Motion of Rocket in Outer Space/Proof 1

Proof
From Motion of Body with Variable Mass:
 * $(1): \quad \mathbf w \dfrac {\d m} {\d t} + \mathbf F = m \dfrac {\d \mathbf v} {\d t}$

where:
 * $\mathbf F$ is the external force being applied
 * $\mathbf w$ is the velocity of the added mass relative to $B$.

In this scenario:
 * there is no external force and so $\mathbf F = \mathbf 0$
 * the velocity of the added mass relative to $B$ is $-\mathbf b$.

Thus $(1)$ becomes:
 * $-\mathbf b \dfrac {\d m} {\d t} = m \dfrac {\d \mathbf v} {\d t}$