Question

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?

• A. 0.50
• B. 1.00
• C. 0.33
• D. 0.67

Hint:

For a pure jet with small losses, remember the compact form \(\eta_p=\dfrac{2V_a}{V_j+V_a}\). Maximum \(\eta_p\) occurs when the jet slip \(V_j-V_a\) is small.

Answer 1

The Correct Option is D

Step 1: Definition of propulsive efficiency.
For an ideal (momentum) jet, the propulsive efficiency is \[ \eta_p \;=\; \frac{\text{useful power}}{\text{jet kinetic power input}} \;=\; \frac{T\,V_a}{\tfrac{1}{2}\dot m (V_j^2 - V_a^2)} , \] where \(T=\dot m (V_j - V_a)\) is the ideal thrust.

Step 2: Simplify.
Substitute \(T\): \[ \eta_p = \frac{\dot m (V_j - V_a)V_a}{\tfrac{1}{2}\dot m (V_j^2 - V_a^2)} = \frac{2 V_a (V_j - V_a)}{(V_j - V_a)(V_j + V_a)} = \frac{2V_a}{V_j + V_a}. \]

Step 3: Insert the given speed ratio.
\(V_j = 2V_a \Rightarrow \eta_p = \dfrac{2V_a}{2V_a + V_a} = \dfrac{2}{3} = 0.67.\)

Final Answer:
\[ \boxed{0.67} \]