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ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/8/b/ii/solution

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Past exam of the short options course of the University of Oxford / 2021 / 8 / b / ii / Solution by

Codex  0 2026-10-25

v / c = tanh θ

and

v^{'} / c = tanh θ^{'}

. Using the addition identity for the hyperbolic tangent and the rapidity sum from (i),

\frac{v ^{''}}{c} = tanh (θ + θ^{'}) = \frac{v / c + v ^{'} / c}{1 + v v ^{'} / c ^{2}} .

(1)

Thus the velocity-addition formula is

v^{''} = \frac{v + v ^{'}}{1 + v v ^{'} / c ^{2}} = \frac{c ^{2} ( v + v ^{'} )}{c ^{2} + v v ^{'}} .

(2)

The denominator is positive for the permitted subluminal signed velocities. For the nonnegative speeds in (iii), the composite is also nonnegative and remains below

c

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