# 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.

**Q1: **

An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a train that is 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.

- A s
- B s
- C s
- D s
- E s

**Q6: **

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

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

- A
- B
- C
- D
- E

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.

- A years
- B years
- C years
- D years
- E years

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

- A years
- B years
- C years
- D years
- E years

**Q7: **

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

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?

**Q8: **

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?

**Q11: **

Spaceship A is moving at speed with respect to another spaceship B. Observers in A and B set their clocks so that the event at 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 and turns it off at , 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.

**Q12: **

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 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?