1. 1
   A train moves east along a straight, level track at a constant 25 m s⁻¹ relative to the platform. Inside, a passenger walks towards the front of the train at 1.5 m s⁻¹ relative to the train. Take east as positive, let frame S be the platform and frame S′ the train, and let the origins of S and S′ coincide at t = t′ = 0.

   (a)

   What is the velocity of the passenger relative to the platform, according to Galilean relativity?

   [1 mark]

   • A23.5 m s⁻¹
   • B1.5 m s⁻¹
   • C26.5 m s⁻¹
   • D25 m s⁻¹

   (b)

   Which observation is consistent with Galilean relativity?

   [1 mark]

   • AA ball dropped by the passenger falls vertically relative to the train, just as it would on the platform
   • BA clock on the train runs more slowly than an identical clock on the platform
   • CThe ball's acceleration measured from the platform is greater than when measured on the train
   • DAn experiment inside the closed train can show that the train is moving

   (c)

   A lamp on the platform flashes at x = 400 m and t = 10 s in frame S. Calculate the coordinates x′ and t′ of this event in frame S′.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A spacecraft travels directly away from Earth at 0.60c relative to Earth. It emits a pulse of light forwards, in its direction of motion. It also launches a probe forwards at 0.50c relative to the spacecraft.

   (a)

   What is the speed of the light pulse measured by an observer on Earth?

   [1 mark]

   • A1.6c
   • Bc
   • C0.40c
   • D0.60c

   (b)

   What is the speed of the probe relative to Earth?

   [1 mark]

   • A1.10c
   • B0.50c
   • C0.10c
   • D0.85c

   (c)

   Outline why the measured speed of the light pulse is incompatible with Galilean relativity.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Muons are created by cosmic rays high in the atmosphere and travel vertically downwards at 0.995c relative to the Earth. The half-life of muons at rest is 1.5 μs. Detector A, at the top of a mountain, counts 1000 muons per minute. Detector B, 3.0 km vertically below A at sea level, counts about 630 muons per minute for the same range of speeds.

   (a)

   Show that, if relativistic effects are ignored, fewer than 10 muons per minute would be expected at detector B.

   [3 marks]

   (b)

   Explain the count measured at B using time dilation in the Earth frame and using length contraction in the muons' frame, showing that both give the same prediction.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In the Earth frame S, two lamps A and B are 900 m apart on a straight track, with A at x = 0 and B at x = 900 m. Both lamps flash at t = 0 in frame S. A rocket travels in the positive x-direction at 0.80c relative to the Earth, passing lamp A at t = 0. Frame S′ is the rocket frame, with the origins of S and S′ coinciding at the moment the rocket passes A. Take c = 3.00 × 10⁸ m s⁻¹.

   (a)

   Using the Lorentz transformations, determine the time interval between the two flashes in the rocket frame, state which lamp flashes first in that frame, and show that the space-time interval between the two flashes is the same in both frames.

   [6 marks]

   (b)

   Discuss what this example shows about simultaneity and about the order of events in different inertial frames.

   [6 marks]

   Total for question 4: 12 marks

End of questions