A.5 Galilean and special relativity: IB Physics HL exam-style questions

A star is 9.6 ly from the Earth in the Earth frame. A spacecraft leaves the Earth and travels to the star at a constant speed of 0.96c relative to the Earth.

Which statements are correct?
I. In the Earth frame the journey takes 10 years.
II. The crew measure the journey to take 2.8 years.
III. According to the crew, a clock on the Earth advances by 0.78 years during the journey.

Muons produced in the upper atmosphere travel towards the ground at a speed close to c. Far more of them reach the ground than would be expected from their mean lifetime at rest and the thickness of the atmosphere measured by an observer on the ground.

Which row gives a correct explanation in the frame of the ground and in the rest frame of the muons?

In an inertial frame, two events are separated by a distance of 5.0 light-years and a time of 3.0 years.

Which statement is correct?

A train moves at constant speed v in the positive x-direction relative to a platform (frame S). The train frame S′ has its origin at the front of the train, and the origins of S and S′ coincide at t = 0. At t = 0 a passenger is at x′ = L and walks towards the rear of the train, in the negative x′-direction, at a constant speed w relative to the train. Both speeds are much less than c.

What is the position x of the passenger in frame S at time t?

In frame S, event Y occurs 4.0 μs after event X and 600 m further along the positive x-direction. Frame S′ moves at 0.60c in the positive x-direction relative to S.

What is the time between X and Y measured in S′?

A particle moves with a speed v for which the Lorentz factor is γ.

Which expression gives v/c?

Two inertial frames move relative to each other at a constant velocity.

Which quantities have the same value in both frames according to the Galilean transformation, but can have different values according to the Lorentz transformation?
I. The time interval between two events
II. The length of a rod lying along the direction of the relative motion
III. The order in time of two events that occur at the same place in one of the frames

A sealed laboratory without windows is on a train that moves along a straight track at constant velocity relative to the ground.

Which experiment, carried out entirely inside the laboratory, could be used to determine the speed of the train relative to the ground?

A train moves at constant velocity past a long platform. Event 1 is the front of the train passing the start of the platform; event 2 is the front of the train passing the end of the platform. Observer F sits at the front of the train. Observer S stands on the platform at its start.

Which row identifies the observer who measures the proper time between events 1 and 2 and the observer who measures the proper length of the platform?

A rectangular solar sail has proper dimensions 40 m × 30 m. It moves at 0.80c relative to an observer, in a direction parallel to its 40 m sides.

What area of the sail does the observer measure?

A spacecraft moves away from the Earth at 0.50c. It fires a projectile straight ahead. An observer on the Earth measures the speed of the projectile to be 0.80c.

What is the speed of the projectile relative to the spacecraft?

Two beacons P and Q, 1200 m apart, are at rest in frame S with P at x = 0 and Q at x = 1200 m. They flash at the same time in S. A spacecraft moves at 0.80c in the positive x-direction, from P towards Q.

Which row gives the flash that happens first according to an observer on the spacecraft, and the time between the flashes measured by that observer?

The space-time diagram shows the world line of a particle. The world line makes an angle of 37° with the ct axis.

Which row gives the speed of the particle and its Lorentz factor γ?

The space-time diagram shows the axes of an inertial frame S and of a frame S′ moving at 0.50c relative to S. Five events A, B, C, D and E are marked.

Which two events are simultaneous in frame S′?

A cargo shuttle of proper length 120 m passes a navigation beacon at a constant speed of 0.80c.

Which row gives the time taken for the shuttle to pass the beacon, as measured by an observer at the beacon and as measured by the crew of the shuttle?

A spacecraft of proper length 200 m moves at constant speed along its length through an open-ended straight hangar, of proper length 160 m, on a space station. In the rest frame of the hangar, there is an instant at which the spacecraft fits exactly inside the hangar.

What is the length of the hangar in the rest frame of the spacecraft?

A rod of proper length L₀ moves at constant speed v along its own length relative to a laboratory. A light pulse is emitted from the rear end of the rod towards the front end. The Lorentz factor for speed v is γ.

What is the time taken for the pulse to reach the front end of the rod, measured in the laboratory frame?

Three observers each use a frame of reference attached to themselves. Treat a frame fixed to the surface of the Earth as inertial.

I. An astronaut in a space probe coasting with its engines off, far from any planet or star
II. A passenger in a car travelling round a roundabout at a constant speed
III. A passenger in a lift that is moving upwards at a constant speed

Which observers are in an inertial frame of reference?

The graph shows how the position x of a cyclist varies with time t in the frame S of the road. Frame S′ moves at a constant velocity of 10 m s⁻¹ in the positive x-direction relative to S, and the origins of S and S′ coincide at t = 0.

The graph of the position x′ of the cyclist in S′ against t is a straight line. Which row gives the gradient and the intercept on the position axis of this line?

In frame S a ball moves with velocity u in the positive x-direction. Frame S′ moves relative to S with speed v in the negative x-direction. Both speeds are much less than c.

