Solved examples of Principle of Relativity problems

📚 Quick Study Guide

• ⏱️ The principle of relativity states that the laws of physics are the same for all observers in uniform motion.
• 🌌 There is no absolute frame of reference in the universe.
• 💡 The speed of light in a vacuum ($c \approx 3 \times 10^8$ m/s) is the same for all inertial observers, regardless of the motion of the light source.
• ⏳ Time dilation: Moving clocks run slower relative to a stationary observer. The time dilation formula is: $t' = \gamma t$, where $t'$ is the dilated time, $t$ is the proper time, and $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor.
• 📏 Length contraction: The length of an object moving relative to an observer is contracted in the direction of motion. The length contraction formula is: $L' = \frac{L}{\gamma}$, where $L'$ is the contracted length and $L$ is the proper length.
• ➕ Velocity addition: Velocities do not simply add linearly at relativistic speeds. If an object has velocity $u$ in a frame that is moving with velocity $v$ relative to another frame, the object's velocity in the other frame is: $w = \frac{u + v}{1 + \frac{uv}{c^2}}$.

Practice Quiz

1. A spaceship travels at 0.6c relative to Earth. A light beam is emitted from the spaceship in the same direction of motion. What is the speed of the light beam as observed from Earth?
   1. 0.4c
   2. 0.6c
   3. 1.0c
   4. 1.6c
2. A clock is moving at a speed of 0.8c relative to an observer. If 10 seconds pass on the moving clock, how much time has passed for the observer?
   1. 6 seconds
   2. 10 seconds
   3. 16.7 seconds
   4. 20 seconds
3. A meter stick is moving parallel to its length at a speed of 0.9c relative to an observer. What is the length of the meter stick as measured by the observer?
   1. 0.44 m
   2. 0.9 m
   3. 1 m
   4. 2.29 m
4. Two spaceships, A and B, are moving towards each other. Spaceship A is moving at 0.7c and spaceship B is moving at 0.8c, both relative to an observer on a planet. What is the velocity of spaceship A as observed from spaceship B?
   1. 0.1c
   2. 0.946c
   3. 1.5c
   4. -0.1c
5. A particle has a lifetime of $2 \times 10^{-8}$ s when measured at rest. If it moves at a speed of 0.95c, what is its lifetime as measured by a laboratory observer?
   1. $2.09 \times 10^{-8}$ s
   2. $6.4 \times 10^{-8}$ s
   3. $1.05 \times 10^{-8}$ s
   4. $3.2 \times 10^{-8}$ s
6. What is the value of Lorentz factor if $v = 0$?
   1. -1
   2. 0
   3. 1
   4. Undefined
7. How does the mass of a particle change as its velocity approaches the speed of light?
   1. Decreases
   2. Remains constant
   3. Increases
   4. Becomes zero