Exercises: Rocket Propulsion Fundamentals

Work these with a calculator and $g_0 = 9.80665\ \text{m/s}^2$ (use $9.81$ where a round value is fine). Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each 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. Keep units on every quantity and sanity-check every final number.

Part A — Warm-ups: read an engine data sheet (⭐)

16.1 † An engine expels propellant at a mass flow rate $\dot m = 280\ \text{kg/s}$ with an effective exhaust velocity $c = 3{,}000\ \text{m/s}$. What thrust does it produce (in kN)?

16.2 An engine produces $F = 1{,}200\ \text{kN}$ of thrust at a mass flow rate of $\dot m = 400\ \text{kg/s}$. Find its effective exhaust velocity and its specific impulse.

16.3 † An engine has a specific impulse of $I_{sp} = 350\ \text{s}$. What is its effective exhaust velocity $c$?

16.4 An attitude thruster has $\dot m = 5\ \text{kg/s}$, exhaust velocity $v_{\text{ex}} = 2{,}800\ \text{m/s}$, exit pressure $p_e = 50\ \text{kPa}$, and exit area $A_e = 0.02\ \text{m}^2$, firing at sea level ($p_a = 101.3\ \text{kPa}$). Compute the momentum thrust, the pressure thrust, and the total thrust.

16.5 † A rocket weighs in at $300\ \text{t}$ fully fueled and its engines make $4{,}000\ \text{kN}$ of thrust at liftoff. What is its thrust-to-weight ratio? Does it leave the pad, and if so, at what initial acceleration?

16.6 A solid motor burns at a constant thrust of $500\ \text{kN}$ for $120\ \text{s}$. What is its total impulse, in N·s?

Part B — Effective exhaust velocity and $I_{sp}$, rigorously (⭐⭐)

16.7 † An engine makes $F_{\text{SL}} = 800\ \text{kN}$ at sea level with $I_{sp,\text{SL}} = 290\ \text{s}$, and has a nozzle exit area $A_e = 1.0\ \text{m}^2$. Estimate its vacuum thrust. (Hint: only the pressure term changes with altitude; $F_{\text{vac}} = F_{\text{SL}} + p_{a,\text{SL}}A_e$.)

16.8 Starting from $c = v_{\text{ex}} + (p_e - p_a)A_e/\dot m$, show algebraically that a perfectly expanded nozzle ($p_e = p_a$) has $c = v_{\text{ex}}$ and therefore $I_{sp} = v_{\text{ex}}/g_0$. In one sentence, say why this is the most efficient expansion condition.

16.9 † In vacuum, an engine's actual exhaust velocity is $v_{\text{ex}} = 3{,}200\ \text{m/s}$, its mass flow is $\dot m = 250\ \text{kg/s}$, and its exit pressure acting over its exit area contributes $p_e A_e = 40\ \text{kN}$ of pressure thrust. Find the effective exhaust velocity and the specific impulse.

16.10 An upper-stage engine is quoted at $I_{sp,\text{vac}} = 452\ \text{s}$. Compute its effective exhaust velocity, and identify which propellant combination (from Appendix H) it must be using — and why that propellant achieves the highest chemical $I_{sp}$.

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. Use the canonical signature where given.

16.11 † Write thrust(mdot, ve, pe, pa, ae) returning $\dot m\, v_{\text{ex}} + (p_e - p_a)A_e$. Trace it for thrust(300, 2900, 70000, 101325, 0.68).

16.12 Write specific_impulse(F, mdot) returning $F/(\dot m\, g_0)$ with G0 = 9.80665. Trace it for specific_impulse(845000, 306), rounded to one decimal.

16.13 † Write total_impulse(F, tb) returning $F\,t_b$ and burn_time(mp, mdot) returning $m_p/\dot m$. Trace total_impulse(845000, 162) and burn_time(43900, 306).

Part D — Find the error (⭐⭐)

16.14 A student writes: "In vacuum there is no air for the exhaust to push against, so a rocket engine must produce less thrust in space than at sea level." Identify the error and state what actually happens to thrust as the rocket climbs, and why.