What is the velocity of the ball in S′, taking the positive x′-direction to be the same as the positive x-direction?

In an inertial frame S, two events are separated by a distance Δx and a time Δt, where Δx < cΔt. An inertial observer travels in a straight line at constant velocity and is present at both events.

What time interval does this observer measure between the two events?

Two spacecraft P and Q have the same proper length. They pass an observer, moving parallel to their lengths, at speeds of 0.60c (P) and 0.80c (Q).

What is (length of P measured by the observer)/(length of Q measured by the observer)?

The space-time diagram, drawn in frame S, shows the world lines of three inertial observers P, Q and R. Event E1 is at the origin and event E2 is on the world line of Q.

Which observer measures the shortest time interval between E1 and E2?

In frame S, event Y occurs 5.0 years after event X and 3.0 ly from it. In another inertial frame S′ the two events are 4.5 ly apart.

What is the time between X and Y in S′?

The space-time diagram shows event O and four other events E1, E2, E3 and E4. The dashed lines are the world lines of light rays through O. The scales on the two axes are the same.

Which event could occur at the same place as O in some inertial frame, and occurs after O in every inertial frame?

The relativistic velocity addition equation is u′ = (u − v)/(1 − uv/c²), where v is the velocity of frame S′ relative to frame S.

Which statements are correct?

I. If u = c, then u′ = c for every value of v less than c.
II. If u and v are both much less than c, then u′ ≈ u − v.
III. Two spacecraft approach each other, each moving at 0.60c relative to a space station. The station observer measures the distance between them decreasing at a rate less than c.

In frame S an event occurs at x = 900 m and t = 2.0 μs. Frame S′ moves at 0.80c in the positive x-direction relative to S, and the origins of S and S′ coincide at t = t′ = 0.

Which row gives the coordinates of the event in S′?

The space-time diagram, drawn in the frame S of a space station, shows the world lines of two spacecraft P and Q that move towards each other along the x-axis. The scales on the two axes are the same.

What is the speed of Q measured by an observer on P?

In frame S, event B occurs 2.0 μs after event A at a point 900 m from A in the positive x-direction.

Which inertial frames, moving in the positive x-direction relative to S, measure event B to occur before event A?

The diagram shows the axes of an inertial frame S and those of a frame S′ that moves at speed v relative to S, drawn on a space-time diagram of S. The angle between the ct′ axis and the x′ axis is 50°.

What is v?

A loudspeaker and a microphone are fixed 1.500 ± 0.002 m apart inside a long straight air duct, with the microphone downstream. Air flows past them at a speed w measured with an anemometer. The loudspeaker emits a short click and a data logger records the time t until the microphone detects it; each reading is ±0.01 ms. When the microphone is pressed against the loudspeaker, the logger reads 0.15 ms.

The speed of the click relative to the duct is u = 1.500 m/(t − 0.15 ms).

A beam of unstable particles, all moving at 0.95c relative to the laboratory (γ = 3.20), travels along a straight vacuum pipe. Detectors at distances x from the production point count the number N of particles that reach them in a fixed time. The particles decay exponentially, so in the laboratory frame ln N = ln N₀ − x/(γvτ₀), where τ₀ is the mean lifetime of the particles in their rest frame.

The uncertainty in each value of ln N is negligible. The graph shows ln N against x with the line of best fit.

In an ion-beam experiment, singly charged ions of mass m, initially at rest, are accelerated through a potential difference V and then move at a constant speed v ≪ c. An atomic transition in the ions acts as a clock. The time interval Δt₀ between two "ticks" in the ions' rest frame is compared with the interval Δt measured in the laboratory, giving the fractional shift f = (Δt − Δt₀)/Δt₀.

For v ≪ c, γ ≈ 1 + v²/(2c²). Each value of f has an uncertainty of ±0.3 × 10⁻⁷.

In an accelerator, neutral pions moving at 0.9990c relative to the laboratory decay into gamma-ray photons. Photons emitted in the direction of motion of the pions are detected at a distance d from the decay region. The time t between a reference signal from the accelerator and the detection of the photons is recorded; each time is ±0.5 ns. Each value of t includes the same fixed delay in the electronics.

The graph shows t against d with the line of best fit.

A student tests the Galilean transformation. A ball rolls in a straight line along a long horizontal bench, slowing down. Camera 1 is fixed to the bench (frame S). Camera 2 is fixed to a motorised trolley that moves along a rail beside the bench at a constant velocity in the positive x-direction (frame S′). Video analysis gives the position x of the ball measured from a mark on the bench and its position x′ measured from a mark on the trolley, at the same times t. At t = 0 the trolley mark is a distance d ahead of the bench mark. Each position reading has an uncertainty of ±0.01 m; the uncertainty in t is negligible.

The Galilean transformation predicts that x − x′ = vt + d, where v is the speed of the trolley. The graph shows the data, with the line of best fit.

