A.5 Galilean and special relativity (HL)

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

Which expression gives v/c?

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 communications satellite moves directly away from a ground station at a speed of 0.50c relative to the station. The satellite fires a laser pulse back towards the station. What is the speed of the pulse measured by an observer at the ground station?

An astronaut on a fast-moving spacecraft wears a heart monitor that records the time between two consecutive heartbeats. Several observers measure this time interval. Which observer measures the proper time between the two heartbeats?

A delivery drone flies along the aisle of a moving cargo train. Observers on the train and on the ground both measure the drone's motion. According to the Galilean transformation, which quantity has the same value for both observers?

A spacecraft approaches Earth at 0.50c. It sends a laser pulse towards Earth. What speed does an observer on Earth measure for the laser pulse?

A long railway carriage moves at constant high speed along a straight track past a platform. A lamp fixed at the midpoint of the carriage emits a brief flash of light. Which statement about the light reaching the two ends of the carriage is correct?

A spacetime diagram for frame S has ct on the vertical axis and x on the horizontal axis, with the same scale on both axes. A particle moves along the x-axis at a constant speed of 0.25c. What is the gradient, Δ(ct)/Δx, of its worldline?

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?

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?

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?

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 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 deep-space probe moves at constant velocity. It leaves a relay station (event 1) and later passes a navigation beacon (event 2). In the frame of the relay station, the events are separated by 5.0 light-years in distance and 13 years in time. What time interval does the probe's own clock record between the two events?

A shuttle has a proper length of 120 m. An observer on a space station measures its length as 72 m as it flies past parallel to its length. What is the speed of the shuttle relative to the station?

Two probes are launched from Earth in opposite directions. Each moves at 0.50c relative to Earth. What is the speed of one probe measured in the rest frame of the other?

Two events are observed from several inertial frames moving relative to one another. Which quantity has the same value in every one of these frames?

A spacetime diagram for frame S has axes ct and x drawn to the same scale. The ct′ and x′ axes of frame S′ are added to the diagram. The angle between the ct′ axis and the x′ axis is 30°. What is the speed of S′ relative to S?

A spacecraft moves away from Earth at 0.50c. It launches a probe backwards, towards Earth, at 0.80c relative to the spacecraft. What is the velocity of the probe relative to Earth?

Muons created high in the atmosphere reach detectors at sea level in much greater numbers than non-relativistic physics predicts. Which statement correctly describes this observation from the muon's rest frame?

Two navigation beacons are at rest in the frame of a space station. Each beacon emits one flash. In the station frame the flashes are 5.0 years apart in time and 3.0 light-years apart in space. A courier ship moves at constant velocity and is present at both flashes. What time interval between the flashes does the ship's clock record?

A spacecraft moves directly away from Earth at 0.60c, as measured from Earth. It launches a probe directly away from Earth at 0.60c relative to the spacecraft. What is the speed of the probe relative to Earth?

Frame S′ moves at 0.80c in the +x direction relative to frame S. The origins coincide at t = t′ = 0. In S, a flare occurs at x = 9.0 × 10⁸ m at t = 0. Take c = 3.00 × 10⁸ m s⁻¹. What is the time of the flare in S′?

Two starting guns at opposite ends of a straight stadium fire at the same instant in the rest frame of the stadium: gun P at x = 0 and gun Q at x = L, where L > 0. A fast spacecraft flies along the +x direction past the stadium. Which statement describes the firing of the guns in the frame of the spacecraft?

Two spacecraft approach each other along a straight line. Each moves at 0.80c relative to a space station between them, as measured in the station frame. What is the speed of one spacecraft as measured from the other?

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.

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?

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′?

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

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?

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 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?

In frame S, event B occurs 5.0 light-years from event A, in the +x direction, and 3.0 years after A. Which statement is correct?

A spacecraft of proper length L₀ moves at 0.60c relative to a planet. A light pulse is sent from the rear of the spacecraft to the front. In the spacecraft frame the pulse takes a time L₀/c. What time does the pulse take according to an observer on the planet?

Two landing lights, one at each end of a straight runway of proper length 600 m, flash simultaneously in the runway frame. A spacecraft flies over the runway at 0.80c, moving from the west end towards the east end. In the spacecraft frame, what is the time interval between the flashes, and which flash happens first?

In frame S, two events are separated by a distance Δx and a time Δt, where Δx > cΔt > 0. Which statement is correct?

In frame S, two flares occur 12 light-minutes apart in space and 13 minutes apart in time. In frame S′, both flares occur at the same position. What is the time interval between the flares in S′?

