Systems with Varying Mass: A Rocket — Question 8

Question 8

One finite ejection is not the differential rocket limit

An initially stationary $10\,\mathrm{kg}$ system consists of an $8\,\mathrm{kg}$ vehicle and a $2\,\mathrm{kg}$ package. It ejects the package backward in one event at $100\,\mathrm{m/s}$ relative to the final vehicle. External impulse is zero.

1. Find the vehicle velocity from a finite momentum balance.

2. Compare with continuously expelling the same mass at constant relative exhaust speed $100\,\mathrm{m/s}$. Explain the difference.

Question 8 - Solution

One finite ejection is not the differential rocket limit

(a) Derivation and calculation

Use the finite event momentum balance, with laboratory package speed $V-100$: $$8V+2(V-100)=0\Rightarrow10V=200.$$ $$\boxed{V=20\ \mathrm{m/s}},\qquad v_{\mathrm{package}}=-80\ \mathrm{m/s}.$$

(b) Result and interpretation

Continuous expulsion instead yields $$\boxed{\Delta v=100\ln(10/8)=22.31\ \mathrm{m/s}}.$$ Each infinitesimal parcel leaves a different instantaneous vehicle state. The differential law integrates those changing reference velocities; it does not represent a single package referenced to the final vehicle.

Physical check

The finite-event momenta $8(20)=160$ and $2(-80)=-160$ kg m/s cancel.