 Solution

The source expression $E=\frac{1}{2}\mu v^2-GMm/r$ is total mechanical energy in the centre-of-mass frame, not kinetic energy alone. Take $v_0$ as the relative speed at a large initial separation where the mutual potential is negligible. At closest approach, the radial relative velocity is zero, so conservation of angular momentum and conservation of energy giveUsing $Mm/\mu =M+m$ eliminates $v_c$:An impact occurs when $r_c\le R$, where $R=r_M+r_m$. Since the right side increases with positive $r_c$, the collision threshold isThe mutual escape velocity is the relative speed at contact required to reach infinite separation with zero residual speed. Setting total mechanical energy to zero givesThis includes target recoil. If the given initial speed were instead at a finite non-negligible separation, its potential energy would have to be retained; $v_0$ would then not equal the asymptotic speed used in the focusing formula.