In frame S, event Q occurs 4.0 light-seconds from event P along the x-axis and 3.0 s after it. A frame S′ moves along the x-axis at speed v relative to S, and in S′ the events P and Q are simultaneous. What is v?

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 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 a proton therapy unit used to treat tumours, protons are accelerated to a speed of 0.45c relative to the treatment room.

A cruise ship moves due east at a constant 6.0 m s⁻¹ relative to the shore. A passenger on deck throws a ball due west at 15 m s⁻¹ relative to the ship. Frame S is the shore and frame S′ is the ship. The x-axes of both frames point east.

A train moves along a straight track at 40 m s⁻¹ relative to the ground. A passenger rolls a ball forward along the aisle at 5.0 m s⁻¹ relative to the train. A second passenger shines a torch beam forward along the aisle. Take c = 3.00 × 10⁸ m s⁻¹.

Frame S′ moves at 0.60c in the +x direction relative to frame S. The origins coincide at t = t′ = 0. A pulsar flash is recorded in S at x = 4.0 × 10⁸ m and t = 2.0 s. Use the Lorentz transformations x′ = γ(x − vt) and t′ = γ(t − vx/c²), with 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.

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.

A navigation satellite orbits Earth at a speed of 3.87 km s⁻¹. Ignore gravitational effects on time and treat the satellite as an inertial frame over short intervals. Take c = 3.00 × 10⁸ m s⁻¹.

Electrons in a linear accelerator travel along a straight evacuated tube that is 3.2 km long in the laboratory frame. At the exit the electrons have a Lorentz factor of 4.0 × 10⁴. Assume this value of γ applies along the whole tube and that the electron speed is effectively c (3.0 × 10⁸ m s⁻¹).

An astronomer records three events in the Earth frame S. All positions lie along the x-axis.

Event | x / ly | t / years
A | 0 | 0
B | 3.0 | 5.0
C | 6.0 | 2.0

Use (Δs)² = (cΔt)² − (Δx)².

Electrons in a hospital radiotherapy linear accelerator are accelerated until their Lorentz factor γ is 2.50. Take c = 3.00 × 10⁸ m s⁻¹.

Students analyse data from a cosmic-ray experiment. Detector A is on a mountain summit and detector B is at sea level. Both detectors count only muons moving vertically downward at the speed given in the table.

Quantity — Value
Height of A above B — 3.20 km
Speed of the muons — 0.980c
Half-life of muons at rest — 1.52 μs
Count rate at A — 600 per hour

Take c = 3.00×10⁸ m s⁻¹.

In a particle physics laboratory, a radioactive nucleus moves at 0.70c relative to the laboratory. It emits an electron.

A spacetime diagram is drawn for the Earth frame S. It has axes ct and x, both in kilometres and drawn to equal scale. A rocket passes the origin at t = 0, moving at 0.50c in the +x direction. Event E1 is at the origin. Event E2 is at x = 4.0 km, ct = 2.0 km.

A maglev train moves along a straight track at 125 m s⁻¹. Frame S is the ground and frame S′ is the train. The origins coincide at t = t′ = 0. A warning light flashes at x = 2500 m, t = 12.0 s in the ground frame. Take c = 3.00×10⁸ m s⁻¹.

A spacecraft moves directly away from Earth at 0.70c, as measured from Earth. It launches two probes along its line of motion, each at 0.50c relative to the spacecraft: probe P is launched forwards (away from Earth) and probe Q is launched backwards (towards Earth). Use u = (u′ + v)/(1 + u′v/c²).

Two flares occur in a distant gas cloud. In frame S, flare X occurs at x = 0, t = 0 and flare Y occurs at x = 5.0 × 10⁸ m, t = 1.0 s. Use c = 3.00 × 10⁸ m s⁻¹, Δt′ = γ(Δt − vΔx/c²) and Δx′ = γ(Δx − vΔt) for a frame S′ moving at speed v along +x.

On a spacetime diagram for frame S, with ct vertical and x horizontal in the same units (metres), event P is at the origin and event Q has coordinates x = 3.0 m and ct = 5.0 m. Use c = 3.00 × 10⁸ m s⁻¹.

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 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.

A beam of pions is produced for a medical research facility. In their rest frame, pions have a half-life of 18 ns. The pions travel at 0.990c along a beamline 25 m long, measured in the laboratory. Take c = 3.00 × 10⁸ m s⁻¹.

Two beacons, P at the rear and Q at the front, are fixed 600 m apart at opposite ends of a space station. They flash simultaneously in the station frame. A ship moves past the station at 0.80c, travelling in the direction from P to Q. Take c = 3.0 × 10⁸ m s⁻¹.