A rocket of mass $m(t)$ moving at speed $v(t)$ and ejecting fuel behind it at a constant speed $u$ relative to the rocket, is subject to an external force $F$. Considering a small time interval $\delta t$, derive the rocket equation

$$m \frac{d v}{d t}+u \frac{d m}{d t}=F$$

In deep space where $F=0$, how much faster does the rocket go if it burns half of its mass in fuel?