Worksheet: Time Dilation

In this worksheet, we will practice applying Lorentz transformation to time measurements made by observers that have relative velocities to each other.


An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a train that is 5.0×10 m long train simultaneously, at the instant the middle of the train passes him at a speed of 50 m/s. Find the time between the lightning strikes as measured by a passenger seated in the middle of the train.

  • A3.7×10 s
  • B3.2×10 s
  • C4.2×10 s
  • D2.8×10 s
  • E2.3×10 s


𝜋-mesons have a proper lifetime of 2.60×10 s. What lifetime do 𝜋-mesons have as measured by an observer moving relative to them at 2.70×10 m/s?

  • A5.30×10 s
  • B5.56×10 s
  • C5.80×10 s
  • D5.77×10 s
  • E5.96×10 s


The lifetime of a neutral 𝜋-meson is measured by an observer to be 1.40×10 s. The proper lifetime of the 𝜋-meson is 8.40×10 s. What is the velocity of the 𝜋-meson relative to the observer?

  • A0.830𝑐
  • B0.800𝑐
  • C0.750𝑐
  • D0.813𝑐
  • E0.779𝑐


Suppose a 𝑊 particle created in a particle detector lives for 5.00×10 s. What distance does it move in this time if it is traveling at 0.900𝑐?


A muon with a velocity of 0.950𝑐 relative to the Earth exists for 2.20 µs in its own reference frame.

How far does the muon travel as measured by an observer on the Earth?

How far would the muon travel as measured by an observer moving with the same velocity as the muon?


All but the closest galaxies are receding from our own Milky Way Galaxy. A galaxy 12.00×10 light years away is receding at 0.9000𝑐.

At what velocity relative to the Milky Way must an exploratory probe travel to approach the other galaxy at 0.9900𝑐 as measured from that galaxy?

  • A0.9999𝑐
  • B0.9990𝑐
  • C0.9992𝑐
  • D0.9997𝑐
  • E0.9995𝑐

How long will it take the probe to reach the other galaxy as measured from Earth? You may assume that the velocity of the other galaxy remains constant.

  • A1.2064×10 years
  • B1.2070×10 years
  • C1.2073×10 years
  • D1.2078×10 years
  • E1.2058×10 years

How long will it then take for a radio signal to be beamed back?

  • A1.2060×10 years
  • B1.2055×10 years
  • C1.2070×10 years
  • D1.2058×10 years
  • E1.2066×10 years


An astronaut travels a distance of 4.300 light-years at a relative speed of 0.9994444𝑐.

How much time does the astronaut’s journey take as measured by an observer on Earth. Assume that the motion of the Earth is negligible.

How much time does the astronaut measure the journey to take?

What is the Lorentz factor associated with the astronauts’ relative speed?


A muon has a proper lifetime of 2.200 µs. The muon’s life span is measured by an observer on Earth that the muon moves relative to at 0.05130 C.

What is the life span of the muon as measured by the observer?

How far does the observer measure the muon to move during its lifetime?

How far does the muon move during its proper lifetime?


A neutral kaon, which is created by cosmic radiation, strikes the atmosphere. It moves at 0.970𝑐 relative to an observer. The proper lifetime of the kaon is 0.890×10 s. What is the observed lifetime of the kaon?

  • A5.26×10 s
  • B2.16×10 s
  • C3.66×10 s
  • D1.51×10 s
  • E5.14×10 s


A clock in a spaceship runs one-twelfth the rate at which an identical clock on Earth runs. What is the speed of the spaceship?

  • A0.9785𝑐
  • B0.9857𝑐
  • C0.9574𝑐
  • D0.9965𝑐
  • E0.9931𝑐


Spaceship A is moving at speed 𝑐3 with respect to another spaceship B. Observers in A and B set their clocks so that the event at (𝑥,𝑦,𝑧,𝑡) of turning on a laser in spaceship B has coordinates (0, 0, 0, 0) in A and also (0, 0, 0, 0) in B. An observer at the origin of B turns on the laser at 𝑡=0.000s and turns it off at 𝑡=0.200s, as measured in the reference frame of spaceship B. What is the time interval between the light turning on and off as measured by an observer in the reference frame of spaceship A? Give your answer to 3 significant figures.


An astronaut has a heartbeat rate of 66 beats per minute as measured during his physical exam on Earth. The heartbeat rate of the astronaut is measured when he is in a spaceship traveling at 0.50𝑐 with respect to Earth by an observer (A) in the ship and by an observer (B) on Earth.

What will be the heartbeat rate of the astronaut reported by observer A?

What will be the heartbeat rate of the astronaut reported by observer B?


A 𝜋 particle traveling at 0.230𝑐 leaves a track in the bubble chamber in which it is produced and moves through for 4.30×10 s as measured by an observer. What is the length of the track in the observer’s frame of reference?

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