Velocity of Rocket in Outer Space

Theorem
Let $B$ be a rocket of mass $m$ be 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 velocity of the rocket's exhaust be constant at $\mathbf b$.

Then the velocity of $B$ at time $t$ is given by:
 * $\mathbf v \left({t}\right) = \mathbf v \left({0}\right) + \mathbf b \ln \dfrac {m \left({0}\right)} {m \left({t}\right)}$

where $\mathbf v \left({0}\right)$ and $m \left({0}\right)$ are the velocity and mass of $B$ at time $t = 0$.

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

Hence: