= Exhaust collection by a second rocket
{title2=$v_{Bf}=-m_Av_{Af}/(m_B+M)$}
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, $(m_B+M)v_{Bf}=-m_Av_{Af}$. 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.
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