Exercises: A History of Rocketry

This is a history chapter, so many questions ask you to reason and place events in context — but the whole point of the chapter is that the history is the physics, so you will also compute. Work the numbers 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. Vehicle masses and specific impulses in these problems are approximate, historically illustrative figures (Tier 2/3).

Part A — Warm-ups: dates, names, and conversions (⭐)

36.1 † Black powder has a specific impulse of about $I_{sp} = 80\ \text{s}$. Convert this to an effective exhaust velocity $v_e$ in m/s.

36.2 Put these five milestones in chronological order: Apollo 11 Moon landing; Sputnik; Goddard's first liquid-fueled rocket flight; Tsiolkovsky's rocket-equation paper; the first Falcon 9 booster landing.

36.3 † The V-2 had a mass ratio of about $m_0/m_f = 3.2$ and an exhaust velocity of about $v_e = 2.0\ \text{km/s}$. What ideal delta-v did it deliver?

36.4 Name the three early theorists of spaceflight discussed in §36.2, and give the single most important contribution of each.

36.5 † Who launched the first artificial satellite, in what year, and on what rocket? What Chapter 3 technique did that rocket use that the V-2 did not?

Part B — Calculate the history (⭐⭐)

36.6 † A gunpowder rocket has $v_e = 0.8\ \text{km/s}$. What mass ratio would it need to reach orbital speed alone, $\Delta v = 7.8\ \text{km/s}$ (ignoring losses)? Comment on whether such a rocket could ever be built.

36.7 The V-2's ideal delta-v was about $2.3\ \text{km/s}$, but it actually burned out at only about $1.6\ \text{km/s}$. Where did the missing $\sim 0.7\ \text{km/s}$ go? Name the two mechanisms (from Chapter 4).

36.8 † The R-7 that launched Sputnik had to supply about $\Delta v = 9.4\ \text{km/s}$ (orbit plus losses). Using an effective $v_e = 3.0\ \text{km/s}$, what overall mass ratio did it need? Compare this to the V-2's mass ratio of 3.2 and comment on what made the difference.

36.9 Compare a single V-2 stage ($v_e = 2.0\ \text{km/s}$, mass ratio 3.2) with a two-stage rocket whose stages each have the same $v_e$ and mass ratio (idealizing away the upper stage's mass). What total delta-v does the two-stage version deliver, and what does the comparison illustrate about §36.3–36.4?

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. Assume you can from rocket import delta_v, isp_to_ve (your Chapter 3 module).

36.10 † Write v2_delta_v() that returns the V-2's ideal delta-v by calling delta_v(isp_to_ve(203), 12_500, 3_900). Hand-trace the numeric result.

36.11 Write gunpowder_mass_ratio(target_dv) returning $e^{\,target\_dv / 785}$ (with $v_e = 785\ \text{m/s}$). Trace it for gunpowder_mass_ratio(9400) — the mass ratio a gunpowder rocket would need to reach orbit.

36.12 † Write timeline(events) that takes a list of (name, year) tuples, returns them sorted by year, and also returns the span in years between the earliest and latest. Trace it for [("Tsiolkovsky", 1903), ("Apollo 11", 1969), ("Sputnik", 1957)].

Part D — Find the error (⭐⭐)

36.13 † A student writes: "The V-2 reached space, so with a bigger tank and more propellant it could easily have reached orbit." Identify the two things wrong with this reasoning, using the rocket equation and the difference between altitude and orbit.

36.14 A popular article claims: "Goddard proved that rockets work in a vacuum, disproving Newton." What is wrong with this sentence, both historically and physically?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

36.15 † (A "Bumper"-style design.) You try to make an orbital rocket in the late 1940s by stacking a small upper stage on a V-2. The first stage has $v_e = 2.0\ \text{km/s}$, the upper stage $v_e = 2.4\ \text{km/s}$, and you split the $9.4\ \text{km/s}$ to orbit evenly ($4.7\ \text{km/s}$ each). Find the mass ratio each stage needs. Using a 1940s structural coefficient $\varepsilon = 0.10$ (so a single stage cannot exceed a mass ratio of $1/\varepsilon = 10$), which stage is infeasible? Explain what this says about why orbit had to wait until 1957.

36.16 (Mission project.) Write the Heritage Note for your MDR (see the Mission Design Checkpoint): for your chosen track, trace your launch vehicle, your staging, your orbit, and — if crewed — your life support back to specific milestones in this chapter.

36.17 † Kennedy committed to the Moon in May 1961 with about fifteen minutes of American crewed spaceflight behind him. Alan Shepard's suborbital Redstone (a V-2 descendant) delivered roughly $2.3\ \text{km/s}$ of delta-v; the Saturn V would need about $12.9\ \text{km/s}$ for LEO plus trans-lunar injection (Chapter 3 case study). By what factor did U.S. delta-v capability have to grow in under a decade?

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

36.18 † Why couldn't the ancient Chinese, who invented the rocket, have reached space with a sufficiently large gunpowder rocket? Answer with the rocket equation, using the word "exponential," and give the rough mass ratio orbit would demand at $v_e \approx 785\ \text{m/s}$ (from $I_{sp} = 80\ \text{s}$) and $\Delta v = 9.4\ \text{km/s}$.

36.19 Why can't you just call the V-2 "the first spacecraft"? Explain what "reaching space" (crossing the $\sim 100\ \text{km}$ Kármán line) does and does not mean, versus reaching orbit.

36.20 † Estimate how few times a Falcon 9 booster must fly to make reuse pay off, given the rough assumptions that the first-stage booster is about 60% of the rocket's build cost and that refurbishing it costs about 10% of a new booster. State your assumptions and what your estimate ignores.

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

36.21 † (Ch. 1) Sputnik circled the Earth at about $7.8\ \text{km/s}$. In your own words — no formula needed — explain why that sideways speed keeps it in orbit instead of falling back like the V-2. Use the phrase "orbit is sideways."

36.22 (Ch. 3) The Saturn V delivered about $12.9\ \text{km/s}$ across three stages, far more than a single giant stage of the same propellant could. State, in one sentence, the Chapter 3 idea that explains why.

36.23 † (Ch. 7) Gagarin, the Apollo crews, and every returning astronaut faced the same deadly problem on the way home. What is it, physically, and which chapter covers it?

36.24 (Ch. 22, synthesis) A reusable Falcon 9 holds back propellant to land its booster. Using the rocket equation qualitatively, explain why this reduces the payload it can deliver to orbit — and why SpaceX accepts that penalty anyway.

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: correct use of the rocket equation, explicit units, a sanity check on the final number, and — for the historical questions — accuracy about what actually happened.