Level 1 — RecognitionRocket Propulsion

Rocket Propulsion

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Chapter: 3.3 Rocket Propulsion Difficulty Level: 1 — Recognition (MCQ + Matching + True/False with justification) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each) [10 marks]

Select the single best answer.

Q1. The Tsiolkovsky rocket equation is written as: (a) $\Delta v = v_e \cdot (m_0 - m_f)$ (b) $\Delta v = v_e \cdot \ln(m_0/m_f)$ (c) $\Delta v = v_e \cdot \ln(m_f/m_0)$ (d) $\Delta v = g_0 \cdot \ln(m_0/m_f)$

Q2. Specific impulse $I_{sp}$ and effective exhaust velocity $c$ are related by: (a) $I_{sp} = c \cdot g_0$ (b) $I_{sp} = c/g_0$ (c) $I_{sp} = c^2/g_0$ (d) $I_{sp} = g_0/c$

Q3. Which propellant combination gives the highest typical vacuum specific impulse among chemical rockets? (a) Solid propellant (~260 s) (b) LOX/RP-1 (~311 s) (c) LOX/LH₂ (~450 s) (d) N₂O₄/UDMH (~320 s)

Q4. The full thrust equation is: (a) $F = \dot{m} v_e$ (b) $F = \dot{m} v_e + (P_e - P_a)A_e$ (c) $F = \dot{m} v_e - P_c A^*$ (d) $F = (P_e - P_a)A_e$

Q5. A nozzle is at optimum expansion (maximum thrust for given ambient) when: (a) $P_e > P_a$ (b) $P_e < P_a$ (c) $P_e = P_a$ (d) $P_e = P_c$

Q6. Vieille's law for solid propellant burn rate is: (a) $r = a P^n$ (b) $r = a/P^n$ (c) $r = a \ln P$ (d) $r = a P + n$

Q7. In an expander cycle, the turbine is driven by: (a) A separate gas generator exhaust (b) Heated hydrogen from regenerative nozzle cooling (c) Oxidizer-rich preburner gas (d) High-pressure tank helium

Q8. Characteristic velocity $c^*$ is defined as: (a) $c^* = F/(P_c A^*)$ (b) $c^* = P_c A^*/\dot{m}$ (c) $c^* = \dot{m}/(P_c A^*)$ (d) $c^* = v_e + (P_e-P_a)A_e/\dot{m}$

Q9. An over-expanded nozzle ( $P_e < P_a$ ) suffers efficiency loss primarily due to: (a) Prandtl-Meyer expansion fans (b) Oblique shocks / flow separation in the plume (c) Chamber pressure collapse (d) Increased throat area

Q10. Ion engines achieve very high $I_{sp}$ (~3000 s) but their key limitation is: (a) Very low thrust (b) Very high propellant density (c) Instability at high frequency (d) Requiring cryogenic storage

Section B — Matching (1 mark each) [10 marks]

Q11–Q15. Match each engine cycle/component (left) to its defining feature (right).

#	Item		Feature
Q11	Gas generator cycle	A	Simplest; propellant pushed by pressurized tanks, no turbopump
Q12	Staged combustion cycle	B	Turbine exhaust dumped overboard; slight $I_{sp}$ penalty, simple
Q13	Pressure-fed cycle	C	All flow through preburner(s) into main chamber; highest efficiency
Q14	Electric pump-fed cycle	D	Battery-driven electric motors run the propellant pumps
Q15	Expander cycle	E	Coolant heated in nozzle jacket expands to drive turbine

Q16–Q20. Match each cooling method (left) to its mechanism (right).

#	Method		Mechanism
Q16	Regenerative cooling	F	Sacrificial surface chars, absorbs heat, blows gas into boundary layer
Q17	Film cooling	G	Coolant flows through wall channels then into combustion
Q18	Ablative cooling	H	Coolant seeps through a porous wall to protect it
Q19	Transpiration cooling	I	Liquid layer injected along wall to shield it from hot gas
Q20	(Free choice — pick unused)	J	(distractor: radiative cooling)

(For Q20, state which mechanism letter is left unused and name it.)

Section C — True / False WITH Justification (2 marks each) [10 marks]

1 mark correct T/F, 1 mark for a correct one-line reason.

Q21. "Increasing the mass ratio $m_0/m_f$ from 3 to 6 doubles the achievable $\Delta v$ for the same $v_e$ ." — True or False? Justify.

Q22. "For a nozzle in isentropic flow, the exit Mach number depends only on the area ratio $\varepsilon = A_e/A^*$ and $\gamma$ ." — True or False? Justify.

Q23. "Optimal series staging with stages of equal $I_{sp}$ requires each stage to have an equal mass ratio." — True or False? Justify.

Q24. "An under-expanded nozzle ( $P_e > P_a$ ) produces exactly the theoretical maximum thrust with no loss." — True or False? Justify.

Q25. "Characteristic velocity $c^*$ increases with higher chamber flame temperature and lower exhaust molecular weight." — True or False? Justify.

