Level 2 — RecallRocket Flight Mechanics

Rocket Flight Mechanics

⏱ 30 minutes50 marksprintable — key stays hidden on paper

Level: 2 (Recall / Standard problems / Short derivations) Time Limit: 30 minutes Total Marks: 50

Use $...$ for inline math and $...$ for displayed equations. Take $g = 9.81\ \text{m/s}^2$ unless stated.

Q1. (4 marks) Define the following coordinate frames used in rocket flight mechanics: (a) Earth-Centered Inertial (ECI) (b) Earth-Centered Earth-Fixed (ECEF) (c) North-East-Down (NED) (d) Body frame

Q2. (5 marks) A finned rocket has centre of pressure at $X_{CP} = 1.85\ \text{m}$ from the nose and centre of gravity at $X_{CG} = 1.40\ \text{m}$ from the nose. The body diameter is $d = 0.15\ \text{m}$ . (a) Compute the static margin (in calibers). (3) (b) State whether the rocket is statically stable and justify. (2)

Q3. (6 marks) State the three principal forces acting on a rocket in atmospheric flight. Give the expression for aerodynamic axial force and normal force in terms of dynamic pressure $q$ , reference area $S$ , and coefficients $C_A$ , $C_N$ .

Q4. (6 marks) A rocket flies at $v = 340\ \text{m/s}$ at an altitude where $\rho = 0.65\ \text{kg/m}^3$ . (a) Compute the dynamic pressure $q$ . (3) (b) Explain physically what "Max-Q" is and why it is a structural design concern. (3)

Q5. (5 marks) Define the ballistic coefficient $\beta$ for reentry. A capsule has mass $m = 4200\ \text{kg}$ , drag coefficient $C_D = 1.4$ , and reference area $A = 12\ \text{m}^2$ . Compute $\beta$ and state qualitatively how a higher $\beta$ affects deceleration altitude.

Q6. (6 marks) Write the 3DOF point-mass equations of motion for a rocket in a vertical plane (flight-path angle $\gamma$ ), including thrust $T$ , drag $D$ , mass $m$ , and gravity $g$ . State one simplifying assumption of the 3DOF (point-mass) model relative to 6DOF.

Q7. (5 marks) (a) Define a gravity-turn trajectory. (2) (b) State the condition on angle of attack that characterizes an ideal gravity turn and explain why this minimizes structural loads. (3)

Q8. (4 marks) Explain the communications blackout during reentry. Name the physical mechanism (plasma sheath) and state one method used to mitigate or work around it.

Q9. (5 marks) A single-stage rocket with exhaust velocity $v_e = 2800\ \text{m/s}$ has initial mass $m_0 = 5000\ \text{kg}$ and final mass $m_f = 2000\ \text{kg}$ . Using the Tsiolkovsky rocket equation, compute the ideal $\Delta v$ (ignore gravity and drag losses).

Q10. (4 marks) Briefly define the following, one sentence each: (a) Aerocapture (b) Aerobraking (c) Suicide burn (terminal propulsive landing) (d) Static stability / weather-cocking

Answer keyMark scheme & solutions

Q1. (4 marks — 1 each)

(a) ECI: Non-rotating frame with origin at Earth's centre, axes fixed relative to the stars (inertial); Newton's laws apply directly. (1)
(b) ECEF: Origin at Earth's centre but rotates with the Earth; convenient for expressing ground positions (lat/long/altitude). (1)
(c) NED: Local topocentric frame with axes pointing North, East, and Down (toward Earth's centre) at the vehicle location. (1)
(d) Body frame: Fixed to the rocket, origin usually at CG, with axes along the roll (longitudinal), pitch, and yaw axes of the vehicle. (1)

Q2. (5 marks) (a) Static margin $= \dfrac{X_{CP}-X_{CG}}{d} = \dfrac{1.85-1.40}{0.15} = \dfrac{0.45}{0.15} = 3.0$ calibers. (3) Why: Static margin is the CP–CG separation normalized by body diameter (caliber). (b) The rocket is statically stable because $X_{CP}$ is behind $X_{CG}$ (positive margin) and the margin (3.0) exceeds the minimum recommended 1 caliber. (2)

Q3. (6 marks)

