Quiz: Staging, Propellant Management, and Reusability
Twenty questions to check your grasp of Chapter 22. Aim for 16/20 (80%) before moving on. Answers, with one-line rationales, are in the key at the end; a topic map follows so you know what to review.
Multiple choice
1. A single stage's best possible delta-v (with zero payload) is: - (a) $v_e / \varepsilon$ - (b) $v_e \ln(1/\varepsilon)$ - (c) $v_e (1 - \varepsilon)$ - (d) $v_e \ln(\varepsilon)$
2. Staging increases total delta-v mainly by: - (a) adding more propellant to the same tanks - (b) increasing each engine's exhaust velocity - (c) discarding empty structure so later engines need not accelerate it - (d) reducing the delta-v the mission requires
3. For two identical stages producing a fixed total delta-v, lift-off mass is minimized when: - (a) the first stage does most of the delta-v - (b) the second stage does most of the delta-v - (c) the delta-v is split equally between them - (d) the split does not affect lift-off mass
4. A stage designed to need a mass ratio $R > 1/\varepsilon$ is: - (a) extremely efficient - (b) physically impossible (its payload fraction is negative) - (c) optimal for the first stage - (d) only possible with hydrogen
5. Parallel (strap-on) staging is chosen primarily to: - (a) improve the vehicle's specific impulse - (b) add lift-off thrust when the core alone cannot lift the fueled stack - (c) eliminate the need for a second stage - (d) reduce propellant slosh
6. An ullage burn just before an engine restart in space is performed to: - (a) cool the engine bell - (b) settle liquid propellant over the tank outlet so the pumps draw gas-free - (c) increase the exhaust velocity - (d) dump excess propellant
7. Anti-slosh baffles are installed in tanks to counter: - (a) longitudinal thrust oscillations - (b) cryogenic boiloff - (c) lateral sloshing of liquid that shifts the center of mass and disturbs control - (d) cavitation at the pump inlet
8. POGO oscillation is best described as: - (a) a lateral bending of the airframe - (b) a longitudinal feedback loop among structure, feed lines, and combustion - (c) the boiling of cryogenic propellant - (d) a failure of the guidance computer
9. The boiloff mass rate of a cryogenic tank with heat leak $Q$ and latent heat $h_{fg}$ is: - (a) $Q\,h_{fg}$ - (b) $h_{fg}/Q$ - (c) $Q/h_{fg}$ - (d) $Q + h_{fg}$
10. Losing an engine is hardest to survive at lift-off because: - (a) the guidance computer is still warming up - (b) the vehicle is fully fueled and heaviest, so $T/W$ is lowest and least forgiving - (c) the atmosphere is thinnest there - (d) the engines are coldest there
11. Keeping a propellant tank pressurized above the propellant's vapor pressure prevents: - (a) POGO - (b) slosh - (c) pump cavitation (loss of net positive suction head) - (d) boiloff
12. Autogenous pressurization (as on Starship) works by: - (a) carrying large helium bottles - (b) vaporizing the vehicle's own propellants and feeding that gas back into the tanks - (c) using solid gas generators - (d) venting the tanks to the atmosphere
13. A Falcon 9 first stage cannot hover during landing because: - (a) its engines cannot relight - (b) even one engine at minimum throttle out-thrusts the nearly empty booster's weight ($T/W_{\min}>1$) - (c) it has no landing legs - (d) the grid fins produce too much drag
14. The entry burn is performed high in the atmosphere in order to: - (a) save propellant for the boostback burn - (b) slow the booster before the air becomes dense, cutting peak heating and dynamic pressure - (c) deploy the landing legs earlier - (d) increase downrange distance
15. In the reuse cost model $C(N) = M/N + R + F$, the cost per flight bottoms out at a floor equal to: - (a) zero (the cost of propellant) - (b) $M$, the manufacturing cost - (c) $R + F$, the refurbishment plus fixed/expended cost - (d) $M/N$ only
16. Grid fins on a returning booster provide: - (a) additional thrust - (b) aerodynamic steering during the descent - (c) thermal protection for the engines - (d) propellant settling
True / False (justify in one sentence)
17. A rocket that cannot reach orbit in one stage can be made to reach orbit simply by scaling it up.
18. A rocket with more engines automatically has better engine-out capability.
19. Reusing the first stage makes even the very first flight cheaper than an expendable one.
Short answer
20. In one or two sentences, explain why the delta-vs of successive stages add ($\Delta v_{\text{total}} = \sum_i v_{e,i}\ln R_i$) rather than combining into a single mass ratio, and connect your answer to what physically happens at staging.
