Special Relativity — Inertial Frames and Postulates

Q1 — Sample Band 6 response (8 marks), annotated

(a) Lorentz factor at $v = 0.98c$:

$\gamma = 1/\sqrt{1 - (0.98)^2} = 1/\sqrt{1 - 0.9604} = 1/\sqrt{0.0396} = 1/0.1990 \approx \mathbf{5.03}$ [1].

(b) Total relativistic energy:

$E = \gamma m_e c^2 = 5.03 \times 511\text{ keV} = 2570\text{ keV} \approx \mathbf{2.57\text{ MeV}}$ [1].

(c) Relativistic kinetic energy:

$E_k = (\gamma - 1)m_e c^2 = (5.03 - 1) \times 511 = 4.03 \times 511 = 2059\text{ keV} \approx \mathbf{2.06\text{ MeV}}$ [1].

(d) Comparison with classical formula:

Classical: $E_k^{\text{cl}} = \tfrac{1}{2}m_e v^2 = \tfrac{1}{2}m_e(0.98c)^2 = \tfrac{1}{2}(0.98)^2 m_e c^2 = 0.480 \times 511 = 245\text{ keV} = 0.245\text{ MeV}$ [1]. Ratio: $E_k^{\text{rel}}/E_k^{\text{cl}} = 2.06/0.245 \approx 8.4$. At $v = 0.98c$, the relativistic formula gives about 8.4 times more kinetic energy than the classical formula. The classical formula assumes mass is constant, ignoring the increase in effective mass (factor $\gamma$) at high speeds; it is not valid for $v$ comparable to $c$ [1].

(e) Proton at $v = 0.87c$:

$E = \gamma m_p c^2 = 2.03 \times 938\text{ MeV} = \mathbf{1904\text{ MeV}} \approx 1.90\text{ GeV}$ [1]. $E_k = (\gamma - 1)m_p c^2 = 1.03 \times 938 = \mathbf{966\text{ MeV}} \approx 0.97\text{ GeV}$ [1].

Marking criteria summary (8 marks): 1 = correct $\gamma$ (5.03); 1 = correct total energy in MeV; 1 = correct KE in MeV; 1 = correct classical KE (0.245 MeV); 1 = correct ratio and explanation of why classical formula fails; 1 = correct proton total energy; 1 = correct proton KE; 1 = consistent units and clear presentation throughout.

Q2 — Sample Band 6 response (7 marks), annotated

Hypothesis and variables: The hypothesis was that light travels through a medium (the luminiferous aether) and Earth moves through this medium at approximately 30 km/s (its orbital speed). IV: the orientation of the interferometer relative to Earth’s direction of motion (rotated through 90°). DV: the observed shift in interference fringes, which would indicate a difference in light travel time along the two arms [1].

Scientific significance of the null result: The expected fringe shift (0.4 fringe) was much larger than the instrumental uncertainty (0.01 fringe), so the null result was not due to poor resolution. Finding no shift meant there was no detectable difference in the speed of light along different directions — inconsistent with an aether wind. This was a major anomaly: either the aether does not exist, or light does not behave as a wave in a medium like sound [1]. It undermined classical mechanics’ assumption of an absolute rest frame [1].

How Einstein’s postulates resolve the null result: Einstein’s second postulate states that the speed of light in vacuum is $c$ for all observers, regardless of the motion of the source or observer. If $c$ is invariant, then rotating the interferometer should produce no fringe shift — exactly what Michelson and Morley observed. There is no need for an aether at all: light does not travel through a medium but propagates as an electromagnetic wave whose speed is determined by $\mu_0$ and $\varepsilon_0$, which are constants of nature [1].

Limitation and improvement: One limitation is that Earth might have been at rest relative to the aether at the time of the experiment — though unlikely, the aether could be dragged along by Earth (the aether drag hypothesis). Improvement: repeat the experiment at different times of year (when Earth is moving in different directions around the Sun) to rule out a coincidental rest condition [1]. (Many later experiments, including those by Miller, Joos, and modern laser interferometers, all yield null results at any time of year.)

Does the result prove Einstein’s postulates? No. Scientific experiments can provide supporting evidence for a hypothesis but cannot prove it universally [1]. The Michelson-Morley result is a necessary condition for Einstein’s second postulate (if $c$ varied, his postulate would be false), but it is not sufficient to prove it — the null result is also consistent with other hypotheses such as aether drag or the Lorentz-FitzGerald length contraction hypothesis (which preceded special relativity). The strength of Einstein’s postulates rests on many independent lines of evidence (time dilation experiments with muons and atomic clocks, energy-mass conversions in nuclear reactions, GPS corrections) rather than on the MM experiment alone [1].

Marking criteria summary (7 marks): 1 = correct hypothesis, IV and DV; 1 = explanation of why null result was significant (expected shift much larger than observed); 1 = links null result to undermining the aether model; 1 = explains how Einstein’s 2nd postulate predicts null result (c invariant, no aether needed); 1 = valid limitation (seasonal repetition, aether drag); 1 = states experiment does not prove postulates, only supports them; 1 = discusses broader evidence base for special relativity beyond MM alone.