In this worksheet, we will practice applying Lorentz transformation to determine the velocities of objects moving relative to each other at relativistic speeds.

**Q1: **

Suppose a spaceship A and a spaceship B are moving towards each another. Spaceship A fires a missile at spaceship B. The speed of the missile, relative to spaceship A, is . The speed of the missile, relative to the spaceship B, is . What is the relative velocity of the two spaceships?

**Q2: **

Suppose a spaceship heading directly away from the Earth at can shoot a canister at relative to the ship.

What is the velocity of the canister relative to Earth if it is shot directly at Earth?

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What is the velocity of the canister relative to Earth if it is shot directly away from Earth?

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**Q4: **

Two planets are on a collision course, heading directly toward each other at a speed of . A spaceship sent from one of the planets approaches the other planet at a speed of as measured in the reference frame of the other planet. What is the speed of the ship relative to its planet of origin?

**Q5: **

A space probe speeding towards the nearest star moves away from Earth at . The probe sends radio waves towards Earth. The radio waves have a frequency of 1.00 GHz in the probeβs reference frame. What is the frequency of these radio waves as measured by a receiver on Earth?

**Q6: **

If a spaceship is approaching Earth at and a message capsule is sent towards it at relative to Earth, what is the speed of the capsule relative to the ship?

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**Q7: **

A moving object initially has a Lorentz factor which later becomes .

At what relative velocity is ?

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At what relative velocity is ?

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**Q8: **

Suppose a spaceship heading straight towards Earth at can shoot a canister at relative to the ship.

What is the velocity of the canister relative to Earth if it is shot directly at it?

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What is the velocity of the canister relative to Earth if it is shot directly away from it?

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**Q9: **

If two spaceships are heading directly towards each other at a relative velocity of , at what speed must a canister be shot from the first ship for the other ship to perceive the speed as ?

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**Q10: **

A muonβs proper lifetime is 2.2 Β΅s. An observer measures the muonβs lifetime as 7.05 Β΅s. What is the relative velocity of the muon to the observer?

**Q11: **

When a missile is shot from one spaceship toward another, it leaves the first at and approaches the other at . What is the relative velocity of the two ships?

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