Special Relativity: Unit Review

u = −2.0 + 30 = 28 m/s east.

No. Its velocity changes direction, so it accelerates; it is non-inertial.

Maxwell's equations give one speed c for light with no reference frame. Galilean addition implies different observers measure different light speeds, so the equations could only hold in one special frame.

They expected a fringe shift of about 0.4 fringe from Earth's motion through the ether; they saw less than one-twentieth of that, consistent with no shift.

Across speed √(9.0 − 1.0) = 2.83 m/s; t = 120/2.83 = 42.4 s.

The predicted effect (0.4 fringe) was far larger than the smallest shift the apparatus could detect (about 0.01 fringe), so the absence of a shift was meaningful.

1. The laws of physics are the same in all inertial frames. 2. Light in vacuum has the same speed c in all inertial frames, regardless of the motion of source or observer.

c.

Chasing a light beam: a frozen light wave is impossible, so no one can catch up to light. (Or the magnet and conductor: only relative motion matters, so the principle of relativity covers electromagnetism.)

The front strike (the one the passenger moves toward).

Back: 10 ÷ (1.60c) = 20.8 ns; front: 10 ÷ (0.40c) = 83.3 ns.

No. Events that can be causally connected (a signal at or below c can travel between them) have the same order in all frames.

γ = 1/√(1 − 0.9025) = 3.20.

Δt = 3.20 × 2.0 = 6.4 years.

30/7.089 = 4.2 years.

v²/(2c²) = 9.0 × 10⁴ ÷ 1.80 × 10¹⁷ = 5.0 × 10⁻¹³; × 86 400 s = 4.3 × 10⁻⁸ s (43 ns).

Δt = 4500/(2.97 × 10⁸) = 15.2 µs = 10.0 half-lives: (1/2)^10.0 ≈ 0.001. With γ = 7.09: 15.2/7.09 = 2.14 µs = 1.41 half-lives: (1/2)^1.41 ≈ 0.38.

Eastbound. In a non-rotating frame the plane's speed added to Earth's rotation, so it moved faster than the ground clocks; time dilation (speed effect) outweighed the gravitational gain.

Yes, 420 lies within 390 to 430.

50/1.667 = 30 m.

6.0/1.25 = 4.8 ly; 4.8/0.60 = 8.0 years.

The spectator (the track is at rest in the spectator's frame).

s = √(169 − 25) = 12 ly, so 12 years.

Earth: 2 × 4.0/0.80 = 10 years; traveller: 10/1.667 = 6.0 years.

Only the travelling twin changes inertial frames (turns around); the stay-at-home twin's worldline is straight.

1.40c/1.49 = 0.940c.

(0.90 − 0.50)c/(1 − 0.45) = 0.40c/0.55 = 0.727c forward.

u = (c + v)/(1 + v/c) = c(c + v)/(c + v) = c.

γ = 2.294; p = 2.294 × 1.67 × 10⁻²⁷ × 2.70 × 10⁸ = 1.03 × 10⁻¹⁸ kg·m/s.

γ = 3 → v/c = √(1 − 1/9) = 0.943.

p = γmv → ∞ as v → c; a finite force can only add finite momentum in a finite time, so v approaches but never reaches c.

0.0010 × 9.00 × 10¹⁶ = 9.0 × 10¹³ J.

γ = 1 + 1.533/0.511 = 4.00; v/c = √(1 − 1/16) = 0.968.

pc = √(1876² − 938²) = √(3 519 376 − 879 844) = 1625 MeV.

Kinetic energy is (γ − 1)mc²; ½mv² is only its low-speed approximation. At 0.9c, (γ − 1)mc² = 1.294mc², while ½mv² = 0.405mc².

0.0500 × 931.5 = 46.6 MeV.

Uranium has a lower binding energy per nucleon (about 7.6 MeV) than its medium-sized fragments (about 8.5 MeV); the products are more tightly bound, have less mass, and the difference is released as energy.

Two photons of 0.511 MeV each, travelling in opposite directions.

Speed (special relativity): about −7 µs/day. Gravity (general relativity): about +45 µs/day. Net: about +38 µs/day.

Examples: particle accelerators (TRIUMF, Canadian Light Source), PET scanners, radiation-therapy linear accelerators, nuclear power, electron microscopes.

Benefit: reliable, low-emission electricity. Risk: long-lived radioactive waste (or accidents, or high costs).