Systems with Varying Mass: A Rocket — Question 10

Question 10

A dry-mass constraint limits the deliverable velocity change

A rocket has $100\,\mathrm{kg}$ of dry structure, $300\,\mathrm{kg}$ of propellant and payload mass $L\ge0$. In force-free straight-line operation its constant relative exhaust speed is $u=2000\,\mathrm{m/s}$. It must supply at least $1500\,\mathrm{m/s}$ of velocity gain.

1. Find the maximum payload mass.

2. Check whether a $200\,\mathrm{kg}$ payload meets the requirement, and explain why the ideal logarithm cannot diverge for this rocket.

Question 10 - Solution

A dry-mass constraint limits the deliverable velocity change

(a) Derivation and calculation

The full-burn capability must satisfy $$2000\ln\frac{400+L}{100+L}\geq1500.$$ With $R=e^{0.75}>1$, $$400+L\geq R(100+L)\Rightarrow\boxed{0\leq L\leq\frac{400-100R}{R-1}=168.58\ \mathrm{kg}}.$$

(b) Result and interpretation

For $L=200$ kg, $$\boxed{\Delta v=2000\ln(600/300)=1386.29\ \mathrm{m/s}<1500\ \mathrm{m/s}}.$$ The final mass obeys $m_f=100+L\geq100$ kg. In fact $$\Delta v\leq2000\ln4=2772.59\ \mathrm{m/s},$$ so the logarithm cannot diverge within the stated physical model.

Physical check

Adding payload decreases the mass ratio and therefore decreases the achievable velocity gain.