Acceleration of Rocket in Outer Space

Theorem
Let $B$ be a rocket travelling in outer space.

Let the velocity of $B$ at time $t$ be $\mathbf v$.

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

Let the exhaust velocity of $B$ be constant at $\mathbf b$.

Then the acceleration of $B$ at time $t$ is given by:
 * $\mathbf a = \dfrac 1 m \paren {- \mathbf b \dfrac {\d m} {\d t} }$

Proof
From Motion of Rocket in Outer Space, the equation of motion of $B$ is given by:
 * $(1): \quad m \dfrac {\d \mathbf v} {\d t} = -\mathbf b \dfrac {\d m} {\d t}$

By definition, the acceleration of $B$ is its rate of change of velocity:
 * $\mathbf a = \dfrac {\d \mathbf v} {\d t}$

The result follows by substituting $\mathbf a$ for $\dfrac {\d \mathbf v} {\d t}$ and dividing by $m$.