Answer Key
| # | Ans | Rationale |
|---|---|---|
| 1 | b | Max mass ratio is $1/\varepsilon$ (zero payload), so max delta-v is $v_e\ln(1/\varepsilon)$. |
| 2 | c | Staging drops dead structure; later engines start with a fresh, favorable mass ratio. |
| 3 | c | For identical stages, a fixed total fixes the product of mass ratios; AM–GM makes the payload fraction largest when they are equal. |
| 4 | b | $\pi = (1-\varepsilon R)/[R(1-\varepsilon)] < 0$ when $R > 1/\varepsilon$ — no room for payload or even all the structure. |
| 5 | b | Boosters add thrust when the core's $T/W$ at lift-off would otherwise be below one. |
| 6 | b | With no gravity to settle it, propellant floats off the outlet; a small forward burn settles it so pumps draw liquid. |
| 7 | c | Baffles damp lateral slosh that would move the center of mass and couple into control. |
| 8 | b | POGO is a longitudinal (axial) structure–feedline–thrust loop; cured by feed-line accumulators. |
| 9 | c | $\dot m = Q/h_{fg}$: the heat leak divided by the latent heat of vaporization. |
| 10 | b | Fully fueled, the vehicle is heaviest, so $T/W$ is lowest and an engine loss is least forgiving. |
| 11 | c | Adequate inlet (ullage) pressure keeps the liquid from flashing to vapor at the pump — prevents cavitation. |
| 12 | b | Autogenous pressurization taps and vaporizes the vehicle's own propellants, eliminating helium. |
| 13 | b | Minimum single-engine thrust exceeds the empty booster's weight, so it cannot balance gravity — hence the hoverslam. |
| 14 | b | Slowing before the dense air keeps peak heating and dynamic pressure survivable without a heavy heat shield. |
| 15 | c | As $N\to\infty$, $M/N\to0$ and cost approaches $R+F$ — the parts you still expend and the ops you still pay. |
| 16 | b | Grid fins are lattice aerodynamic control surfaces that steer the falling booster to its target. |
| 17 | False | The ceiling $v_e\ln(1/\varepsilon)$ depends only on mass fractions; scaling up leaves $\varepsilon$ and the ceiling unchanged. |
| 18 | False | More engines means more chances one fails; capability requires isolation, fast detection, and adaptive guidance (the N1 had 30 engines and no such management, and failed every flight). |
| 19 | False | The first flight of a reusable design costs more — you paid for recovery hardware and gave up payload; the savings come only from reusing it. |
| 20 | — | Each stage's delta-v comes from its own mass ratio $v_e\ln R_i$; at staging the spent structure is left behind, so the next stage's mass ratio is computed on a lighter vehicle, and the independent contributions simply sum. |
Topics to review by question
- Q1–4, 20: §22.1 (why stage, the ceiling, stage delta-vs add) and §22.2 (payload ratios).
- Q3: §22.2 (optimal split; the equal-stage proof).
- Q5: §22.2 (serial vs. parallel).
- Q6–9: §22.3 (ullage, slosh, POGO, boiloff).
- Q10–12: §22.4 (engine-out, tank pressurization, autogenous pressurization).
- Q13, 14, 16: §22.5 (propulsive landing: hoverslam, entry burn, grid fins).
- Q15, 19: §22.6 (reuse economics and the cost floor).
- Q17, 18: §22.1 (scaling and the ceiling) and §22.4 (engine-out is designed, not automatic).