Exercises: Aerodynamics of Ascent

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, sea-level density $\rho_0 = 1.225\ \text{kg/m}^3$, and scale height $H = 8\ \text{km}$ unless told otherwise. 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. Carry units through every step and sanity-check every number against $g$ or against sea level.

Part A — Warm-ups: density, pressure, and force (⭐)

5.1 † A rocket passes $v = 100\ \text{m/s}$ at sea level. Compute the dynamic pressure $q = \tfrac12\rho v^2$ in pascals and in kilopascals.

5.2 Using $\rho(h) = \rho_0 e^{-h/H}$, find the air density at $16\ \text{km}$. What fraction of sea-level density is it?

5.3 † At $11\ \text{km}$ a vehicle is flying at $450\ \text{m/s}$. Compute the dynamic pressure. How does it compare to sea-level atmospheric pressure ($\approx 101\ \text{kPa}$)?

5.4 A payload fairing is $5.2\ \text{m}$ in diameter. Compute its frontal cross-sectional area $A$.

5.5 † At max-Q a vehicle sees $q = 30\ \text{kPa}$. With a drag coefficient $C_d = 0.4$ and frontal area $A = 10.75\ \text{m}^2$, compute the drag force $D = q\,C_d\,A$. Express it as a weight (divide by $g_0$) for intuition.

5.6 A launch vehicle at max-Q masses $400\ \text{t}$ with $C_d = 0.5$ and $A = 10.75\ \text{m}^2$. Compute its ballistic coefficient $\beta = m/(C_d A)$, and the drag deceleration $q/\beta$ at $q = 32\ \text{kPa}$.

Part B — Dynamic pressure and max-Q (⭐⭐)

5.7 † Using $v_{\text{maxQ}} = \sqrt{2aH}$, estimate the speed at max-Q for a vehicle accelerating at $a = 12\ \text{m/s}^2$ (with $H = 8\ \text{km}$). Roughly what Mach number is that near the tropopause (speed of sound $\approx 295\ \text{m/s}$)?

5.8 A profile gives these points: $(10\ \text{km}, 400\ \text{m/s})$, $(13\ \text{km}, 500\ \text{m/s})$, $(16\ \text{km}, 600\ \text{m/s})$. Compute $q$ at each and state which is nearest max-Q.

5.9 † Show algebraically that for a vehicle under constant net acceleration $a$, rising vertically from rest (so $v^2 = 2ah$), the max-Q condition $v_{\text{maxQ}}^2 = 2aH$ implies max-Q occurs at altitude $h = H$ — exactly one scale height. Why is real max-Q usually a bit higher than one scale height?

5.10 A vehicle throttles down so its net acceleration through the dense air drops from $20\ \text{m/s}^2$ to $12\ \text{m/s}^2$. Using $v_{\text{maxQ}} = \sqrt{2aH}$, by what factor does the max-Q speed change, and hence (at a fixed altitude/density) by what factor does the peak $q$ change?

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.

5.11 † Write density(h) returning $\rho_0 e^{-h/H}$ and dynamic_pressure(h, v) returning $\tfrac12\rho v^2$. Trace dynamic_pressure(12000, 485).

5.12 Write max_q_speed(a, H) returning $\sqrt{2aH}$. Trace max_q_speed(18, 8000).

5.13 † Write ballistic_coefficient(m, cd, A) returning $m/(C_d A)$ and drag_decel(q, beta) returning $q/\beta$. Trace them for m=350000, cd=0.5, A=10.75, q=33000.

Part D — Find the error (⭐⭐)

5.14 † A student computes the drag force on a $500\ \text{kg}$ sounding rocket at $q = 33\ \text{kPa}$ with $C_d = 0.5$, $A = 0.071\ \text{m}^2$, getting $D \approx 1{,}170\ \text{N}$, then writes "deceleration $= 1{,}170\ \text{m/s}^2$." Identify the error, give the correct deceleration, and say why the wrong number is obviously impossible.

