 Exhaust collection by a second rocket

Two initially stationary rockets exchange a fuel load $M$: A ejects it with constant rearward relative speed $u_0$ and B captures all of it. A has final velocity $v_{Af}=u_0\ln [(m_A+M)/m_A]$. Since final momentum is zero, $(m_B+M)v_{Bf}=-m_A{v}_{Af}$. No elastic-capture assumption is needed. With negligible flight time, transferred mass $q$ gives the same balance with instantaneous masses $m_A+M-q$ and $m_B+q$; their opposite speeds are equal when those masses agree. Final complete collection is a condition on the model, not a guarantee that every ballistic exhaust trajectory intercepts B.