Exercises: The Future

This is a survey chapter, so its exercises lean less on new formulas and more on reasoning with the physics you already have — the rocket equation above all — and on the professional habit this chapter exists to teach: separating the possible from the probable from the speculative, and tiering every projected number. Use $g_0 = 9.81\ \text{m/s}^2$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold first. For "implement it" problems, do not run the code: hand-trace and record the result in an # Expected output: comment.

Part A — Warm-ups: know the landscape (⭐)

39.1 † In one sentence each, define Artemis, the Gateway, and the Human Landing System, and say which one is a station, which is a program, and which is a lander.

39.2 What orbit does the Gateway occupy, and which Lagrange point (from Chapter 15) is it associated with?

39.3 † State the difference between suborbital and orbital spaceflight in terms of sideways speed, not altitude.

39.4 Name the four remaining challenges of a crewed Mars mission listed in §39.2, and identify the one the chapter calls the crux.

39.5 † What is the near-term prize of asteroid mining, and why is it more valuable in space than on Earth?

39.6 Today's space economy is dominated by which category of activity — launch, or satellite services? Roughly what annual size is it (give the attributed range)?

Part B — Calculate it (⭐⭐)

39.7 † A suborbital vehicle reaches a peak speed of $1.0\ \text{km/s}$; an orbital vehicle must reach $7.8\ \text{km/s}$. Compute the specific kinetic energy ($\tfrac{1}{2}v^2$) of each, and the ratio between them. Comment on why the vehicle difference is even larger than this energy ratio.

39.8 A Mars Ascent Vehicle must supply $\Delta v = 4.0\ \text{km/s}$ using methane/oxygen at $I_{sp} = 370\ \text{s}$. Find $v_e$ and the required mass ratio $m_0/m_f$. If the empty ascent stage plus crew masses $5\ \text{t}$, estimate the propellant mass.

39.9 † The Sabatier reaction is $\mathrm{CO_2} + 4\,\mathrm{H_2} \rightarrow \mathrm{CH_4} + 2\,\mathrm{H_2O}$. Using molar masses ($\mathrm{CO_2}=44$, $\mathrm{H_2}=2$, $\mathrm{CH_4}=16$, $\mathrm{H_2O}=18\ \text{g/mol}$), how many kilograms of hydrogen must you bring from Earth to make $1\ \text{kg}$ of methane (taking Martian $\mathrm{CO_2}$ as free)? How much water do you get as a bonus?

39.10 Your MDR vehicle has a launch (wet) mass of $8{,}000\ \text{kg}$. Compute the launch cost at $\$54{,}000/\text{kg}$ (Shuttle era), $\$2{,}700/\text{kg}$ (Falcon 9), and $\$150/\text{kg}$ (a Starship target, Tier 3). By what factor does the cost fall across the three?

39.11 † From the delta-v map of Chapter 3, trans-Mars injection from LEO is about $3.6\ \text{km/s}$ and escaping Earth entirely is about $3.2\ \text{km/s}$. Use these two numbers to argue quantitatively against the claim "we have never gone to Mars because it is too far."

Part C — Implement it in Python (⭐⭐)

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

39.12 † Write launch_cost(wet_mass_kg, dollars_per_kg) returning the product. Trace it for launch_cost(6200, 2700).

39.13 Write energy_ratio(v_orbital, v_suborbital) returning the ratio of specific kinetic energies $(v_\text{orb}/v_\text{sub})^2$. Trace it for energy_ratio(7800, 1000).

39.14 † Write hydrogen_for_methane(kg_ch4) returning the hydrogen mass needed via Sabatier ($0.5\ \text{kg H}_2$ per $\text{kg CH}_4$). Trace it for hydrogen_for_methane(10000).

39.15 Write economy_2040(current_billion, annual_growth, years) that compounds a starting size at a constant annual growth rate. Trace it for economy_2040(500, 0.06, 17) and round to the nearest billion. Label the result's tier.

Part D — Find the error (⭐⭐)

39.16 † A pitch reads: "A metallic asteroid holds $\$20$ trillion in platinum. Capture even 0.1% and that's $\$20$ billion — the best business in history." Identify the two independent errors (one about physics/delivery, one about markets).

39.17 A commentator says: "Hundreds of tourists have flown to space on New Shepard, so affordable trips to orbit for ordinary people are just a few years away." Diagnose the error using the rocket equation.

39.18 † A student writes: "The space economy will pass $\$1$ trillion by 2040, so within fifteen years most of that money will come from asteroid mining and Mars colonies." Explain what the projection actually assumes, and why this conclusion misreads it.

Part E — Design it & your mission (⭐⭐ / ⭐⭐⭐)

39.19 † (Mission project) Write the "Where this approach is heading" forward-look for your track (A/B/C/D), following the Mission Design Checkpoint. Name one trend from this chapter that would change your design and one it would leave untouched, and save it to your MDR.

39.20 A lunar base wants to produce $100\ \text{kg}$ of oxygen per day from regolith or ice. If the process needs about $10\ \text{kWh}$ of energy per kilogram of oxygen (an illustrative figure, Tier 3), what continuous electrical power (in kW) must the base's power system (Chapter 25) supply, assuming the plant runs 24 hours a day? Comment on what the two-week lunar night does to this requirement.

39.21 † You are asked to justify, or reject, a Gateway-style station for a Mars program by analogy to the lunar one. Give one physical reason a libration-point staging station could help a Mars architecture and one reason the analogy might not carry. (Keep it qualitative; cite Chapter 15.)

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

39.22 † Why can't you just bring an asteroid's platinum back to Earth and get rich? Frame the answer in delta-v (the cost of returning mass down a gravity well) and in economics (what a large delivery does to the price).

39.23 Why can't you just resupply a Mars crew from Earth every few months, the way we resupply the ISS? Estimate, in orders of magnitude, why the transit time and launch windows (Chapter 11) make this impossible, and what that forces the life-support system (Chapter 28) to be.

39.24 † Why can't you just skip the Gateway and fly crews straight from Earth to the lunar surface, as Apollo did? Give the argument for the Gateway (what it buys) and the argument against (what it costs), and state which mission goal tips the balance.

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

39.25 † (Ch. 3) Explain, in two sentences, why the tyranny of the rocket equation is the single reason behind three separate claims in this chapter: that Mars missions must use ISRU, that asteroid platinum should stay in space, and that orbital tourism is far harder than suborbital.

39.26 (Ch. 11 & 15) The Voyager Grand Tour is the chapter's closing inspiration. Explain how the gravity assist (Ch. 11) and the interplanetary superhighway (Ch. 15) each embody the principle "cooperate with the physics instead of fighting it," and why that principle — not a repeal of the rocket equation — is what the future runs on.

39.27 † (Ch. 38) State the one change from Chapter 38 that this chapter calls the "master variable," and explain in your own words why falling launch cost — rather than any new physics — is what turns items in the "possible" column into the "probable" column.

39.28 (Ch. 28 & 34, synthesis) Of the four Mars challenges, two (radiation, the return trip) are mainly mass/cost problems and two (closed-loop life support, ISRU at scale) are mainly not-yet-demonstrated problems. Pick one from each group and explain, in a few sentences, what a robotic precursor mission (like a Track C Mars orbiter) could do to retire some of its risk before a crew flies.

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 the design and "why can't you just…" problems, the rubric rewards: correctly identifying which constraint (physics, economics, schedule) binds; explicit tiering of any projected number; and a sanity check against a real mission or vehicle.