Exercises: Why Is Space So Hard?

This chapter opens the book, so these problems lean on physical reasoning, unit conversions, and order-of-magnitude estimates more than heavy algebra — the habits you will use on every page after. Work them with a calculator and these constants: Earth's gravitational parameter $\mu = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$, Earth's radius $R_E = 6{,}371\ \text{km}$, standard gravity $g_0 = 9.81\ \text{m/s}^2$, sea-level sound speed $343\ \text{m/s}$.

Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and all 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.

Part A — Warm-ups: convert and compute (⭐)

1.1 † Compute the circular orbital velocity at an altitude of $300\ \text{km}$ (so $r = R_E + 300\ \text{km}$), using $v = \sqrt{\mu/r}$. Give your answer in km/s and compare it to the ~$7.8\ \text{km/s}$ quoted for LEO.

1.2 Express orbital velocity ($7.8\ \text{km/s}$) as a multiple of the speed of sound. Roughly what Mach number is that?

1.3 † Escape velocity at a given radius is $\sqrt{2}$ times the circular orbital velocity there. If the circular velocity at some altitude is $7.7\ \text{km/s}$, what is the escape velocity from that altitude?

1.4 A launch to LEO needs about $9.4\ \text{km/s}$ of delta-v. A jetliner cruises at about $250\ \text{m/s}$. By what factor does the orbital delta-v exceed the jet's cruising speed?

1.5 † The ISS orbits at ~$400\ \text{km}$ with a period of about $92$ minutes. How many times does it circle the Earth in one day? (There are $1{,}440$ minutes in a day.)

1.6 A micrometeoroid massing $0.5\ \text{g}$ strikes a spacecraft at a closing speed of $12\ \text{km/s}$. Compute its kinetic energy, $\tfrac12 m v^2$, in joules.

Part B — The environment and its numbers (⭐⭐)

1.7 † Compute the local gravitational acceleration at geostationary altitude ($35{,}786\ \text{km}$ above the surface), using $g = \mu/r^2$. What fraction of surface gravity is it? What does your answer say about the phrase "zero gravity"?

1.8 A spacecraft viewport measures $0.3\ \text{m} \times 0.4\ \text{m}$. With one atmosphere of cabin pressure ($101\ \text{kPa}$) inside and vacuum outside, what outward force (in newtons, and in tonnes-force) does the glass hold back?

1.9 † Natural background radiation at sea level is about $3\ \text{mSv}$ per year; an astronaut on the ISS receives roughly $0.7\ \text{mSv}$ per day. Over a $6$-month ($\approx 182$-day) stay, what dose does the astronaut accumulate, and how many years of ground-level background is that equivalent to?

1.10 Deep space (the cosmic microwave background) sits at about $2.7\ \text{K}$. Convert this to degrees Celsius and degrees Fahrenheit. Why does a spacecraft's shadowed side tend toward this value?

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. Reference code is in code/exercise-solutions.py.

1.11 † Write circular_velocity(mu, r) returning $\sqrt{\mu/r}$. Trace it for Earth at $500\ \text{km}$ altitude: circular_velocity(3.986e14, 6.871e6).

1.12 Write escape_velocity(mu, r) returning $\sqrt{2\mu/r}$. Trace it at Earth's surface: escape_velocity(3.986e14, 6.371e6).

1.13 † Write mass_ratio(dv, ve) returning $e^{\Delta v/v_e}$ (a preview of Chapter 3). Trace it for the orbital case mass_ratio(9400, 3400), and say in one line what the number means.

Part D — Find the error (⭐⭐)

1.14 † A student writes: "At $400\ \text{km}$ the ISS is in zero gravity, and that is why astronauts float." Identify the error and give the correct explanation, including the approximate value of gravity at that altitude.

1.15 A blog post claims: "Getting to space is easy — you just need to get above the $100\ \text{km}$ Kármán line. A big enough balloon, or a cannon pointed straight up, would put a payload into orbit." Two things in that sentence are wrong. Name and correct them.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

1.16 † (Mission project — start your MDR.) Choose your mission track — A (GEO comsat), B (lunar lander), C (Mars orbiter), or D (asteroid rendezvous) — and write a one-paragraph mission statement: what it does, why it exists, and a bulleted list of the major "legs" you expect to cost delta-v. Save it as the first page of your Mission Design Review document.

1.17 For the track you chose in 1.16, rank the challenges of §1.3–1.4 — vacuum, radiation, temperature, debris/micrometeoroids, microgravity, and the no-repair rule — from most to least design-driving for your mission, with a one-sentence justification for your top three. (A Mars orbiter and a GEO comsat will rank them very differently.)

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

1.18 † Estimate, for a $1\ \text{kg}$ mass, the kinetic energy of orbital motion ($v \approx 7.7\ \text{km/s}$) and the potential energy of merely lifting it to $400\ \text{km}$ (use the exact $\mu\,(1/R_E - 1/r)$, or approximate with $\overline{g}\,h$ for $\overline{g}\approx 9\ \text{m/s}^2$). Take the ratio and state, in one sentence, what it proves about where a launch's effort goes.

1.19 Why can't you just fly a jet aircraft, or float a balloon, to orbit? Frame your answer in terms of the two things an orbit requires and which one neither vehicle can supply.

1.20 † Why can't you just build a bigger rocket to reach a higher delta-v (to go faster)? Answer using the mass ratio $m_0/m_f$ explicitly (a preview of Chapter 3's reasoning).

Part G — Synthesis (this chapter's big ideas)

1.21 † In two or three sentences, explain how the two themes introduced this chapter — the tyranny of the rocket equation and space is unforgiving — pull spacecraft design in opposite directions. Give one concrete example of the tension (e.g., shielding vs. mass).

1.22 The chapter echoes Heinlein's line that "once you are in orbit, you are halfway to anywhere." Using the checkpoint numbers (LEO $\approx 9.4\ \text{km/s}$; trans-Mars from LEO $\approx 3.6\ \text{km/s}$; Earth escape from LEO $\approx 3.2\ \text{km/s}$), explain quantitatively what that saying means.

1.23 † A friend argues space "isn't really hard anymore — rockets launch every week now." Marshal two quantified facts from this chapter to argue it is still genuinely hard, while fairly acknowledging what has changed (theme 5, reusability).

1.24 Sort these four maneuvers from cheapest to most expensive in delta-v, and justify: (a) a suborbital hop to $100\ \text{km}$ and back; (b) reaching $7.8\ \text{km/s}$ orbital speed; (c) one year of GEO station-keeping (~$0.05\ \text{km/s}$); (d) escaping Earth starting from LEO (~$3.2\ \text{km/s}$).

1.25 † A Falcon 9 masses about $549\ \text{t}$ on the pad and delivers roughly $15\ \text{t}$ to LEO. Compute the payload fraction (payload ÷ lift-off mass) and explain, in one sentence, how it illustrates the theme mass is the enemy.

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: explicit units on every quantity, a stated assumption for every estimate, and a sanity check on the final number — the same discipline the chapter models.