a boost from a simultaneity line: the slope Δ(ct)/Δx of a line parallel to x′ is v/c, giving c/3 for the plotted P–Q pair.
State the worldline speed as a multiple of c or convert it to SI units, as 0.60c = 1.8 × 10⁸ m/s.

[Maximum number: 1]

The space-time diagram shows the coordinate axes (x, c t) of an inertial reference frame S. Two events, P and Q, are plotted.

Events P and Q occur simultaneously in an inertial reference frame $S^{\prime}$ . The clocks of S and $S^{\prime}$ are synchronized at $x=x^{\prime}=0$ .

Determine the speed of $S^{\prime}$ relative to S.

$\left\langle\frac{v}{c}=\frac{1}{3}\right.$ so $\| v=0.33 \mathrm{c}$ or $1.0 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$

Watch for ECF from bi.

Marking guidance:

Allow fraction as the answer, i.e. $\frac{c}{3}$

1

Exam points

Explain the evidence chain: short proper lifetime predicts few ground arrivals classically, but more are detected because Earth-frame lifetimes are longer.
Establish the classical baseline using d/v or vt₀: a 10 km flight at 0.98c takes about 15 proper mean lifetimes, so few muons should arrive.
For the 2 km experiment, compare the 6.8 μs Earth transit time with the dilated lifetime γt₀ = 11 μs.
Alternatively compare distances: time dilation increases the muon’s characteristic Earth-frame travel distance from about 650 m to 3.2 km.
Explain the same experiment in the muon frame using a shorter journey time or contracted atmospheric distance: 6.8/5 ≈ 1.4 μs or 2000/5 = 400 m.
When half the laboratory muons survive, use one dilated half-life as the travel time: γ × 1.56 μs = 3.12 μs, giving D ≈ 811 m.
In the muon frame, show the detector’s travel distance using vT₁/₂ or D/γ; both routes give about 405 m and require working.

[Maximum number: 3]

This question is about time dilation and relativistic mass.

Muons are produced in the upper atmosphere of Earth and travel towards the surface of Earth where they are detected. Explain how, with reference to the situation described in (b), the production and detection of muons provide evidence for time dilation.

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A.5.2 (HL)—Galilean relativity4 questions

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