Motion of Rocket in Outer Space/Proof 2

Proof
From Newton's Second Law:
 * $\mathbf F = \dfrac{\mathrm d} {\mathrm d t} \left({m \mathbf v}\right)$

At time $t + \Delta t$, let:
 * the mass of $B$ be $m + \Delta m$
 * the velocity of $B$ be $\mathbf v + \Delta \mathbf v$.

The fuel is being consumed, so the increase in mass of the fuel during time $\Delta t$ is $-\Delta m$.

Thus the exhaust products, therefore of mass $-\Delta m$, are expelled at a velocity $-\mathbf b$ relative to $B$.

Thus this material is actually moving at a velocity $\mathbf v - \mathbf b$.

By Conservation of Momentum, the total linear momentum is constant.

Thus: