 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/8/b/iii/solution

When

0 \leq v, v^{'} ≪ c

, expand the denominator:

v^{''} = v + v^{'} - \frac{v v ^{'} ( v + v ^{'} )}{c ^{2}} + O (\frac{v ^{2} v ^{'2} ( v + v ^{'} )}{c ^{4}}) .

The leading term is ordinary Galilean velocity addition. When both

v, v^{'} \to c

from below, the exact formula instead gives

v^{''} \to c

, rather than

2 c

. Indeed, writing

a = v / c

b = v^{'} / c

, we have

1 - v^{''} / c = (1 - a) (1 - b) / (1 + ab) > 0

whenever

a, b < 1

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