Exercises: The Space Shuttle

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, and the chapter's Tier-2 figures unless a problem gives its own. This is a systems-and-history chapter, so the problems mix real calculations (cost, thrust, energy, reliability) with conceptual and organizational analysis — both are engineering. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. 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.

Part A — Warm-ups: the promise, the engine, the stack (⭐)

37.1 † The Shuttle program's lifetime cost is estimated at about \$209 billion (Tier 2) across 135 flights, each carrying up to $27{,}500\ \text{kg}$ to low orbit. Compute the average cost per flight and the average cost per kilogram to LEO. Compare the cost per kilogram with the program's original promise of roughly \$220/kg, and state the factor by which it missed.

37.2 The SSME had a vacuum specific impulse of about $452\ \text{s}$. Compute its effective exhaust velocity $v_e = I_{sp}\,g_0$ in km/s, and say where that sits relative to the "best chemical $v_e \approx 4.5\ \text{km/s}$" figure from Chapter 3.

37.3 † At liftoff, two SRBs produced about $12.5\ \text{MN}$ of thrust each and three SSMEs about $1.8\ \text{MN}$ each (sea level). Compute the fraction of total liftoff thrust supplied by the solid boosters. What does your answer say about which part of the propulsion the crew could and could not control?

37.4 Using the total liftoff thrust from 37.3 and a stack liftoff mass of about $2{,}030\ \text{tonnes}$, compute the liftoff thrust-to-weight ratio $T/W$ (use $g_0 = 9.81\ \text{m/s}^2$). Is it in the healthy range for a launch vehicle (Chapter 16)?

37.5 † Of 135 flights, 2 ended in loss of crew and vehicle. Express the demonstrated catastrophic-failure rate as a percentage and as a "1 in $N$" figure. Compare it with the working engineers' pre-Challenger estimate (~1 in 100) and management's claim (~1 in 100,000).

37.6 A piece of external-tank foam of mass $0.76\ \text{kg}$ struck Columbia's wing at a relative speed of about $230\ \text{m/s}$. Compute its kinetic energy $\tfrac12 m v^2$. Comment on whether "it was only foam" is a good defense.

Part B — Reusability economics (⭐⭐)

37.7 † Model cost per flight as $c = m + F/n$, where $m$ is the marginal cost of one flight, $F$ is the fixed annual program cost, and $n$ is the flight rate (flights per year). Take $m = \$0.45\ \text{B}$ and $F = \$4.0\ \text{B/yr}$ (illustrative, Tier 3). Compute $c$ at $n = 4$, $n = 10$, and $n = 40$ flights per year. What single variable is doing almost all the work?

37.8 A reusable booster costs \$60 million to build and can be reflown many times; each reflight needs a refurbishment costing \$R. An expendable booster of the same capability costs \$60 million and is built new every flight. (a) Write the reusable cost per flight as a function of the number of flights $n$ and $R$. (b) With cheap refurbishment $R = \$10\ \text{M}$, how many flights are needed to beat the expendable? (c) With expensive, Shuttle-like refurbishment $R = \$45\ \text{M}$, what does the reusable cost per flight approach as $n \to \infty$, and how much does reuse save then?

37.9 † In one paragraph, use your results from 37.7 and 37.8 to explain the chapter's claim that "reusability is an operations problem, not a hardware problem." Which two conditions must hold for reuse to save money, and which did the Shuttle fail?

Part C — The engine and the boosters (⭐⭐)

37.10 † Explain, in terms of Chapter 17's engine cycles, why the SSME's fuel-rich staged-combustion cycle achieves a higher specific impulse than a gas-generator cycle of the same propellants. What does staged combustion make harder — and why is that difficulty worse for a reusable engine than a single-use one?

37.11 A solid rocket motor cannot be throttled or shut down once ignited. The Shuttle's SRBs burned for about $124\ \text{s}$. Explain what this implies for crew abort options during the first two minutes of flight, and contrast it with the SSMEs, which could be throttled between 67% and 109% and shut down.

37.12 † On ascent, the SSMEs were throttled down to about 65–72% through the region of maximum dynamic pressure. Referring to max-Q from Chapter 5, explain why the vehicle throttles down there, and why a solid booster's inability to throttle is a disadvantage in that phase (the SRB thrust profile is instead shaped by the propellant grain geometry).

Part D — 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.

37.13 † Write cost_per_flight(marginal, fixed_annual, flights_per_year) returning $m + F/n$. Trace it for cost_per_flight(0.45, 4.0, 4) and cost_per_flight(0.45, 4.0, 40) (all in \$B).

37.14 Write solid_fraction(srb_each, n_srb, ssme_each, n_ssme) returning the fraction of total liftoff thrust from the solids. Trace it for solid_fraction(12.5, 2, 1.8, 3).

