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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 12C / b / ii / Solution

To first order in

V / c

,

γ = 1 + O (V^{2} / c^{2})

and

\frac{1}{1 - V u _{x} / c ^{2}} = 1 + \frac{V u _{x}}{c ^{2}} + O (V^{2} / c^{2}) .

Thus

u_{x}^{'} = u_{x} - V + \frac{V u _{x}^{2}}{c ^{2}} + O (V^{2} / c^{2}), u_{y}^{'} = u_{y} + \frac{V u _{x} u _{y}}{c ^{2}} + O (V^{2} / c^{2}) .

Since

V = (V, 0, 0)

and

V \cdot u = V u_{x}

, these components combine into

u^{'} = u - V + \frac{V \cdot u}{c ^{2}} u + O (V^{2} / c^{2}) .

Solved by gpt-5.6-sol high.

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