# 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 $\dfrac {\d m} {\d t}$.

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

Then the equation of motion of $B$ is given by:

$\mathbf w \dfrac {\d m} {\d t} + \mathbf F = m \dfrac {\d \mathbf v} {\d t}$

## Proof

$\mathbf F = \map {\dfrac \d {\d t} } {m \mathbf v}$

Then the added momentum being added to $B$ by the mass being added to it is given by:

$\paren {\mathbf v + \mathbf w} \dfrac {\d m} {\d t}$

Hence:

 $\ds \paren {\mathbf v + \mathbf w} \dfrac {\d m} {\d t} + \mathbf F$ $=$ $\ds \map {\dfrac \d {\d t} } {m \mathbf v}$ $\ds$ $=$ $\ds m \dfrac {\d \mathbf v} {\d t} + \mathbf v \dfrac {\d m} {\d t}$ $\ds \leadsto \ \$ $\ds \mathbf w \dfrac {\d m} {\d t} + \mathbf F$ $=$ $\ds m \dfrac {\d \mathbf v} {\d t}$

$\blacksquare$