16.15 † A student evaluates an upper stage's thrust-to-weight by dividing its vacuum thrust of $900\ \text{kN}$ by the entire launch vehicle's mass of $520\ \text{t}$, gets $T/W = 0.176$, and concludes "this stage can never fire — it can't lift the rocket." Two things are wrong with this reasoning. Name both.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

16.16 (Mission project.) For your chosen mission (Track A/B/C/D), list every propulsive maneuver in your Chapter 3 delta-v budget and label each thrust-critical (needs $T/W > 1$ against some gravity — e.g. launch, powered landing) or efficiency-critical (done in free-fall, where a low thrust is fine — e.g. orbit-raising, cruise). Save the annotated table to your MDR; it will drive your engine choice in Chapters 17–19.

16.17 † A stage must deliver a total impulse of $I_t = 3.0\times10^{8}\ \text{N·s}$ using a propellant with $I_{sp} = 320\ \text{s}$. How much propellant mass must it carry? (Hint: $I_t = I_{sp}\,g_0\,m_p$.)

16.18 You are sizing a lunar-lander descent engine. The lander masses $2{,}000\ \text{kg}$ at the start of descent, lunar gravity is $g_{\text{Moon}} = 1.62\ \text{m/s}^2$, and you want an initial thrust-to-weight of $2.0$ (so you can decelerate briskly and hover with margin). What thrust do you need? If the engine has $I_{sp} = 311\ \text{s}$, what mass flow rate does that thrust imply?

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

16.19 † Estimate, to an order of magnitude, the total jet power of a Falcon 9 first stage at liftoff. Use $\dot m \approx 306\ \text{kg/s}$ per engine, nine engines, and $c \approx 2{,}765\ \text{m/s}$. State your result in watts and compare it to a large ($\sim 1\ \text{GW}$) power station.

16.20 Why can't you just bolt an ever-larger nozzle onto a vacuum engine to get unlimited pressure thrust? Explain what happens to the exit pressure $p_e$ as the nozzle grows, and why the pressure thrust $p_e A_e$ does not increase without bound. (Forward-looks to Chapter 19.)

16.21 † Why can't you just use a high-$I_{sp}$ ion engine ($I_{sp} \approx 3{,}000\ \text{s}$, thrust $\sim 0.1\ \text{N}$) to launch from Earth? Frame your answer entirely in terms of thrust-to-weight for a modest $500\ \text{kg}$ vehicle.

Part G — Interleaved & synthesis (from Chapters 3–4) (⭐⭐ / ⭐⭐⭐)

16.22 † (Ch. 3.) A stage has effective exhaust velocity $c = 3{,}050\ \text{m/s}$, initial mass $m_0 = 120\ \text{t}$, and final mass $m_f = 30\ \text{t}$. Compute its delta-v. Then state which of the quantities in this problem is set by the engine and which by the vehicle's construction.

16.23 (Ch. 3.) Using total impulse and burn time, explain in two or three sentences why thrust does not appear anywhere in the rocket equation, even though it is obviously essential to a real launch.

16.24 † (Ch. 4 connection.) A first stage could be designed with a liftoff $T/W$ of $1.4$ or of $3.0$. Explain qualitatively the competing effects: how does a higher $T/W$ change gravity losses (Ch. 4) and how does it change aerodynamic/structural loads? Why do real first stages cluster near $T/W \approx 1.2$–$1.5$ rather than much higher?

16.25 (Synthesis.) An engine's specific impulse rises from $282\ \text{s}$ at sea level to $311\ \text{s}$ in vacuum. By what percentage does its effective exhaust velocity rise? For a stage flying a fixed mass ratio, by what percentage does its delta-v rise? Are the two percentages the same, and why?

16.26 † (Synthesis, Ch. 3.) Two engines, A and B, have the same effective exhaust velocity $c$ but engine A has ten times the thrust of engine B. A stage is built twice, once with each engine, with identical propellant and dry mass. Which version has the greater delta-v? Which has the shorter burn? Explain using $\Delta v = c\ln(m_0/m_f)$ and $t_b = m_p/\dot m$.

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 set-up of the governing relation, explicit units and conversions, and a sanity check on the final number against a real engine or vehicle.