 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/section-b/7/e/solution

Equating the lander and return-signal positions gives

\frac{35}{37} t_{D} = 2 - (t_{D} - \frac{25}{12}), t_{D} = \frac{1813}{864} yr = 2.09838 yr .

This is measured from launch, so about 25.18 months pass on Earth-frame synchronized clocks. D occurs slightly before the lander reaches Earth at

t = 74/35 yr

The two events A and D lie on the inertial lander's own worldline, and their proper time difference is

Δ τ = \frac{t _{D}}{γ _{u}} = \frac{49}{72} yr = 0.68056 yr ≃ 8.17 months .

Equivalently, applying the Lorentz transformation with

Δ x = u Δ t

gives

Δ t^{'} = γ_{u} Δ t (1 - u^{2} / c^{2}) = Δ t / γ_{u}

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