 Equal-intake-fuel mass-loading rocket

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.