Rocket Propulsion – Propulsive Efficiency Formula If V1 denotes the jet (exhaust) velocity relative to the vehicle and V0 is the vehicle flight velocity, which expression gives the propulsive efficiency of an ideal rocket?

Introduction / Context:
Propulsive efficiency measures how effectively a propulsion device converts the power in its jet into useful thrust power on the vehicle. For rockets (self-contained propellant), the classical one-dimensional relation links the vehicle speed V0 and jet speed V1. Knowing this formula helps reason about why very high jet speeds are inefficient at low flight speeds.

Given Data / Assumptions:

• Steady, 1-D ideal rocket with fully expanded exhaust.
• Neglect pressure thrust and nozzle inefficiencies for simplicity.
• Jet speed V1 is relative to the rocket; flight speed is V0.

Concept / Approach:

Propulsive efficiency is defined as useful power divided by rate of energy output in the jet. Useful power = T * V0, where thrust T ≈ m_dot * (V1 − V0) in the ideal model (neglecting pressure terms). Jet power available = 0.5 * m_dot * (V1^2 − V0^2) when referenced to the vehicle frame. Taking the ratio and simplifying yields eta_p = 2 * V0 / (V1 + V0), a compact expression commonly cited for rockets.

Step-by-Step Solution:

Verification / Alternative check:

Limits check: as V0 → V1, eta_p → 1 (ideal matching). As V0 → 0 (static firing), eta_p → 0, meaning no useful propulsive power though thrust exists—consistent with intuition.

Why Other Options Are Wrong:

Other forms either invert the ratio or omit the factor of 2, giving incorrect limiting behavior or values outside [0, 1].

Common Pitfalls:

Confusing propulsive efficiency with overall efficiency (which also includes conversion of chemical energy to jet kinetic energy), or using air-breathing formulas directly for rockets.

Final Answer:

eta_p = 2 * V0 / (V1 + V0)