Systems with Varying Mass: A Rocket — Question 7

Question 7

Discarding empty structure helps only subsequent propulsion

An ideal two-stage rocket in force-free space starts at $1000\,\mathrm{kg}$. The first burn expels $400\,\mathrm{kg}$, then $100\,\mathrm{kg}$ of empty structure is detached with no relative speed. The second burn expels $300\,\mathrm{kg}$, leaving $200\,\mathrm{kg}$. Both stages have $u=2000\,\mathrm{m/s}$.

1. Find total velocity gain.

2. Compare with retaining the empty structure through the second burn. Does jettison itself create an instantaneous speed increase?

Question 7 - Solution

Discarding empty structure helps only subsequent propulsion

(a) Derivation and calculation

Sum the two powered increments using their own mass endpoints: $$\Delta v_1=2000\ln(1000/600),\qquad\Delta v_2=2000\ln(500/200).$$ $$\boxed{\Delta v=2000\ln(25/6)=2854.23\ \mathrm{m/s}}.$$

(b) Result and interpretation

Retaining the empty structure gives $$\Delta v_{\mathrm{retain}}=2000[\ln(1000/600)+\ln(600/300)]=2407.95\ \mathrm{m/s}.$$ $$\boxed{\Delta v_{\mathrm{advantage}}=446.29\ \mathrm{m/s}}.$$ For equal-velocity detachment, $600v=500v+100v$: jettison creates no instantaneous speed change, but reduces the mass accelerated afterward.

Physical check

The unpowered mass ratio $600/500$ must not be inserted as a rocket thrust increment.