# Motion of Body with Constant Mass

## Theorem

Let $B$ be a body with constant mass $m$ undergoing a force $\mathbf F$.

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

$\mathbf F = m \mathbf a$

where $\mathbf a$ is the acceleration of $B$.

## Proof

 $\ds \mathbf F$ $=$ $\ds \map {\dfrac \d {\d t} } {m \mathbf v}$ Newton's Second Law of Motion $\ds$ $=$ $\ds m \dfrac {\d \mathbf v} {\d t} + \mathbf v \dfrac {\d m} {\d t}$ $\ds$ $=$ $\ds m \dfrac {\d \mathbf v} {\d t} + 0$ Derivative of Constant $\ds$ $=$ $\ds m \mathbf a$ Definition of Acceleration

$\blacksquare$