Unstable particles are produced when a proton beam strikes a target. Particles moving at a chosen speed v are selected and sent along an evacuated beam pipe. For each speed, the mean lifetime τ of the particles in the laboratory frame is found from the decrease in the number of particles along the pipe. Each value of τ has an uncertainty of ±1.0 ns; the uncertainty in v is negligible.

Special relativity predicts that τ = γτ₀, where τ₀ is the mean lifetime of the particles in their rest frame. The graph shows τ against γ with error bars, for all six speeds.

Cosmic-ray muons that come to rest in a large block of plastic scintillator later decay. For each stopped muon, the detector records the time t between its arrival and its decay. The table gives the number of decays N recorded in time bins 1.0 μs wide, centred on t, for six selected bins. Random coincidences add a constant background of B = 40 counts to every bin, found from bins at very long times. For the decays, N − B = N₀e^−t/τ₀, where τ₀ is the mean lifetime of muons at rest.

The graph shows ln(N − B) against t with the line of best fit.

A beam of pions travels at 0.980c along a straight 40.0 m beam line in a laboratory. Detector 1 at the start of the beam line and detector 2 at the end count the pions passing them. The half-life of pions at rest is 18.0 ns.

A spacecraft passes the Earth at 0.60c, travelling directly towards the Moon. Take the Earth and the Moon to be at rest in frame S, with the Earth at x = 0 and the Moon on the positive x-axis. Frame S′ is the spacecraft frame. As the spacecraft passes the Earth, at t = t′ = 0, a laser pulse is fired from the Earth towards the Moon (event E1). The pulse arrives at the Moon (event E2).

The Earth–Moon distance in S is 3.84 × 10⁸ m.

A nearby star has a parallax angle of 0.817 arc-seconds measured from the Earth. A spacecraft travels from the Earth to the star at a constant speed of 0.800c relative to the Earth. Treat the Earth and the star as at rest relative to each other, and consider only the outward journey.

In the inertial frame S of a laboratory, event 1 occurs at x = 0, t = 0 and event 2 occurs at x = 1500 m, t = 8.00 μs. Frame S′ moves at 0.500c in the positive x-direction relative to S, and the origins coincide at t = t′ = 0.

Clock P is at rest at the origin of frame S, the rest frame of a space station. An identical clock Q is carried by a probe that moves at 0.60c in the positive x-direction relative to S. The probe passes the station at event O, when both clocks read zero. The probe frame is S′.

The space-time diagram, drawn in S, shows the world line of the probe (the ct′ axis) and the x′ axis. Event A is clock P reading 5.0 μs. Event B is clock Q reading 4.0 μs. Take c = 3.00 × 10⁸ m s⁻¹.

A train moves along a straight horizontal track at a constant velocity of 60.0 m s⁻¹ in the positive x-direction relative to the ground. Frame S is attached to the ground and frame S′ to the train; their origins coincide at t = t′ = 0. The air is still relative to the ground, and the speed of sound in the air is 343 m s⁻¹.

A courier ship passes a space station at event O and continues at a constant speed of 0.50c in the positive x-direction. Frame S is the station frame and frame S′ the ship frame; O is the origin of both. The space-time diagram, drawn in S, shows the ct′ and x′ axes of the ship frame.

Event E₁ is the flash of a beacon at rest in S at x = 900 m, ct = 450 m. Event E₂ is an alarm on the ship at x = 400 m, ct = 800 m. Take c = 3.00 × 10⁸ m s⁻¹.

In a laboratory, a beam of unstable particles B is selected by passing it between parallel plates 5.0 mm apart with a potential difference of 12 kV between them. A uniform magnetic field of flux density 0.010 T acts perpendicular to both the beam and the electric field. Only particles that pass straight through are used. Beam B travels in the negative x-direction. A second beam of particles A travels at 0.60c in the positive x-direction.

Two gliders move towards each other along a straight horizontal air track. Frame S is attached to the track. In S, glider A of mass 0.40 kg moves at +0.60 m s⁻¹ and glider B of mass 0.20 kg moves at −0.30 m s⁻¹. The gliders collide and stick together. Friction and air resistance are negligible.

A navigation satellite moves in a circular orbit of radius 2.66 × 10⁷ m around the Earth. An atomic clock on the satellite is compared with an identical clock at rest in a frame centred on the Earth, which may be treated as inertial. Consider only the effect of the motion of the satellite (special-relativistic time dilation).

Mass of the Earth = 5.97 × 10²⁴ kg. For v ≪ c, γ ≈ 1 + v²/(2c²).

A probe passes the Earth and travels away from it in a straight line at a constant speed of 0.80c relative to the Earth. It carries a sample of a radioactive isotope. The graph shows how the activity A of the sample, measured by instruments on the probe, varies with the time t shown by the probe clock. Both the probe clock and an Earth clock read zero as the probe passes the Earth.

1 year = 3.16 × 10⁷ s.