Discussion
An aircraft is cruising with a forward speed \(V_{a}\) and the jet exhaust speed relative to the engine at the exit is \(V_{j}\). If \(V_{j}/V_{a} = 2\), what is the propulsive efficiency?
Answer: D
The propulsive efficiency is 0.67.
For a jet engine with negligible fuel mass flow and a fully expanded exhaust, propulsive efficiency is thrust power divided by the rate of increase of the air’s kinetic energy:
\[ \begin{aligned} \eta_{p} &= \frac{2V_{a}}{V_{a} + V_{j}} = \frac{2}{1 + V_{j}/V_{a}} \\ &= \frac{2}{1 + 2} = 0.67 \end{aligned} \]
0.50 is the ratio \(V_{a}/V_{j}\), not the efficiency. 1.00 would require the jet to leave at flight speed, which produces no thrust. 0.33 is \(1/(1 + V_{j}/V_{a})\), the formula without its factor 2.
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