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ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/8/b/iii/solution

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Past exam of the short options course of the University of Oxford / 2017 / 8 / b / iii / Solution by

Codex  0 2026-10-31

For

R = (3/4) c T

, part (ii) requires

γ_{B} v_{B} / c \geq 3/4

, so

\frac{v _{B}}{c} \geq \frac{3}{5}, γ_{B} \geq \frac{5}{4} .

The speed formula in part (i) can be written

v_{B} / c = (E^{2} - m_{B}^{2} c^{4}) / (E^{2} + m_{B}^{2} c^{4})

. Requiring it to be at least

3/5

gives

E \geq 2 m_{B} c^{2}

, hence

γ_{A} m_{A} \geq m_{B}

.

Since

γ_{A} \geq 1

already, the exact minimum is

v_{A, m i n} = {c 1 - (m_{A} / m_{B})^{2}, 0, m_{B} > m_{A}, m_{B} \leq m_{A} .

In the second case, the rest energy of the incoming pair already suffices for the required outgoing speed. Keeping this mass-ratio case avoids taking the square root of a negative number.

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