5.15 A blog claims "max-Q happens at the moment of maximum speed, just before orbit, because dynamic pressure grows with $v^2$." Diagnose the error using $q = \tfrac12\rho v^2$ and the exponential atmosphere.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

5.16 † Your new vehicle, flown at full thrust, would reach $q = 45\ \text{kPa}$ at its max-Q altitude, but its airframe is qualified only to $35\ \text{kPa}$. At a fixed altitude (fixed $\rho$), $q \propto v^2$. By what factor must you reduce the vehicle's speed at that altitude to hit the limit? Explain in one sentence how throttling down achieves this.

5.17 (Mission project) Add an Ascent Loads & Environment note to your MDR (Track A/B/C/D): record the peak dynamic pressure your payload must survive, the $q\alpha$/lateral-load case, the acoustic/buffeting environment, and the fairing jettison condition. Save it; you will match it to a real vehicle in Chapter 30.

5.18 † A fairing is jettisoned when the free-molecular heating rate, modeled here as proportional to $\rho v^3$, falls below a threshold $\dot q_{\text{th}}$. Given the profile $(100\ \text{km}, 2{,}500\ \text{m/s})$, $(120\ \text{km}, 3{,}000\ \text{m/s})$, $(140\ \text{km}, 3{,}500\ \text{m/s})$ and using $\rho(h) = \rho_0 e^{-h/H}$, compute the relative heating $\rho v^3$ at each altitude and state at which altitude jettison is safest (lowest heating). (You will find the exponential utterly dominates — that is the point.)

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

5.19 † Why can't you just fly slowly up through the atmosphere to keep the dynamic pressure low and avoid max-Q entirely? Frame your answer in delta-v: what does flying slowly cost, and which loss from Chapter 4 does it run up?

5.20 Why can't you just build the rocket strong enough to ignore max-Q, so you never have to throttle down? Answer in terms of theme #4 (mass is the enemy) and the rocket equation.

5.21 † Estimate the sideways (normal) force on a launch vehicle if a wind gust drove it to an angle of attack $\alpha = 5^\circ$ at max-Q ($q = 33\ \text{kPa}$, $A = 10.75\ \text{m}^2$). Use a normal-force slope $C_{N\alpha} \approx 2\ \text{per radian}$, so the normal-force coefficient is $C_N \approx C_{N\alpha}\,\alpha$ and $N = q\,C_N\,A$. Comment on why this is dangerous for a slender tube.

Part G — Interleaved & synthesis (Chapters 1–4)

5.22 † (Ch. 4) Drag loss to orbit is only $\sim 0.1\ \text{km/s}$ out of a $\sim 9.4\ \text{km/s}$ launch delta-v. Compute that fraction. Then, in one sentence, explain why max-Q is nonetheless a top-tier design driver despite drag's tiny delta-v cost.

5.23 (Ch. 3) Throttling through max-Q adds roughly $50\ \text{m/s}$ of gravity loss. With an effective $v_e = 3{,}000\ \text{m/s}$, by what factor does the required mass ratio $e^{\Delta v/v_e}$ grow for that extra $50\ \text{m/s}$? Is throttling a cheap or expensive way to protect the vehicle?

5.24 † (Ch. 2, synthesis) The exponential atmosphere follows from hydrostatic balance. Argue qualitatively why a roughly constant temperature atmosphere gives an exponential density profile (hint: think about the fractional pressure drop across a thin layer whose weight is proportional to the local density).

5.25 (Ch. 1) Chapter 1 called space "an unforgiving environment." Name two ways the atmosphere is unforgiving during ascent (from this chapter) and one way the vacuum above it is, aerodynamically speaking, actually easier.

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 with explicit units, a sanity check against $g$ or sea level, and a clear statement of which trade-off (delta-v vs. structure, protection vs. mass) is being made.