Systems with Varying Mass: A Rocket — Question 6

Question 6

The ideal speed gain can ignore the throttle history

Two identical rockets in force-free space burn from $m_i=600$ to $m_f=400\,\mathrm{kg}$ with fixed exhaust direction and constant relative exhaust speed $u=1500\,\mathrm{m/s}$. One burns steadily; the other varies its positive mass flow.

1. Compare their velocity gains.

2. Explain which aspects of their motion need not agree, and why adding gravity changes the comparison.

Question 6 - Solution

The ideal speed gain can ignore the throttle history

(a) Derivation and calculation

For either positive throttle history, $dm/dt=-q(t)$ and $$dv=\frac{uq(t)}{m(t)}dt=-u\frac{dm}{m}.$$ $$\boxed{\Delta v=1500\int_{400}^{600}\frac{dm}{m}=1500\ln1.5=608.20\ \mathrm{m/s}}.$$ Thus both obtain the same velocity gain in force-free space.

(b) Result and interpretation

Their instantaneous accelerations and displacements depend on the history: $$a(t)=\frac{uq(t)}{m(t)},\qquad\Delta x=\int_0^{t_b}v(t)\,dt.$$ In vertical uniform gravity, $$\Delta v=u\ln(m_i/m_f)-gt_b.$$ Different burn durations then generally produce different final velocities.

Physical check

Endpoint independence requires constant relative exhaust speed, fixed direction, and zero external force.