= Rocket equation with stationary mass accretion
{title2=$M\dot v=u\dot m-v\dot D$}
For a <rocket equation> with exhaust mass rate $\dot m$ and incoming stationary mass rate $\dot D$, mass balance is $\dot M=\dot D-\dot m$. A short-interval <momentum conservation> calculation gives $M\dot v=u\dot m-v\dot D$. Constant time rates $\dot D=\alpha$, $\dot m=\beta$ give $M\dot v+\alpha v=u\beta$, unlike a constant accretion rate per distance, whose incoming mass rate is proportional to speed. For unequal rates and initial rest, $v=(u\beta/\alpha)[1-(M/M_0)^{\alpha/(\beta-\alpha)}]$.
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