Exercises: SpaceX and the Reusability Revolution

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 write the result in an # Expected output: comment, exactly as the chapter does. Every performance and cost figure here is approximate and version-dependent (Tier 2), and several are round illustrative numbers (Tier 3) — treat them as planning intuition, not specifications.

Part A — Warm-ups: read the vehicles (⭐)

38.1 † Falcon 9's first stage has nine Merlin engines each producing about $845\ \text{kN}$ at sea level. What is the total sea-level thrust of the first stage, in MN?

38.2 A recovered Falcon 9 lifts about $17\ \text{t}$ to LEO; expended, about $22.8\ \text{t}$. What percentage of its payload does the vehicle give up to recover the booster?

38.3 † A Falcon 9 lists at about $\$62\ \text{M}$ and lifts up to $22.8\ \text{t}$ to LEO. Compute its cost per kilogram to LEO. How does it compare to the Shuttle's ~$\$54{,}500/\text{kg}$?

38.4 Super Heavy clusters 33 Raptor engines at about $2{,}300\ \text{kN}$ each (sea level). What is the cluster's total liftoff thrust, in MN?

38.5 † Name the propellant and the engine cycle of (a) Merlin, (b) Raptor, (c) the RS-25. Which one is full-flow staged combustion, and which was the first of its cycle ever to fly?

38.6 Raptor 2 runs at a chamber pressure of about $300\ \text{bar}$; Merlin at about $97\ \text{bar}$. By what factor is Raptor's chamber pressure higher, and (from Chapter 17) name one thing higher chamber pressure buys.

Part B — The cost of reuse: calculate it (⭐⭐)

38.7 † A first stage carries $400\ \text{t}$ of propellant and masses $25\ \text{t}$ empty. It lifts an upper stack (fueled second stage + payload) of $120\ \text{t}$, with $v_e = 2{,}850\ \text{m/s}$. (a) Compute its ascent delta-v when all propellant is burned. (b) Now reserve $12\%$ of the propellant for landing; recompute the ascent delta-v. (c) How much delta-v did recovery cost?

38.8 Using the reuse cost model $C_{\text{reuse}}(N) = M/N + R + F$ with $M = \$30\ \text{M}$, $R = \$1\ \text{M}$, $F = \$18\ \text{M}$: compute the cost per flight at $N = 1, 2, 5, 10$, and state the floor as $N \to \infty$.

38.9 † For the numbers in 38.8, an expendable version of the same vehicle costs $C_{\text{expend}} = M + F = \$48\ \text{M}$ per flight. At what flight number $N$ does reuse first become cheaper than expendable? (Solve $M/N + R + F < M + F$ for $N$.)

38.10 Subcooling raises a stage's propellant load from $395\ \text{t}$ to $427\ \text{t}$ in the same tanks, with dry+upper burnout mass fixed at $137\ \text{t}$ and $v_e = 2{,}840\ \text{m/s}$. How much ascent delta-v does densification add compared with the un-densified $395\ \text{t}$ load?

38.11 † A fully reusable upper stage has dry mass $120\ \text{t}$, carries $1{,}200\ \text{t}$ of propellant, and burns methalox at $v_e = 3{,}700\ \text{m/s}$. Fully fueled in LEO with a $100\ \text{t}$ payload, what delta-v can it produce? Is that enough for trans-Mars injection (~$3.6\ \text{km/s}$ from LEO)?

Part C — Implement it in Python (⭐⭐)

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

38.12 † Write payload_penalty(dv_expend, dv_reuse) returning the delta-v lost to recovery, dv_expend - dv_reuse. Trace it for payload_penalty(3862, 3140).

38.13 Write cost_per_kg(cost_musd, payload_kg) returning dollars per kilogram, cost_musd * 1e6 / payload_kg. Trace it for cost_per_kg(62, 22800) and cost_per_kg(1500, 27500).

38.14 † Write reuse_cost(M, R, F, N) returning M/N + R + F, and a loop that prints the cost for N in (1, 2, 5, 10) with M=30, R=1, F=18. Trace the four outputs.

