Worksheet: Time Dilation

In this worksheet, we will practice calculating the change in the amount of time between two events as measured in different reference frames.

Q1:

The Minkowski diagram shows two inertial reference frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 5 𝑐 relative to frame 𝑆 . The red line shows the world line of a light bulb that is stationary in frame 𝑆 and is initially off. At 𝐴 and 𝐵 , the light bulb flashes for a fraction of a second. In frame 𝑆 , the amount of time between the flashes, Δ 𝑡 , is 10 seconds. How much time passes between the flashes in frame 𝑆 ? Give your answer to 3 significant figures.

Q2:

A space probe moving away from Earth at a speed of 0 . 2 𝑐 sends out transponder signals back to Earth at regular intervals. On Earth, the signals are received every 153.1 minutes. What is the interval between transponder signals that are sent out in the reference frame of the space probe? Give your answer to 4 significant figures.

Q3:

A muon has a mean lifetime of 2 . 2 0 × 1 0 s in its rest frame. What is the mean lifetime of a muon, as measured in the reference frame of a laboratory, moving at a speed of 0 . 9 9 𝑐 relative to the laboratory? Give your answer to 3 significant figures.

  • A 2 . 2 0 × 1 0 s
  • B 4 . 3 8 × 1 0 s
  • C 3 . 1 0 × 1 0 s
  • D 1 . 1 1 × 1 0 s
  • E 1 . 5 6 × 1 0 s

Q4:

A spaceship carrying passengers departs Earth for the Alpha Centauri system, moving at a constant speed of 0 . 6 𝑐 relative to Earth. Exactly 1 hour after departing, in the reference frame of the spaceship, a transponder sends a signal back to Earth. In the reference frame of Earth, how long after departing was the transponder signal sent?

Q5:

A pion traveling at 0 . 8 𝑐 relative to a laboratory has a mean lifetime of 4 . 3 3 × 1 0 . What is the mean lifetime of the pion in its rest frame? Give your answer to 3 significant figures.

  • A 1 . 5 6 × 1 0 s
  • B 8 . 6 6 × 1 0 s
  • C 1 . 9 4 × 1 0 s
  • D 7 . 2 2 × 1 0 s
  • E 2 . 6 0 × 1 0 s

Q6:

A kaon is a particle with the mean lifetime of 1 . 2 4 × 1 0 s in its rest frame. How many meters per second less than the speed of light would a kaon have to move relative to an observer to have a mean lifetime of 0.1 m/s? Give your answer to three significant figures.

Q7:

The Sirius star system is located approximately 8.6 light years away from Earth. A spaceship is sent to the Sirius star system carrying several astronauts. The spaceship travels at a constant speed of 0 . 4 𝑐 .

From the reference frame of Earth, will the astronauts appear to age slower or faster than the people on Earth?

  • ASlower
  • BFaster
  • CAt the same rate

From the reference frame of the spaceship, will the people on Earth appear to age slower or faster than the astronauts?

  • ASlower
  • BFaster
  • CAt the same rate

Q8:

A Lambda baryon is a type of particle that is unstable and decays in a fraction of a second. If Lambda baryons are produced in a particle accelerator with speeds of 0 . 2 𝑐 , will they, on average, take more time to decay, less time to decay, or the same amount of time to decay, in the reference frame of the accelerator?

  • ALess time
  • BMore time
  • CSame amount of time

Q9:

A space probe is launched from Earth and sent toward deep space. It moves at a constant speed of 0 . 4 𝑐 . The probe flashes a laser beam back toward Earth twice. The first flash occurs when the probe is 2 . 1 6 × 1 0 m away from Earth. The second flash occurs 1 hour after that, in the reference frame of the probe.

How long does it take the light emitted in the first flash of laser to reach Earth? Give your answer in hours.

How far does the space probe travel between the first and the second flash, in the reference frame of Earth?

  • A 4 . 7 1 × 1 0 m
  • B 1 . 0 8 × 1 0 m
  • C 5 . 4 0 × 1 0 m
  • D 2 . 1 6 × 1 0 m
  • E 8 . 6 4 × 1 0 m

How long does it take the light emitted in the second flash of laser to reach Earth? Give your answer in hours.

In the reference frame of Earth, how much time elapses between the two flashes? Give your answer to 3 significant figures.

Q10:

The Minkowski diagram shows two inertial reference frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 4 𝑐 relative to frame 𝑆 . The red line shows the world line of a light bulb that is stationary in frame 𝑆 and is initially off. At 𝐴 and 𝐵 , the light bulb flashes for a fraction of a second. In frame 𝑆 , the amount of time between the flashes, Δ 𝑡 , is 80 seconds. How much time passes between the flashes in frame 𝑆 ? Give your answer to 3 significant figures.

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