Consider a conventional subsonic fixed-wing airplane. $e$ is the Oswald efficiency factor and $AR$ is the aspect ratio. Corresponding to the minimum $\left( \frac{C_D}{C_L^{3/2}} \right)$, which of the following relations is true?
Scientific schematic diagram reproduced from official IIT paper
Options:
(A) $\frac{C_D}{C_L^2} = \frac{1}{\pi e AR}$
(B) $\frac{C_D}{C_L^2} = \frac{4}{3\pi e AR}$
(C) $\frac{C_D}{C_L} = \frac{1}{\pi e AR}$
(D) $\frac{C_D}{\sqrt{C_L}} = \frac{1}{\sqrt{\pi e AR}}$
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.40)**
**1. Minimum Power Required Condition**:
Minimizing $\frac{C_D}{C_L^{3/2}}$ (minimum power required / maximum endurance for jet):
$$C_D = C_{D,0} + K C_L^2, \quad K = \frac{1}{\pi e AR}$$
Minimizing $\frac{C_{D,0} + K C_L^2}{C_L^{3/2}}$ with respect to $C_L$:
$$\frac{d}{dC_L} \left( C_{D,0} C_L^{-3/2} + K C_L^{1/2} \right) = -\frac{3}{2} C_{D,0} C_L^{-5/2} + \frac{1}{2} K C_L^{-1/2} = 0$$
$$\frac{1}{2} K = \frac{3}{2} \frac{C_{D,0}}{C_L^2} \implies C_{D,0} = \frac{1}{3} K C_L^2$$
**2. Total Drag Coefficient**:
$$C_D = C_{D,0} + K C_L^2 = \frac{1}{3} K C_L^2 + K C_L^2 = \frac{4}{3} K C_L^2 = \frac{4}{3} \frac{C_L^2}{\pi e AR}$$
$$\frac{C_D}{C_L^2} = \frac{4}{3\pi e AR}$$
- **Official Answer Key**: **B**