Part D — Find the error (⭐⭐)

38.15 † A blog post claims: "Because Falcon 9 recovers its first stage — most of the rocket's cost — the marginal cost of a launch is basically the price of propellant, a few hundred thousand dollars. Everything above that is pure profit." Identify the flaw using the reuse cost model, and name the dominant term the writer ignored.

38.16 A student computes Starship's beyond-LEO capability by fueling it on the ground, filling all $1{,}200\ \text{t}$ of tanks, and getting a huge delta-v — concluding no orbital refueling is needed. What did they get wrong about the state of the vehicle when it reaches LEO?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

38.17 † You are choosing a recovery mode for a Falcon 9-class booster. A droneship landing reserves $10\%$ of first-stage propellant; RTLS reserves $18\%$ (the extra pays for the boostback burn). Using a first stage with $400\ \text{t}$ propellant, $25\ \text{t}$ dry, upper stack $120\ \text{t}$, and $v_e = 2{,}850\ \text{m/s}$, compute the ascent delta-v for each mode. Which loses more payload, and why would you still sometimes choose it?

38.18 (Mission project) Add the "launch economics" note to your Mission Design Review (see the Mission Design Checkpoint). State your track's payload, your expected flight rate, whether reuse fits, and an estimated cost per kilogram. Justify in three sentences.

38.19 † Design a tanker campaign. A Mars-bound Starship needs $1{,}200\ \text{t}$ of propellant in LEO, and each tanker delivers $120\ \text{t}$ per flight. (a) How many tanker flights are needed? (b) If each tanker launch costs $\$25\ \text{M}$, what is the propellant-delivery cost of one Mars mission? (c) Why is such an architecture only affordable with reusable launch?

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

38.20 † Why can't you just make Falcon 9's second stage reusable too, and get to the propellant-cost floor immediately? Frame your answer in terms of the mass-ratio penalty (§38.3, §38.5) and where that penalty hurts most on an upper stage.

38.21 Estimate, to an order of magnitude, how many Falcon 9 launches it would take to deploy a $4{,}000$-satellite constellation if each launch carries $\sim 50$ satellites. Then argue in two sentences why such a constellation was economically implausible at Shuttle-era launch prices. (Ties to Chapter 33.)

38.22 † Why can't you just build Starship out of lightweight carbon fiber like other modern rockets, since mass is the enemy? Give the two reasons steel wins for a reusable, re-entering vehicle (§38.5).

38.23 Why can't you just land a solid-rocket booster propulsively the way Falcon 9 lands? Answer from the engine's capabilities (§38.3 and Chapter 17).

Part G — Interleaved & synthesis (Chapters 3, 17, 22, 37)

38.24 † (Ch. 3) A recovered booster produces $722\ \text{m/s}$ less ascent delta-v than expended. Using $\Delta v = v_e\ln(m_0/m_f)$ with $v_e = 2{,}840\ \text{m/s}$, by what factor did the reserved landing propellant reduce the first stage's mass ratio? (Hint: $\Delta(\ln R) = \Delta v/v_e$.)

38.25 (Ch. 17) Explain, in terms of the engine cycle, why Raptor can run at $\sim 300\ \text{bar}$ and still be built for reuse, while a gas-generator engine like Merlin runs near $\sim 100\ \text{bar}$. Which cycle feature "wastes" propellant, and which recovers it?

38.26 † (Ch. 22, Ch. 37) Both the Space Shuttle and Falcon 9 were "reusable." In two or three sentences, contrast why reuse failed to make the Shuttle cheap but succeeded for Falcon 9, using the cost model $C_{\text{reuse}} = M/N + R + F$ (comment on $R$, $F$, and $N$ for each).

38.27 (Synthesis) Trace one anchor thread through the book in your own words: pick either Falcon 9 or Starship and name, in order, at least four earlier chapters whose physics or engineering this chapter braids together, saying what each contributed.

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 setup of $m_0$ and $m_f$ (with reserved propellant placed in the final mass), explicit units, labeling every figure's tier, and a sanity check on the final number.