# Motion of Body with Variable Mass

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## 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

From Newton's Second Law of Motion:

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

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:

\(\ds \paren {\mathbf v + \mathbf w} \dfrac {\d m} {\d t} + \mathbf F\) | \(=\) | \(\ds \map {\dfrac \d {\d t} } {m \mathbf v}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds m \dfrac {\d \mathbf v} {\d t} + \mathbf v \dfrac {\d m} {\d t}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \mathbf w \dfrac {\d m} {\d t} + \mathbf F\) | \(=\) | \(\ds m \dfrac {\d \mathbf v} {\d t}\) |

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): Miscellaneous Problems for Chapter $2$