= Equal-intake-fuel mass-loading rocket
{title2=$m\dot u=\mu(2u_0-u)$}
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.
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