Worksheet: Relativistic Velocity Transformation

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 0 . 7 5 0 𝑐 . The speed of the missile, relative to the spaceship B, is 0 . 9 5 0 𝑐 . What is the relative velocity of the two spaceships?

Q2:

Suppose a spaceship heading directly away from the Earth at 0 . 7 5 0 𝑐 can shoot a canister at 0 . 5 0 0 𝑐 relative to the ship.

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

  • A 0 . 3 6 7 𝑐
  • B 0 . 3 3 3 𝑐
  • C 0 . 3 8 3 𝑐
  • D 0 . 4 0 0 𝑐
  • E 0 . 4 1 1 𝑐

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

  • A 0 . 9 0 9 𝑐
  • B 0 . 9 4 9 𝑐
  • C 0 . 8 3 8 𝑐
  • D 0 . 7 9 0 𝑐
  • E 0 . 9 8 5 𝑐

Q3:

Suppose the speed of light were only 3 0 0 0 m/s. A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1 0 0 0 m/s. What are the bullets’ velocity relative to the target?

Q4:

Two planets are on a collision course, heading directly toward each other at a speed of 0 . 2 5 0 𝑐 . A spaceship sent from one of the planets approaches the other planet at a speed of 0 . 7 5 0 𝑐 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 0 . 2 5 0 𝑐 . The probe sends radio waves towards Earth. The radio waves have a frequency of 1.00 GHz in the probe’s 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 0 . 2 0 0 𝑐 and a message capsule is sent towards it at 0 . 2 0 0 𝑐 relative to Earth, what is the speed of the capsule relative to the ship?

  • A 0 . 3 6 2 𝑐
  • B 0 . 4 0 0 𝑐
  • C 0 . 4 4 1 𝑐
  • D 0 . 3 8 5 𝑐
  • E 0 . 4 1 7 𝑐

Q7:

A moving object initially has a Lorentz factor 𝛾 = 2 . 5 0 which later becomes 𝛾 = 1 0 . 0 .

At what relative velocity is 𝛾 = 2 . 5 0 ?

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

At what relative velocity is 𝛾 = 1 0 . 0 ?

  • A 0 . 9 9 5 𝑐
  • B 0 . 9 4 9 𝑐
  • C 0 . 9 9 0 𝑐
  • D 0 . 9 9 9 𝑐
  • E 0 . 9 8 5 𝑐

Q8:

Suppose a spaceship heading straight towards Earth at 0 . 6 5 0 𝑐 can shoot a canister at 0 . 4 0 0 𝑐 relative to the ship.

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

  • A 0 . 7 9 5 𝑐
  • B 0 . 8 5 2 𝑐
  • C 0 . 8 4 3 𝑐
  • D 0 . 8 3 3 𝑐
  • E 0 . 7 8 5 𝑐

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

  • A 0 . 3 3 8 𝑐
  • B 0 . 3 6 2 𝑐
  • C 0 . 4 1 7 𝑐
  • D 0 . 4 0 0 𝑐
  • E 0 . 4 4 1 𝑐

Q9:

If two spaceships are heading directly towards each other at a relative velocity of 0 . 7 0 0 0 𝑐 , at what speed must a canister be shot from the first ship for the other ship to perceive the speed as 0 . 9 9 8 0 𝑐 ?

  • A 0 . 4 1 7 3 𝑐
  • B 0 . 2 9 8 0 𝑐
  • C 0 . 7 1 5 3 𝑐
  • D 0 . 9 8 8 7 𝑐
  • E 0 . 9 6 5 4 𝑐

Q10:

A muon’s proper lifetime is 2.2 Β΅s. An observer measures the muon’s 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 0 . 9 6 0 𝑐 and approaches the other at 0 . 8 5 0 𝑐 . What is the relative velocity of the two ships?

  • A 0 . 5 6 2 𝑐
  • B 0 . 6 8 5 𝑐
  • C 0 . 1 1 0 𝑐
  • D 0 . 5 9 8 𝑐
  • E 0 . 3 9 3 𝑐

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