Exercises: The Rocket Equation

Work these 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.

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

3.1 † An engine has a specific impulse of $I_{sp} = 311\ \text{s}$. What is its effective exhaust velocity $v_e$ in m/s?

3.2 A rocket stage has a mass ratio $m_0/m_f = 6$ and an exhaust velocity of $3.2\ \text{km/s}$. What delta-v does it deliver?

3.3 † A maneuver needs $\Delta v = 2.5\ \text{km/s}$ and the engine has $v_e = 3.4\ \text{km/s}$. What mass ratio must the stage have?

3.4 A stage is 85% propellant by mass. What is its mass ratio $m_0/m_f$?

3.5 † Convert an exhaust velocity of $v_e = 4{,}400\ \text{m/s}$ to a specific impulse in seconds. Which propellant combination (from the chapter's table) is this closest to?

3.6 A stage masses $120\ \text{t}$ fueled and $30\ \text{t}$ empty, with $I_{sp} = 300\ \text{s}$. Find its delta-v.

Part B — The exponential (⭐⭐)

3.7 † A rocket has $v_e = 3.0\ \text{km/s}$. Compute the mass ratio it needs for $\Delta v = 3\ \text{km/s}$, then for $\Delta v = 6\ \text{km/s}$. By what factor did the mass ratio grow when the delta-v doubled? Explain why it is not a factor of 2.

3.8 Starting from $\Delta v = v_e \ln(m_0/m_f)$, show algebraically that the propellant mass fraction is $m_p/m_0 = 1 - e^{-\Delta v / v_e}$.

3.9 † A deep-space mission needs $\Delta v = 10\ \text{km/s}$ from a single stage with the best chemical engine, $v_e = 4.5\ \text{km/s}$. Find the mass ratio and propellant fraction. In one sentence, why does this explain the appeal of electric propulsion for such missions?

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.

3.10 † Write delta_v(ve, m0, mf) returning $v_e \ln(m_0/m_f)$. Trace it for delta_v(3400, 500000, 100000).

3.11 Write propellant_fraction(dv, ve) returning $1 - e^{-\Delta v/v_e}$. Trace it for propellant_fraction(9400, 3400).

3.12 † Write stage_delta_v(stages) that sums the delta-v of a list of (ve, m0, mf) tuples. Trace it for a two-stage rocket: [(2900, 500000, 150000), (3400, 120000, 25000)].

Part D — Find the error (⭐⭐)

3.13 † A blogger writes: "Since $\Delta v = v_e \ln(m_0/m_f)$, and the logarithm grows without bound, you can reach any delta-v just by making the rocket big enough." Identify the flaw in physical terms, not just mathematical.

3.14 A student sizes a two-stage rocket by lumping both stages' dry mass and both stages' propellant into one mass ratio, and concludes staging gives no delta-v benefit. What did they get wrong, and which mass did they incorrectly carry to the end of the burn?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

3.15 † You must deliver a $1{,}000\ \text{kg}$ payload with a single stage that provides $\Delta v = 5\ \text{km/s}$ using $v_e = 3.4\ \text{km/s}$. Assume the dry (structural) mass equals 8% of the propellant mass. Find the required propellant mass and the total lift-off mass. (Hint: set up the mass ratio with $m_f = m_L + 0.08\,m_p$ and $m_0 = m_f + m_p$, then solve for $m_p$.)

3.16 (Mission project) Using the delta-v map in §3.6, write the delta-v budget for your chosen mission (Track A/B/C/D). List each maneuver, its delta-v, a margin, and the total. Save it to your MDR.

3.17 † You have $\Delta v = 9\ \text{km/s}$ to achieve and choose to split it evenly between two stages, each with $v_e = 3.3\ \text{km/s}$. What mass ratio must each stage have? If each stage's dry mass is 10% of its propellant mass, is such a stage physically buildable? Comment.

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

3.18 † Estimate, to an order of magnitude, the lift-off mass of a rocket that puts a $1{,}000\ \text{kg}$ payload into LEO ($\Delta v \approx 9.4\ \text{km/s}$), assuming a two-stage vehicle with an overall effective $v_e \approx 3.2\ \text{km/s}$ and a generous overall structural fraction. State your assumptions.

3.19 Why can't you just hover a rocket straight up to space and stop? Frame your answer in terms of delta-v: what is the (rough) delta-v cost of "hovering," and why does the rocket equation make it prohibitive? (Preview of Chapter 4.)

3.20 † Why can't you just make the rocket ten times bigger to go ten times as far? Answer using the mass ratio explicitly.

Part G — Interleaved & synthesis (from Chapters 1–2)

3.21 † (Ch. 2) Earth's escape velocity is about $11.2\ \text{km/s}$. Using the delta-v map, is the LEO-to-escape delta-v larger or smaller than $11.2\ \text{km/s}$? Explain why they differ. (Hint: you are already moving in LEO.)

3.22 (Ch. 1) The chapter uses $9.4\ \text{km/s}$ to reach orbit, but orbital speed in LEO is only $7.8\ \text{km/s}$. Where does the extra $\sim 1.6\ \text{km/s}$ go? Name the two culprits (you will meet them in Chapter 4).

3.23 † (Ch. 2, synthesis) A rocket exhausts $250\ \text{kg/s}$ of propellant at $v_e = 3{,}000\ \text{m/s}$. Using conservation of momentum, what thrust does it produce? Does thrust appear anywhere in the rocket equation? Explain the relationship between this thrust and the stage's delta-v.

3.24 (Synthesis) Two stages each deliver $4\ \text{km/s}$. Explain why the total is $8\ \text{km/s}$ (they add) rather than something you would get from combining their mass ratios into one. Tie your answer to what physically happens at staging.

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 $m_0$ and $m_f$, explicit units, and a sanity check on the final number.