Systems with Varying Mass: A Rocket — Question 3

Question 3

Constant thrust gives increasing acceleration

A rocket in force-free space starts with mass $1000\,\mathrm{kg}$ and expels propellant at constant $q=5\,\mathrm{kg/s}$ with relative exhaust speed $u=1800\,\mathrm{m/s}$. The burn lasts $40\,\mathrm s$.

1. Find thrust, initial acceleration, final acceleration and velocity gain.

2. Compare with using the initial acceleration for the whole burn.

Question 3 - Solution

Constant thrust gives increasing acceleration

(a) Derivation and calculation

Mass and thrust during the burn are $$m(t)=1000-5t,\qquad\boxed{T=uq=1800(5)=9000\ \mathrm N}.$$ $$\boxed{a_i=9000/1000=9.0},\qquad\boxed{a_f=9000/800=11.25\ \mathrm{m/s^2}}.$$ $$\boxed{\Delta v=1800\ln(1000/800)=401.66\ \mathrm{m/s}}.$$

(b) Result and interpretation

Holding acceleration at its initial value would give $$\Delta v_{\mathrm{est}}=9(40)=360\ \mathrm{m/s},$$ $$\boxed{\Delta v-\Delta v_{\mathrm{est}}=41.66\ \mathrm{m/s}}.$$ Because $a(t)=9000/(1000-5t)$ rises monotonically, the initial-acceleration estimate is a lower bound.

Physical check

The exact velocity gain lies between $9(40)=360$ and $11.25(40)=450$ m/s.