37.15 † Write prob_no_loss(p, n) returning the probability of completing $n$ flights with zero losses given a per-flight loss probability $p$, i.e. $(1-p)^n$. Trace it for prob_no_loss(0.015, 135) and interpret the number: was flying 135 times without a loss likely at that rate?

Part E — Reliability & organizational analysis (⭐⭐ / ⭐⭐⭐)

37.16 † (Series reliability, Chapter 32) Suppose a vehicle has $k = 2{,}000$ independent Criticality-1 items, each with a per-flight reliability of $r = 0.999995$. Compute the whole-system reliability $R = r^{k}$ and the probability of a catastrophic failure $1 - R$. Compare with the Shuttle's demonstrated ~1-in-68 rate, and comment on what "independent" is hiding.

37.17 The SRB field joint had a secondary O-ring intended as redundancy behind the primary. Using Chapter 32's idea of common-cause failure, explain why this redundancy did not deliver the reliability gain that independent redundancy would, and what specific shared cause defeated both seals on Challenger.

37.18 † (Analysis) Read the following (fictional but realistic) log: "Anomaly seen again on this flight — same as the last six. Never caused a problem before. Recommend we keep flying and monitor." Identify the failure mode by name, explain what rule the team should hold to instead, and state what the Shuttle's history says happens if they do not.

37.19 (Mission project) Write the Reusability & Reliability Reflection for your mission track (A/B/C/D): is your launch vehicle reusable and does that pay off at your flight rate; what are your Criticality-1 single points of failure; and one specific warning sign you will refuse to normalize. Save it to your MDR.

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

37.20 Why can't you just bolt a launch-escape system onto the Shuttle orbiter, the way Apollo had an escape tower? Frame your answer around the architecture of §37.3 (where the crew sits, what is above and beside them), not around cost.

37.21 † Why can't you just make reuse cheap by reusing every part of the rocket? Using the Shuttle as the example, explain why reusing the hardest hardware (orbiter, engines, tiles) did not make access cheap, and what has to be true of the refurbishment and flight rate for reuse to win.

37.22 Feynman reported a factor-of-1,000 disagreement about the failure probability: ~1 in 100 (engineers) versus ~1 in 100,000 (management). Using the demonstrated 2-in-135 rate, state which estimate reality vindicated, and explain in two or three sentences what the disagreement itself — not just the wrong number — reveals about the organization.

Part G — Interleaved & synthesis (⭐⭐ / ⭐⭐⭐)

37.23 † (Ch. 7) Using the stagnation-heating scaling $\dot q \propto \sqrt{\rho/R_n}\,v^3$ from Chapter 7, explain why the wing leading edge (small $R_n$) is the most heat-critical surface on the orbiter and why it was armored with reinforced carbon-carbon rather than tiles. Then explain why a hole there is catastrophic rather than merely damaging.

37.24 (Ch. 3) The Shuttle needed both the solids (~80% of liftoff thrust) and the SSMEs. Using the rocket equation and the roles of thrust versus efficiency, explain why a launch vehicle wants high thrust low down (the solids) and high specific impulse higher up (the hydrogen SSMEs), and why neither alone would do.

37.25 † (Ch. 32) Classify each of the following Shuttle elements as Criticality 1 (loss of vehicle/crew if it fails, no backup) or not, and justify: (a) an SRB field-joint O-ring during first-stage burn; (b) one of three SSMEs failing after SRB separation, given engine-out abort modes; (c) a single belly tile among thousands; (d) an RCC wing-leading-edge panel during re-entry.

37.26 (Ch. 22) The Shuttle's SRBs are an example of parallel staging. Contrast parallel with serial staging (Chapter 22), and give one advantage and one safety drawback that the parallel solids brought to the Shuttle.

37.27 † (Ch. 16) Suppose one SSME fails at liftoff. Recompute the liftoff thrust and $T/W$ (two SSMEs at $1.8\ \text{MN}$ plus two SRBs at $12.5\ \text{MN}$, stack mass $2{,}030\ \text{t}$). Does the vehicle still leave the pad? What does this reveal about why the solids, not the engines, dominate the liftoff margin?

37.28 (Ch. 2) Explain, using $E_k = \tfrac12 m v^2$ and the idea of relative velocity, why a light piece of foam could carry ~20 kJ into Columbia's wing. Why is the closing speed between the shed foam and the accelerating orbiter the number that matters, rather than the foam's density or weight on the ground?

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 and analysis problems, the rubric rewards: a correctly set-up calculation with units and a sanity check; for organizational problems, naming the failure mode precisely and proposing a rule the team could actually hold to.