Solved Examples: Calculating Kinetic Energy at Relativistic Speeds

📚 Quick Study Guide

• ⚛️ Relativistic effects become significant when an object's speed approaches a substantial fraction of the speed of light.
• 💡 The classical kinetic energy formula, $KE = \frac{1}{2}mv^2$, is not accurate at relativistic speeds.
• 📝 The relativistic kinetic energy formula is given by: $KE = mc^2(\gamma - 1)$, where $m$ is the rest mass, $v$ is the velocity, $c$ is the speed of light, and $\gamma$ is the Lorentz factor.
• ⏱️ The Lorentz factor, $\gamma$, is calculated as $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.
• 🔢 When $v$ is much smaller than $c$, the relativistic kinetic energy approximates the classical kinetic energy.
• 📌 Remember to use consistent units, typically kilograms for mass, meters per second for velocity, and meters per second for the speed of light ($c \approx 3 \times 10^8$ m/s).
• 🧮 Always calculate the Lorentz factor first before calculating the kinetic energy.

Practice Quiz

1. A particle with a rest mass of $9.11 \times 10^{-31}$ kg is moving at a speed of $0.6c$. What is its relativistic kinetic energy?

   1. $2.05 \times 10^{-14}$ J
   2. $4.10 \times 10^{-14}$ J
   3. $6.15 \times 10^{-14}$ J
   4. $8.20 \times 10^{-14}$ J

2. What is the Lorentz factor for a particle moving at a speed of $0.8c$?

   1. 1.25
   2. 1.67
   3. 2.00
   4. 2.50

3. An electron has a kinetic energy of $0.511$ MeV. What is its speed, expressed as a fraction of $c$?

   1. 0.5c
   2. 0.6c
   3. 0.7c
   4. 0.8c

4. A proton has a rest mass of $1.67 \times 10^{-27}$ kg and is moving with a kinetic energy of $8.35 \times 10^{-11}$ J. What is its speed?

   1. $0.1c$
   2. $0.2c$
   3. $0.3c$
   4. $0.4c$

5. If the kinetic energy of a particle is equal to its rest energy, what is the Lorentz factor?

   1. 0
   2. 1
   3. 2
   4. 3

6. A particle's kinetic energy is measured to be $3mc^2$. What is the Lorentz factor for this particle?

   1. 2
   2. 3
   3. 4
   4. 5

7. What happens to the kinetic energy of a particle as its velocity approaches the speed of light?

   1. It approaches zero.
   2. It remains constant.
   3. It approaches infinity.
   4. It decreases slightly.