Training Special Relativity Placement Test Practice — Special Relativity
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Placement Test Practice — Special Relativity

25 min Special Relativity

Placement Test Practice — Special Relativity

This comprehensive practice test covers all the topics from this module: the Lorentz factor, time dilation, length contraction, relativistic momentum and energy, velocity addition, and spacetime intervals. The problems range from straightforward formula applications to multi-step derivations, mirroring the style of college-level physics problems. Work through them carefully, showing all steps, and check your answers against the solution key.

Practice Test — 25 Questions

1. Calculate $\gamma$ for $v = 0.75c$.
2. A clock on a spaceship at $0.9c$ ticks for 3 years (ship time). How many years pass on Earth?
3. A 200 m spaceship travels at $0.95c$. What is its length as seen from Earth?
4. Find the rest energy of a proton ($m_p = 1.67 \times 10^{-27}$ kg) in joules and MeV.
5. Two ships approach each other at $0.8c$ each. What is their relative speed?
6. A particle at $v = 0.6c$ has rest mass 2 kg. Find its relativistic momentum.
7. If $\gamma = 5$, find $v/c$.
8. A muon's rest half-life is 1.56 μs. At $0.994c$, what is the observed half-life?
9. At what speed is an object contracted to 25% of its rest length?
10. Event A: $(0, 0)$. Event B: $(4.5 \times 10^8 \text{ m}, 2 \text{ s})$. Classify the spacetime interval.
11. Calculate the kinetic energy of a 1 kg object at $0.5c$.
12. A ship at $0.9c$ fires a laser forward. What speed does Earth see for the laser?
13. How much mass is converted to energy in a 1 megaton nuclear explosion ($4.2 \times 10^{15}$ J)?
14. Twins: one at rest, one travels at $0.99c$ for 2 ship-years. What is the age difference when reunited?
15. Find the proper time between events with $\Delta t = 8$ s and $\Delta x = 1.8 \times 10^9$ m.
16. A rocket has $\gamma = 3$. By what factor has its momentum increased compared to the classical value $mv$?
17. Derive: at what speed is relativistic KE equal to rest energy?
18. A ship at $0.4c$ fires a probe at $0.5c$ backward (relative to ship). What does Earth see?
19. Show that $\gamma(0.866c) = 2$.
20. The LHC accelerates protons to $\gamma = 7{,}000$. What is $v/c$ to 8 decimal places?
21. A 50 m pole at $0.866c$ passes through a 25 m barn. In the barn's frame, does the pole fit? In the pole's frame?
22. Calculate $E^2 - (pc)^2$ for a 1 kg object at $0.8c$ and verify it equals $(mc^2)^2$.
23. A photon has energy 2 eV. What is its momentum?
24. At $v = 0.1c$, by what percentage do the relativistic and classical kinetic energies differ?
25. Explain why the spacetime interval is called "Lorentz invariant."
Show Answer Key

1. $\gamma = 1/\sqrt{1-0.5625} = 1/\sqrt{0.4375} \approx 1.512$

2. $\gamma \approx 2.294$. Earth time $= 2.294 \times 3 \approx 6.88$ years.

3. $\gamma \approx 3.203$. $L = 200/3.203 \approx 62.4$ m.

4. $E_0 = 1.67 \times 10^{-27} \times 9 \times 10^{16} = 1.503 \times 10^{-10}$ J $= 938.3$ MeV.

5. $u = (0.8+0.8)/(1+0.64) = 1.6/1.64 \approx 0.976c$.

6. $\gamma = 1.25$. $p = 1.25 \times 2 \times 0.6 \times 3 \times 10^8 = 4.5 \times 10^8$ kg·m/s.

7. $v/c = \sqrt{1-1/25} = \sqrt{24/25} \approx 0.9798c$.

8. $\gamma = 1/\sqrt{1-0.988} \approx 9.13$. Observed half-life $\approx 9.13 \times 1.56 \approx 14.2$ μs.

9. $L/L_0 = 0.25 = 1/\gamma$, so $\gamma = 4$. $v = c\sqrt{1-1/16} = c\sqrt{15/16} \approx 0.968c$.

10. $s^2 = 9 \times 10^{16} \times 4 - (4.5 \times 10^8)^2 = 3.6 \times 10^{17} - 2.025 \times 10^{17} = 1.575 \times 10^{17} > 0$. Timelike.

11. $\gamma \approx 1.155$. $K = (1.155-1) \times 1 \times 9 \times 10^{16} = 0.155 \times 9 \times 10^{16} = 1.39 \times 10^{16}$ J.

12. $c$ (light always travels at $c$).

13. $\Delta m = 4.2 \times 10^{15} / 9 \times 10^{16} \approx 0.047$ kg $= 47$ g.

14. $\gamma \approx 7.089$. Earth time $= 7.089 \times 2 \approx 14.18$ years. Age difference $\approx 14.18 - 2 = 12.18$ years.

15. $s^2 = 9 \times 10^{16} \times 64 - 3.24 \times 10^{18} = 5.76 \times 10^{18} - 3.24 \times 10^{18} = 2.52 \times 10^{18}$. $\Delta\tau = \sqrt{2.52 \times 10^{18}}/(3 \times 10^8) \approx 5.29$ s.

16. Factor of $\gamma = 3$. Relativistic momentum is 3 times the classical value.

17. $K = mc^2 \implies \gamma - 1 = 1 \implies \gamma = 2 \implies v = c\sqrt{3}/2 \approx 0.866c$.

18. $u = (0.4c - 0.5c)/(1 - 0.2) = -0.1c/0.8 = -0.125c$ (moving backward at $0.125c$ from Earth's view).

19. $\gamma = 1/\sqrt{1 - 0.75} = 1/\sqrt{0.25} = 1/0.5 = 2$ ✓

20. $v/c = \sqrt{1-1/\gamma^2} = \sqrt{1 - 1/49000000} \approx 0.99999999c$.

21. Barn frame: $\gamma = 2$, pole is $25$ m — it fits. Pole frame: barn is $12.5$ m — pole does NOT fit. This is the relativity of simultaneity at work.

22. $\gamma = 5/3$. $E = (5/3)(9 \times 10^{16}) = 1.5 \times 10^{17}$. $p = (5/3)(0.8)(3 \times 10^8) = 4 \times 10^8$. $E^2 - (pc)^2 = 2.25 \times 10^{34} - 1.44 \times 10^{34} = 8.1 \times 10^{33} = (mc^2)^2$ ✓

23. $p = E/c = 2 \times 1.602 \times 10^{-19}/(3 \times 10^8) = 1.07 \times 10^{-27}$ kg·m/s.

24. $\gamma \approx 1.00504$. Relativistic $K = 0.00504mc^2$. Classical $K = 0.005mc^2$. Difference $\approx 0.8\%$.

25. "Lorentz invariant" means the quantity has the same value in every inertial reference frame. Different observers may measure different $\Delta t$ and $\Delta x$, but the combination $c^2\Delta t^2 - \Delta x^2$ is identical for all of them.

Spacetime Intervals and Minkowski Diagrams Back to Module