Ballistic exhaust interception condition ($v_A-u_0<v_B$)

For exhaust moving freely from a rocket A toward a collector B behind it along a common axis, the exhaust must overtake B in the rearward direction. At negligible separation and flight time this gives the displayed condition. In the ideal simultaneous-collection solution, write $a=m_A+M-q$, $b=m_B+q$ and $v_A=u_0\ln [(m_A+M)/a]$, $v_B=-a{v}_A/b$. The criterion becomes $[(a+b)/b]\ln [(m_A+M)/a]<1$. A formal endpoint obtained by assuming all fuel is collected need not satisfy this interception test throughout a large burn.