Section 10: Relativity
1. Length Contraction
At what speed must a 1-meter-long ruler travel for its length to be observed as 0.5 meters due to length contraction?
2. Time Dilation
Two identical devices, each playing a series of 1-hour episodes of a TV show, were built and synchronized on Earth. One device stayed on Earth, while the other was placed on a spaceship that was immediately accelerated to relativistic speed. How fast must the spaceship be moving for the device on Earth to have played 2 episodes while the device on the spaceship has played only 1 episode?
3. Time Dilation in Real Life
A muon has the average lifetime of \(2.20\,\mu\text{s}\) in its own rest frame. If it travels at \(0.990c\), what will its average lifetime be as measured by an observer on Earth?
4. High-Energy Physics
How much energy (in GeV) is required to accelerate a proton from rest to a speed of \(0.99c\)? The rest mass of a proton is approximately \(938 \text{ MeV/c}^2\).
5. Relativistic Mass
What is the relativistic mass of a 50 kg supermodel traveling at 99.9% of the speed of light?
6. Mass-Energy
The "Little Boy" atomic bomb converted approximately 0.7 grams of mass into energy. Using \(E=mc^2\), calculate the energy released in Joules. (1 kiloton of TNT is approximately \(4.184 \times 10^{12}\) J).
7. Velocity Transformation
A rocket traveling at \(0.8c\) away from Earth shoots a probe forward at \(0.5c\) relative to the rocket. What is the probe's speed as measured by an observer on Earth according to classical mechanics, and what is it according to special relativity?
8. Adding Velocities
What would happen to the relative velocities of two objects moving away from each other if one of the objects is moving at the speed of light?
9. Twin Paradox
An astronaut spends one year on the International Space Station (ISS), which travels at about \(7.66 \text{ km/s}\). How much younger would the astronaut be than their twin on Earth due to the kinetic time dilation?
10. Gravitational Time Dilation
How much younger would the astronaut be than their twin on Earth purely due to gravitational time dilation after one year? Assume the ISS orbits at an altitude of 400 km above Earth's surface.