Rocket velocity relative to decrease in mass due to fuel consumption

Question

A rocket of mass $m$ emits mass backwards at speed $u$ relative to the rocket at a constant rate $k$. Ignoring gravity and air resistance find the speed $v$ at time $t$ if at $t=0$ it has speed $v_0$ and mass $M_0= M + m_0$, where $m_0$ is the amount of fuel for burning. Your expression for $v$ need only be valid till the fuel runs out.

Answer 1

some suggestions, before this gets deleted or migrated to pastures new,

Using momentum, in the absence of any external forces, $$(m+\delta m)(v+\delta v)-\delta m(v+\delta v-u)-mv=0$$

$$\implies m\delta v+u\delta m=0$$

$$\implies m\frac{dv}{dm}=-u$$

Now solve this with your initial conditions, together with $$\frac {dm}{dt}=-k\implies m=m_0-kt$$