Systems with Varying Mass: A Rocket — Question 1

Question 1

Deriving thrust from a short momentum balance

An ideal rocket moves along $+x$. At an instant its mass is $m$ and velocity is $v$. In time $dt$ it expels positive mass $d\mu$ backward at speed $u$ relative to the rocket. External force is zero; ignore terms of second order.

1. Derive the rocket acceleration and thrust in terms of $q=d\mu/dt>0$.

2. Explain why setting $d(mv)/dt=0$ for the rocket alone fails even though the full system is isolated.

Question 1 - Solution

Deriving thrust from a short momentum balance

(a) Derivation and calculation

Follow the fixed collection consisting of the rocket and the newly expelled mass. To first order, $$mv=(m-d\mu)(v+dv)+d\mu(v-u).$$ $$0=m\,dv-u\,d\mu+O(d\mu\,dv)\Rightarrow\boxed{m\dot v=uq},\qquad\boxed{T=uq},\quad q=\frac{d\mu}{dt}>0.$$

(b) Result and interpretation

The remaining rocket has $\dot m=-q$, hence $$\frac{d(mv)}{dt}=m\dot v+v\dot m=uq-vq=\boxed{q(u-v)}.$$ Its momentum is not generally constant because exhaust crosses its boundary. Conservation applies to the fixed isolated collection of rocket plus all exhaust.

Physical check

Thrust points forward even when the exhaust still moves forward in the laboratory.