Answer keyMark scheme & solutions

Section A

Q1 — (b) $\Delta v = v_e \ln(m_0/m_f)$ . (1 mark) Derived from integrating momentum conservation; log of the mass ratio, with initial mass on top. Options (a) is linear (wrong), (c) has ratio inverted (gives negative), (d) uses $g_0$ not $v_e$ .

Q2 — (b) $I_{sp}=c/g_0$ . (1 mark) By definition $I_{sp}$ (in seconds) = effective exhaust velocity divided by standard gravity $g_0=9.81\,\text{m/s}^2$ .

Q3 — (c) LOX/LH₂ (~450 s). (1 mark) Low molecular weight of H₂O/H₂ exhaust ⇒ high exhaust velocity ⇒ highest chemical $I_{sp}$ .

Q4 — (b) $F=\dot m v_e+(P_e-P_a)A_e$ . (1 mark) Momentum thrust plus pressure thrust term.

Q5 — (c) $P_e=P_a$ . (1 mark) Pressure thrust term vanishes at design; this is the condition for maximum thrust at given ambient pressure.

Q6 — (a) $r=aP^n$ . (1 mark) Vieille's (Saint-Robert's) law; $a$ temperature-dependent coefficient, $n$ pressure exponent.

Q7 — (b) Heated hydrogen from regenerative nozzle cooling. (1 mark) Expander cycle uses coolant expansion energy to spin the turbine — no preburner.

Q8 — (b) $c^*=P_c A^*/\dot m$ . (1 mark) Characteristic velocity; measure of combustion/energy release efficiency. (a) is $C_F$ ; (d) is effective exhaust velocity $c$ .

Q9 — (b) Oblique shocks / flow separation. (1 mark) Ambient higher than exit pressure compresses plume, forming shocks and possibly separating flow.

Q10 — (a) Very low thrust. (1 mark) Ion engines trade thrust for $I_{sp}$ ; typical thrust in mN range.

Section B — Matching

Q11 → B, Q12 → C, Q13 → A, Q14 → D, Q15 → E (1 mark each)

Gas generator: dumps turbine exhaust overboard (B).
Staged combustion: all/most flow reaches main chamber (C).
Pressure-fed: no pumps, tank pressure feeds (A).
Electric pump-fed: battery + electric motor pumps (D).
Expander: nozzle-heated coolant drives turbine (E).

Q16 → G, Q17 → I, Q18 → F, Q19 → H (1 mark each)

Regenerative: wall channels, coolant then burned (G).
Film: liquid layer along wall (I).
Ablative: charring sacrificial layer + blowing (F).
Transpiration: coolant through porous wall (H).

Q20 → J unused: radiative cooling (1 mark) J (radiative cooling — hot wall re-radiates heat away) is the leftover mechanism.

Section C — True/False with justification

Q21 — FALSE. (1 + 1) $\Delta v \propto \ln(m_0/m_f)$ , not linearly. Going 3→6: ratio of $\Delta v$ = $\ln 6/\ln 3 = 1.79/1.10 \approx 1.63$ , not 2.

Q22 — TRUE. (1 + 1) The area–Mach relation $\varepsilon=\frac{1}{M_e}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M_e^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$ links $\varepsilon$ and $M_e$ uniquely for a given $\gamma$ (supersonic branch).

Q23 — TRUE. (1 + 1) For equal $I_{sp}$ (and equal structural coefficients), the payload-optimal solution via Lagrange multipliers gives identical mass ratios per stage, splitting $\Delta v$ equally.

Q24 — FALSE. (1 + 1) Under-expansion means gas continues expanding outside via Prandtl-Meyer fans; that expansion energy is not converted to axial thrust, so there is a loss. Max thrust needs $P_e=P_a$ .

Q25 — TRUE. (1 + 1) $c^*=\sqrt{\frac{R_u T_c/M}{\gamma}}\big/\left[\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}\right]$ ; $c^*\propto\sqrt{T_c/M}$ , so higher $T_c$ and lower $M$ raise it.

[
  {"claim": "Q21: dv ratio for mass ratio 3->6 is about 1.63, not 2",
   "code": "import sympy as sp; r=sp.log(6)/sp.log(3); result = abs(float(r)-1.6309)<0.001 and float(r)<2"},
  {"claim": "Q2: Isp = c/g0 gives ~459s for c=4500 m/s",
   "code": "c=4500; g0=9.81; Isp=c/g0; result = abs(Isp-458.7)<1.0"},
  {"claim": "Q22: area ratio from Me=3, gamma=1.2 is positive & >1 (supersonic)",
   "code": "import sympy as sp; Me=3; g=sp.Rational(12,10); eps=(1/Me)*((2/(g+1))*(1+(g-1)/2*Me**2))**((g+1)/(2*(g-1))); result = float(eps)>1"},
  {"claim": "Q25: c* proportional to sqrt(Tc/M): doubling Tc raises c* by sqrt(2)",
   "code": "import sympy as sp; ratio=sp.sqrt(2); result = abs(float(ratio)-1.4142)<0.001"}
]