How to Calculate Relativistic Velocity: Step-by-Step Examples

📚 Quick Study Guide

• ⏱️ Relativistic velocities occur when objects move at a significant fraction of the speed of light.
• 💡 Classical velocity addition ($v = v_1 + v_2$) doesn't work at relativistic speeds.
• 🍎 We use the relativistic velocity addition formula: $v = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}$, where $v$ is the relative velocity, $v_1$ and $v_2$ are the velocities of the two objects, and $c$ is the speed of light (approximately $3 \times 10^8$ m/s).
• 📏 Remember that the velocities $v_1$ and $v_2$ are measured relative to a stationary observer or frame of reference.
• 🧪 It's crucial to use consistent units (usually meters per second) for accurate calculations.

Practice Quiz

1. What is the relativistic velocity addition formula?
   1. A) $v = v_1 + v_2$
   2. B) $v = \frac{v_1 + v_2}{1 - \frac{v_1v_2}{c^2}}$
   3. C) $v = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}$
   4. D) $v = v_1 - v_2$
2. Two spaceships are moving towards each other. Spaceship A is moving at $0.6c$ and Spaceship B is moving at $0.7c$ relative to Earth. What is the relative velocity of Spaceship A as observed from Spaceship B?
   1. A) $1.3c$
   2. B) $0.93c$
   3. C) $0.2c$
   4. D) $c$
3. What happens to the relativistic velocity as either $v_1$ or $v_2$ approaches $c$?
   1. A) It exceeds $c$
   2. B) It approaches $c$
   3. C) It approaches $0$
   4. D) It stays constant
4. A particle is moving at $0.8c$ relative to an observer. The observer is moving at $0.5c$ in the same direction relative to a lab. What is the particle's velocity relative to the lab?
   1. A) $1.3c$
   2. B) $0.93c$
   3. C) $0.62c$
   4. D) $0.3c$
5. What is the speed of light, denoted by 'c', approximately equal to?
   1. A) $3 \times 10^5$ m/s
   2. B) $3 \times 10^6$ m/s
   3. C) $3 \times 10^7$ m/s
   4. D) $3 \times 10^8$ m/s
6. If two objects are moving in the same direction, how does relativistic velocity addition affect their combined speed compared to classical addition?
   1. A) Relativistic speed is always greater.
   2. B) Relativistic speed is always smaller.
   3. C) Relativistic speed is the same.
   4. D) Relativistic speed can be greater or smaller depending on the frame of reference.
7. An electron is traveling at $0.9c$ relative to a lab. A photon is emitted in the same direction as the electron. What is the photon's velocity relative to the lab?
   1. A) $1.9c$
   2. B) $0.1c$
   3. C) $c$
   4. D) $0.9c$