= Dust-collecting rocket
{title2=$m\dot v=\alpha u-\beta v^2$}
A classical rocket ejecting fuel at rate $\alpha>0$ and relative exhaust <speed> $u>0$, while collecting stationary dust of <mass> $\beta\,dx$ per traveled distance, obeys
$$
\dot m=\beta v-\alpha,\qquad m\dot v=\alpha u-\beta v^2.
$$
These are <mass conservation> and <momentum conservation> for the rocket, incoming dust and outgoing gas. The dust-loading term supplies the <momentum> needed to accelerate incoming stationary material. Put $\lambda=\sqrt{\alpha/(\beta u)}$ and $V=\lambda u$. While the engines operate and $0\leq v<V$, an initially stationary rocket satisfies
$$
\frac m{m_0}=\left(\frac{V-v}{V}\right)^{(\lambda-1)/2}\left(\frac{V+v}{V}\right)^{-(\lambda+1)/2}.
$$
For $\lambda>1$, continuing to $V$ would drive total <mass> to zero. For $0<\lambda<1$, total <mass> instead grows without bound and $V$ is approached only after infinite time under indefinitely maintained thrust. In both cases a finite fuel supply runs out before that limiting <speed>. Total stored <mass> must not be confused with remaining fuel.
Back to article page
