Exercises: Staging, Propellant Management, and Reusability

Work these with a calculator and $g_0 = 9.81\ \text{m/s}^2$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each one cold before you peek. For the "implement it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does. Unless told otherwise, use a structural coefficient $\varepsilon = m_s/(m_s+m_p)$ and the payload-ratio relations from §22.2.

Part A — Warm-ups: ceilings, ratios, and rates (⭐)

22.1 † A single stage has $v_e = 3.2\ \text{km/s}$ and structural coefficient $\varepsilon = 0.09$. What is its maximum possible delta-v (with zero payload)? Is it enough to reach orbit alone?

22.2 A stage flies with mass ratio $R = 4$ and $\varepsilon = 0.08$. What is its payload ratio $\pi = (1 - \varepsilon R)/[R(1-\varepsilon)]$? What fraction of its lift-off mass is useful load?

22.3 † Two identical stages must together deliver $\Delta v = 8\ \text{km/s}$. According to optimal staging, how much delta-v should each provide, and why?

22.4 A landing booster has a dry mass of $20\ \text{t}$; its single landing engine cannot throttle below $300\ \text{kN}$. Can it hover? Show the thrust-to-weight comparison and explain what this forces the landing to look like.

22.5 † Heat leaks into a liquid-oxygen tank at $Q = 50\ \text{W}$ (latent heat $h_{fg} \approx 213\ \text{kJ/kg}$). What is the boiloff rate in kg/day?

Part B — Optimal staging (⭐⭐)

22.6 † Two identical stages, $v_e = 3.3\ \text{km/s}$ and $\varepsilon = 0.07$, must deliver a $500\ \text{kg}$ payload with $\Delta v_{\text{total}} = 9\ \text{km/s}$. Using an even split, find each stage's mass ratio, the overall payload fraction, and the lift-off mass.

22.7 Starting from $m_0 = m_L + m_s + m_p$, $m_f = m_L + m_s$, the definitions $\pi = m_L/m_0$ and $\varepsilon = m_s/(m_s + m_p)$, derive the relation $R = m_0/m_f = 1/[\varepsilon + \pi(1-\varepsilon)]$.

22.8 † A mission needs $\Delta v_{\text{total}} = 10\ \text{km/s}$ with $v_e = 3.5\ \text{km/s}$ and $\varepsilon = 0.08$ per stage. Compute the overall payload fraction for $N = 2$ and $N = 3$ equal stages. By how much does the third stage improve things, and what does it cost that the numbers don't show?

22.9 † For unequal stages the optimum obeys $R_i = (v_{e,i} - \mu)/(v_{e,i}\,\varepsilon_i)$ for a single constant $\mu$. Explain physically why a stage with higher $v_e$ ends up with a larger $R_i$ and therefore does more of the delta-v.

Part C — Implement it in Python (⭐⭐)

Write each function, hand-trace it, and record the result in an # Expected output: comment. Do not run it.

22.10 † Write payload_fraction(dv_total, ve, eps, n) for $n$ identical stages splitting the delta-v equally (return nan if a stage would exceed its ceiling $1/\varepsilon$). Trace payload_fraction(9000, 3300, 0.07, 2).

22.11 Write stage_mass(m_above, dv, ve, eps) using $m_{\text{stage}} = m_{\text{above}}(R-1)/(1-\varepsilon R)$ with $R = e^{dv/v_e}$. Trace stage_mass(1000, 4500, 3400, 0.08).

22.12 † Write can_hover(thrust_min, dry_mass) returning True only if the minimum thrust does not exceed the booster's weight. Trace can_hover(338000, 25600).

Part D — Find the error (⭐⭐)

22.13 † A commentator writes: "The N1 had thirty engines, so it had superb engine-out capability — losing one of thirty is nothing." Given that all four N1 flights failed, what is wrong with this reasoning?

22.14 A business plan claims: "We recover the first stage, which is most of the rocket, so our cost per flight will drop to essentially the price of propellant." Identify the flaw using the cost model $C_{\text{reuse}}(N) = M/N + R + F$.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

22.15 † Design a two-stage launcher to put $m_L = 2{,}000\ \text{kg}$ into LEO ($\Delta v_{\text{total}} = 9.4\ \text{km/s}$), both stages with $v_e = 3.3\ \text{km/s}$ and $\varepsilon = 0.08$. Use an even split, size each stage top-down with $m_{\text{stage}} = m_{\text{above}}(R-1)/(1-\varepsilon R)$, and report the lift-off mass. Check that each $R < 1/\varepsilon$.

22.16 (Mission project) Turn your mission's delta-v budget (begun in Chapter 3) into a staging plan: number of stages, the split, and — top-down — each stage's mass, ending with lift-off mass. Save it to your MDR.

22.17 † Take the first stage you sized in 22.15 and make it reusable: recovery hardware (legs, grid fins) and residual landing propellant raise its effective structural coefficient from $\varepsilon = 0.08$ to $\varepsilon = 0.10$. Recompute that stage's mass ratio ceiling $1/\varepsilon$ and comment on how the change eats into payload. (Qualitative reasoning plus the two ceiling numbers is enough.)

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

22.18 Why can't you just build a single-stage-to-orbit rocket with ordinary chemical engines? Frame your answer entirely in terms of the ceiling $\Delta v_{\max} = v_e\ln(1/\varepsilon)$ and the delta-v to orbit.

22.19 † Why can't you just bring a booster down on parachutes instead of landing it propulsively? Give three physical or operational reasons, and name one thing propulsive landing needs that parachutes don't.

22.20 Estimate the payload "tax" of return-to-launch-site reuse. If a booster would normally stage at $\Delta v \approx 2.4\ \text{km/s}$ but must instead reserve propellant for boostback + entry + landing worth roughly $2\ \text{km/s}$ of extra delta-v it will not spend on the payload, argue qualitatively (no exact number needed) why RTLS reuse costs more payload than a downrange droneship landing.

Part G — Interleaved & synthesis (Chapters 3, 16, 17) (⭐⭐/⭐⭐⭐)

22.21 † (Ch. 3) A stage has $m_0 = 500\ \text{t}$, $m_f = 150\ \text{t}$, and $v_e = 2.9\ \text{km/s}$. What delta-v does it deliver? Does the number of stages appear anywhere in this single-stage calculation?

22.22 (Ch. 16) An engine expels $\dot m = 300\ \text{kg/s}$ at $v_e = 3{,}000\ \text{m/s}$. What is its (momentum) thrust? If a first stage needs nine such engines to lift off, estimate the stack's lift-off weight limit for $T/W = 1.3$.

22.23 † (Ch. 17) Propulsive landing requires an engine you can restart and deeply throttle. Which class of engine — a liquid bipropellant engine or a solid motor — offers these, and why does the other rule itself out of landing?

22.24 (Ch. 3, synthesis) Two stages each deliver $4.5\ \text{km/s}$. What is the total, and why do the delta-vs add rather than combining through a single mass ratio? Tie your answer to what physically happens at staging.

22.25 † (Ch. 17, synthesis) A cryogenic upper stage must coast for six hours before its main burn. Name the two propellant-management problems it faces during that coast (one that threatens the restart, one that steadily loses propellant) and the fix for each.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: correct top-down set-up of each stage's $m_{\text{above}}$, an explicit check that $R < 1/\varepsilon$, units throughout, and a sanity check on the final lift-off mass.