 Solution

Let $dm$ be the gas mass ejected in a short interval, and let $dD$ be the stationary dust mass acquired. The rocket mass change is $dM=dD-dm$. Before the interval the momentum of the rocket and this dust is $Mv$, because the incoming dust is at rest. Afterwards it isto first order; changing the gas ejection speed by $dv$ affects only second-order terms. Momentum conservation, with no external force, therefore givesDividing by the time interval produces the rocket equation with stationary mass accretion:The incoming material's zero initial momentum is essential: accelerating it to the rocket's speed consumes part of the thrust.