Motion of Body with Variable Mass

Theorem
Let $B$ be a body undergoing a force $\mathbf F$.

Let $B$ be travelling at a velocity $\mathbf v$ at time $t$.

Let mass travelling at a velocity $\mathbf v + \mathbf w$ be added to $B$ at a rate of $\dfrac {\mathrm d m} {\mathrm d t}$.

Let $m$ be the mass of $B$ at time $t$.

Then the equation of motion of $B$ is given by:
 * $\mathbf w \dfrac {\mathrm d m} {\mathrm d t} + \mathbf F = m \dfrac {\mathrm d \mathbf v} {\mathrm d t}$

Proof
From Newton's Second Law:
 * $\mathbf F = \dfrac{\mathrm d} {\mathrm d t} \left({m \mathbf v}\right)$

Then the added momentum being added to $B$ by the mass being added to it is given by:
 * $\left({\mathbf v + \mathbf w}\right) \dfrac {\mathrm d m} {\mathrm d t}$

Hence:
 * $\left({\mathbf v + \mathbf w}\right) \dfrac {\mathrm d m} {\mathrm d t} + \mathbf F = \dfrac {\mathrm d} {\mathrm d t} \left({m \mathbf v}\right)$