AP Physics C: Time Dilation Questions and Detailed Solutions

📚 Quick Study Guide

• ⏱️ Time dilation is the phenomenon where time passes differently for observers in different reference frames, especially at relativistic speeds.
• ⚛️ The formula for time dilation is: $t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$, where:
  • $t'$ is the time observed in the moving frame.
  • $t$ is the proper time (time in the rest frame).
  • $v$ is the relative velocity between the frames.
  • $c$ is the speed of light.
• 🚀 Proper time is always the shortest time interval measured between two events.
• 💡 Remember that as $v$ approaches $c$, time dilation becomes more significant, and $t'$ becomes much larger than $t$.
• 🌌 Time dilation is a real, experimentally verified phenomenon. It's not just theoretical!

Practice Quiz

1. Question 1: A spaceship travels at 0.8c relative to Earth. An astronaut on the spaceship measures 2 hours passing. How much time has passed on Earth during that time?
   1. 1.2 hours
   2. 2.0 hours
   3. 3.33 hours
   4. 5.0 hours
2. Question 2: What is the relative speed of a moving clock if it is observed to run half as fast as a clock at rest?
   1. 0.25c
   2. $\sqrt{0.75}c$
   3. 0.5c
   4. 0.866c
3. Question 3: A muon has a proper lifetime of $2.2 \mu s$. It is observed to have a lifetime of $6.6 \mu s$ in a detector. How fast is the muon moving relative to the detector?
   1. 0.33c
   2. 0.67c
   3. 0.91c
   4. 0.95c
4. Question 4: If an observer on Earth sees a spaceship moving at 0.6c, and the spaceship shines a light beam forward, what speed does the observer on Earth measure for the light beam?
   1. 0.4c
   2. 0.6c
   3. 1.0c
   4. 1.6c
5. Question 5: A clock is moving at a speed of 0.9c. By what factor is its time dilated?
   1. 0.44
   2. 2.29
   3. 1.00
   4. 0.90
6. Question 6: Two events occur at the same location in an inertial frame. The time interval between them is measured to be 4 seconds. In another inertial frame moving at 0.6c relative to the first, what is the time interval between the two events?
   1. 2.5 seconds
   2. 3.2 seconds
   3. 5.0 seconds
   4. 6.4 seconds
7. Question 7: At what speed must a rocket travel so that its length is observed to be shortened to 60% of its proper length by an Earth-based observer?
   1. 0.40c
   2. 0.60c
   3. 0.80c
   4. 0.92c