Solved Examples of Length Contraction: A Step-by-Step Approach

📚 Quick Study Guide

• 📏 Length Contraction: The phenomenon where the length of an object moving at relativistic speeds appears shorter in the direction of motion to an observer.
• ➗ Formula: $L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$, where:
  • $L$ is the observed length.
  • $L_0$ is the proper length (length in the object's rest frame).
  • $v$ is the relative velocity between the observer and the object.
  • $c$ is the speed of light (approximately $3 \times 10^8$ m/s).
• ➡️ Direction: Contraction occurs only along the direction of motion.
• ⚠️ Important Note: The observer must be in a different inertial frame of reference than the object.

Practice Quiz

1. What is the length of a meter stick moving at 0.8c relative to an observer?
   1. 0.4 m
   2. 0.5 m
   3. 0.6 m
   4. 0.7 m
2. A spaceship has a proper length of 100 m. If it moves past you at 0.6c, what length do you observe?
   1. 60 m
   2. 70 m
   3. 80 m
   4. 90 m
3. At what speed must an object move for its length to appear half its proper length?
   1. 0.5 c
   2. $\sqrt{0.5}$ c
   3. $\sqrt{0.75}$ c
   4. 0.75 c
4. A train is moving at 0.9c. An observer measures its length to be 50 m. What is the train's proper length?
   1. 50 m
   2. 100 m
   3. 114.5 m
   4. 120 m
5. What happens to the width of a moving object as its length contracts?
   1. It increases
   2. It decreases
   3. It stays the same
   4. It oscillates
6. A rod has a proper length of 2m. If its length is measured to be 1m, what is its velocity?
   1. $c/2$
   2. $(\sqrt{3}/2)c$
   3. $c/4$
   4. $(\sqrt{2}/2)c$
7. Which of the following is NOT a consequence of length contraction?
   1. Objects appear shorter in the direction of motion
   2. Time dilation
   3. The speed of light remains constant for all observers
   4. The object's mass increases