AP Physics C Questions on Relativistic Energy and Momentum

📚 Quick Study Guide

• ⏱️ Relativistic energy and momentum become significant when dealing with objects moving at a considerable fraction of the speed of light.
• 📐 The Lorentz factor, $\gamma$, is given by $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the object's speed and $c$ is the speed of light.
• ⚛️ Relativistic momentum is defined as $p = \gamma mv$, where $m$ is the rest mass of the object.
• 💡 Relativistic energy is defined as $E = \gamma mc^2$. This can also be expressed as $E = K + mc^2$, where $K$ is the kinetic energy and $mc^2$ is the rest energy.
• ⚗️ The relationship between relativistic energy, momentum, and rest mass is given by $E^2 = (pc)^2 + (mc^2)^2$.
• ⚡ The kinetic energy is given by $K = (\gamma - 1)mc^2$.
• 🔭 Remember to always check if the speed is significant compared to the speed of light before using relativistic formulas!

Practice Quiz

1. Which of the following equations correctly relates relativistic energy (E), momentum (p), and rest mass (m)?
   1. A. $E = pc$
   2. B. $E^2 = p^2 + m^2c^4$
   3. C. $E^2 = (pc)^2 + (mc^2)^2$
   4. D. $E = mc^2$
2. What happens to the relativistic momentum of a particle as its velocity approaches the speed of light?
   1. A. It decreases to zero.
   2. B. It remains constant.
   3. C. It increases without bound.
   4. D. It decreases slightly.
3. A particle has a rest mass of $m$ and is moving at a speed of $0.6c$. What is its relativistic kinetic energy?
   1. A. $0.25mc^2$
   2. B. $0.36mc^2$
   3. C. $0.5mc^2$
   4. D. $0.75mc^2$
4. What is the Lorentz factor ($\gamma$) for a particle moving at a speed of $0.8c$?
   1. A. 1.25
   2. B. 1.67
   3. C. 2.00
   4. D. 2.50
5. If the total relativistic energy of a particle is twice its rest energy, what is its speed?
   1. A. $0.5c$
   2. B. $\frac{\sqrt{3}}{2}c$
   3. C. $0.75c$
   4. D. $\frac{1}{2}c$
6. A proton has a kinetic energy of $938 \text{ MeV}$ (approximately equal to its rest energy). What is its approximate total relativistic energy?
   1. A. $469 \text{ MeV}$
   2. B. $938 \text{ MeV}$
   3. C. $1876 \text{ MeV}$
   4. D. $2814 \text{ MeV}$
7. Which of the following is conserved in all relativistic collisions?
   1. A. Kinetic Energy only
   2. B. Momentum only
   3. C. Both Kinetic Energy and Momentum independently
   4. D. Total relativistic energy and momentum