= Constant mass-loss propulsion with quadratic drag
{title2=$m\dot u=QV-\alpha u^2$}
A body ejecting <mass> at constant rate $Q>0$ and relative exhaust <speed> $V>0$, against <quadratic drag> $\alpha u^2$, has remaining <mass> $m=m_0-Qt$. A short-interval <momentum> balance gives $m\dot u=QV-\alpha u^2$. From rest, <separation of variables> gives
$$
u(t)=\sqrt{\frac{QV}{\alpha}}\tanh\left(\sqrt{\frac{\alpha V}{Q}}\log\frac{m_0}{m_0-Qt}\right).
$$
For final <mass> $m_f>0$, this formula applies until $t=(m_0-m_f)/Q$. The zero-drag limit is the <rocket equation> $u_f=V\log(m_0/m_f)$. Two <dimensionless variables> are $\alpha V/Q$ and $m_0/m_f$.
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