Relativistic Velocity Addition Quiz with Answers and Explanations

📚 Quick Study Guide

• ⏱️ Classical Velocity Addition: In everyday scenarios, velocities simply add linearly. If a boat is moving at 10 m/s and a person walks forward at 2 m/s, their combined speed is 12 m/s.
• 🚀 Einstein's Postulates: Relativity is based on two key ideas: (1) The laws of physics are the same for all observers in uniform motion, and (2) the speed of light in a vacuum ($c$) is the same for all observers, regardless of the motion of the light source.
• ✨ Relativistic Velocity Addition Formula: When velocities are a significant fraction of the speed of light, we use the following formula to correctly add them: $v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}$ where:
  • $v$ is the relative velocity.
  • $v_1$ and $v_2$ are the velocities of the two objects.
  • $c$ is the speed of light (approximately $3 \times 10^8$ m/s).
• 💡 Important Note: If either $v_1$ or $v_2$ is equal to $c$, then $v$ will always be equal to $c$. No matter how fast you go, you can never exceed the speed of light!
• 📝 Approximation: For small velocities ($v_1, v_2 << c$), the relativistic formula approaches the classical addition ($v = v_1 + v_2$).

🧪 Practice Quiz

1. A spaceship moves away from Earth at $0.6c$. It launches a probe forward at $0.8c$ relative to the spaceship. What is the probe's velocity relative to Earth?
   1. $0.2c$
   2. $1.4c$
   3. $0.946c$
   4. $c$
2. Two spaceships, A and B, approach each other. Ship A moves at $0.7c$ and ship B moves at $0.9c$ relative to an observer on Earth. What is the velocity of ship A as observed from ship B?
   1. $1.6c$
   2. $0.2c$
   3. $0.982c$
   4. $c$
3. A particle moves to the right at $0.5c$ in a lab frame. An observer moves to the right at $0.5c$ relative to the lab. What is the particle's velocity as observed by the moving observer?
   1. $c$
   2. $0$
   3. $0.8c$
   4. $0.6c$
4. If two photons are traveling in the same direction, what is their relative speed?
   1. $2c$
   2. $0$
   3. $c$
   4. Insufficient information
5. A spaceship is chasing another. The lead ship moves at $0.5c$ and the chasing ship moves at $0.6c$, both relative to Earth. What is the velocity of the chasing ship relative to the lead ship?
   1. $0.1c$
   2. $1.1c$
   3. $0.167c$
   4. $c$
6. An electron is accelerated to $0.9c$ in a particle accelerator. A physicist starts running alongside the accelerator at $0.5c$ (relative to the lab). What speed does the physicist observe for the electron?
   1. $1.4c$
   2. $0.4c$
   3. $0.714c$
   4. $c$
7. A spaceship travels at $0.95c$ relative to Earth. It fires a laser beam forward. What is the speed of the laser beam relative to Earth?
   1. $1.95c$
   2. $0.05c$
   3. $c$
   4. $0.95c$