Three forces: Thrust (propulsive), Aerodynamic force (axial + normal, i.e. drag/lift), and Gravity. (3, 1 each)
Axial force: $F_A = q S C_A$ where $q=\tfrac12\rho v^2$ . (1.5)
Normal force: $F_N = q S C_N$ . (1.5)

Q4. (6 marks) (a) $q = \tfrac12 \rho v^2 = \tfrac12 (0.65)(340)^2 = 0.5 \times 0.65 \times 115600 = 37570\ \text{Pa} \approx 37.6\ \text{kPa}$ . (3) (b) Max-Q is the point in ascent where dynamic pressure $q=\tfrac12\rho v^2$ reaches its maximum; density falls with altitude while speed rises, so the product peaks at intermediate altitude. It is critical because aerodynamic loads (and bending moments) on the structure are greatest there; vehicles often throttle down near Max-Q. (3)

Q5. (5 marks) Definition: $\beta = \dfrac{m}{C_D A}$ (mass per unit drag area). (2) $\beta = \dfrac{4200}{1.4 \times 12} = \dfrac{4200}{16.8} = 250\ \text{kg/m}^2$ . (2) Higher $\beta$ ⇒ the body penetrates deeper before decelerating, so peak deceleration occurs at lower altitude (and higher heating). (1)

Q6. (6 marks) Planar 3DOF point-mass EOM: $m\dot v = T\cos\alpha - D - mg\sin\gamma$ $mv\dot\gamma = T\sin\alpha + L - mg\cos\gamma$ (often written with $\alpha\approx0$ : $m\dot v = T-D-mg\sin\gamma$ , $mv\dot\gamma=-mg\cos\gamma$ ). (4) Assumption: The point-mass model treats the rocket as a point (no rotational dynamics / attitude), ignoring moments, inertia tensor, and body-rate dynamics captured by 6DOF. (2)

Q7. (5 marks) (a) A gravity turn is an ascent trajectory in which gravity alone (not control forces) gradually rotates the velocity vector, letting the vehicle pitch over naturally after an initial kick. (2) (b) Condition: angle of attack $\alpha = 0$ throughout the turn (thrust/velocity aligned with body axis). (2) This makes aerodynamic normal force $\approx 0$ , minimizing side/bending loads on the structure. (1)

Q8. (4 marks) During reentry, extreme heating ionizes the air around the vehicle, forming a plasma sheath of free electrons that reflects/absorbs radio waves, cutting off communication. (2) Mitigation: use higher-frequency links, relay through a satellite behind the vehicle (e.g. TDRSS via aft antenna), or aerodynamic shaping/electrophilic injection to reduce electron density. (2)

Q9. (5 marks) Tsiolkovsky: $\Delta v = v_e \ln\!\dfrac{m_0}{m_f} = 2800 \ln\!\dfrac{5000}{2000} = 2800 \ln 2.5$ . (3) $\ln 2.5 = 0.91629$ , so $\Delta v = 2800 \times 0.91629 = 2565.6\ \text{m/s} \approx 2.57\ \text{km/s}$ . (2)

Q10. (4 marks — 1 each)

(a) Aerocapture: A single deep atmospheric pass used to decelerate a spacecraft directly into a captured orbit. (1)
(b) Aerobraking: Repeated shallow atmospheric passes gradually lowering an orbit's apoapsis using drag. (1)
(c) Suicide burn: A single, precisely timed maximum-thrust deceleration burn that brings the vehicle to zero velocity exactly at touchdown. (1)
(d) Static stability / weather-cocking: The tendency of a rocket with CP aft of CG to rotate back into the relative wind (align with the airflow), restoring zero angle of attack. (1)

[
  {"claim":"Static margin Q2 = 3.0 calibers","code":"result = ((1.85-1.40)/0.15) == 3.0"},
  {"claim":"Dynamic pressure Q4 = 37570 Pa","code":"result = abs(0.5*0.65*340**2 - 37570) < 1"},
  {"claim":"Ballistic coefficient Q5 = 250 kg/m^2","code":"result = abs(4200/(1.4*12) - 250) < 1e-6"},
  {"claim":"Delta v Q9 approx 2565.6 m/s","code":"import sympy as sp; result = abs(2800*sp.log(sp.Rational(5000,2000)) - 2565.6) < 1"}
]