Rocket equation with stationary mass accretion ($M\dot{v}=u\dot{m}-v\dot{D}$)

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, $v=(u\beta /\alpha )[1-(M/M_0{)}^{\alpha /(\beta -\alpha )}]$.

A Newtonian rocket collecting stationary matter at rate $\mu$ and combining it with the same rate of carried fuel ejects mass at rate $2\mu$. If relative exhaust speed is $u_0$, momentum conservation gives the title equation and mass conservation gives $\dot{m}=-\mu$. With constant $\mu$ and initial rest, $m=m_i-\mu t$ and $u=2u_0\mu t/m_i$. The limiting speed $2u_0$ would require zero remaining mass in this ideal solution; positive dry mass terminates it sooner. A constant collection rate per time is distinct from a constant collection cross-section through a uniform medium.

If stationary mass arrives at the same constant time rate as fuel is ejected, total rocket mass stays fixed. The rocket equation with stationary mass accretion is then $M_0\dot{v}+\alpha v=\alpha u$. For initial rest, speed approaches the relative exhaust speed exponentially with relaxation time $M_0/\alpha$. This is the regular equal-rate limit of the unequal-rate mass-power formula.