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

For $R=(3/4)cT$, part (ii) requires ${\gamma }_B{v}_B/c\ge 3/4$, soThe speed formula in part (i) can be written $v_B/c=({\mathcal{E}}^2-m_B^2{c}^4)/({\mathcal{E}}^2+m_B^2{c}^4)$. Requiring it to be at least $3/5$ gives $\mathcal{E}\ge 2m_B{c}^2$, hence ${\gamma }_A{m}_A\ge m_B$.

Since ${\gamma }_A\ge 1$ already, the exact minimum isIn 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.