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 toward each other. Spaceship A fires a missile at spaceship B. The speed of the missile, relative to spaceship A, is 0.750𝑐. The speed of the missile, relative to spaceship B, is 0.950𝑐. What is the relative velocity of the two spaceships?

Q2:

Suppose a spaceship heading directly away from the Earth at 0.750𝑐 can shoot a canister at 0.500𝑐 relative to the ship.

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

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

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

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

Q3:

Suppose the speed of light were only 3,000 m/s. A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1,000 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 0.250𝑐. A spaceship sent from one of the planets approaches the other planet at a speed of 0.750𝑐 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 toward the nearest star moves away from Earth at 0.250𝑐. The probe sends radio waves toward 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 0.200𝑐 and a message capsule is sent toward it at 0.200𝑐 relative to Earth, what is the speed of the capsule relative to the ship?

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

Q7:

A moving object initially has a Lorentz factor 𝛾=2.50 which later becomes 𝛾=10.0.

At what relative velocity is 𝛾=2.50 ?

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

At what relative velocity is 𝛾=10.0?

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

Q8:

Suppose a spaceship heading straight toward Earth at 0.650𝑐 can shoot a canister at 0.400𝑐 relative to the ship.

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

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

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

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

Q9:

If two spaceships are heading directly toward each other at a relative velocity of 0.7000𝑐, at what speed must a canister be shot from the first ship for the other ship to perceive the speed as 0.9980𝑐?

  • A 0 . 2 9 8 0 𝑐
  • B 0 . 4 1 7 3 𝑐
  • C 0 . 9 8 8 7 𝑐
  • D 0 . 7 1 5 3 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 0.960𝑐 and approaches the other at 0.850𝑐. What is the relative velocity of the two ships?

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

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