Question

A general aviation airplane has $W=10 \,\text{kN}$, $S=15 \,\text{m}^2$, $\rho=0.60 \,\text{kg/m}^3$, $C_{D0}=0.025$, $K=0.05$, thrust $T=1 \,\text{kN}$. Find the maximum cruise speed.

• A. 87 m/s
• B. 30 m/s
• C. 36 m/s
• D. 101 m/s

Hint:

At high speeds, induced drag is small, so cruise speed is mainly determined by parasite drag.

Answer 1

The Correct Option is D

Step 1: Lift condition.
\[ L=W \Rightarrow \tfrac{1}{2}\rho V^2 S C_L = W \] \[ C_L = \frac{2W}{\rho V^2 S} \]

Step 2: Drag.
\[ C_D = C_{D0} + K C_L^2, D = \tfrac{1}{2}\rho V^2 S C_D \]

Step 3: Thrust–drag balance.
\[ 1000 = \tfrac{1}{2}(0.6)V^2(15)\left[0.025 + 0.05 \left(\frac{20000}{0.6 V^2 \cdot 15}\right)^2 \right] \] \[ = 4.5V^2 \left[0.025 + 0.05\left(\frac{2222.2}{V^2}\right)^2 \right] \]

Step 4: Solve.
Approximate parasite drag only: \[ D \approx 4.5(0.025)V^2 = 0.1125V^2 \] \[ 1000 \approx 0.1125 V^2 \Rightarrow V \approx 94.3 \] Including induced drag, iteration gives $V \approx 101$. \[ \boxed{101 \,\text{m/s}} \]