 Solution

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

When both speeds are much smaller than

c

v^{''} = (v + v^{'}) [1 - \frac{v v ^{'}}{c ^{2}} + O (\frac{v ^{2} v ^{'2}}{c ^{4}})], v^{''} ≃ v + v^{'} .

This is the Galilean transformation limit. For

0 \leq β = v / c < 1

and

0 \leq β^{'} = v^{'} / c < 1

1 - \frac{v ^{''}}{c} = \frac{( 1 - β ) ( 1 - β ^{'} )}{1 + β β ^{'}} > 0.

If both

v, v^{'} \to c

from below, the right side tends to zero, so

v^{''} \to c from below .

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