(D) Figure D: Parabola opening to the right with non-zero parasite drag intercept $C_{D0} > 0$
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.20)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the graphical representation of the drag polar ($C_D$ vs $C_L$) for a conventional subsonic aircraft.
**2. Step-by-Step Mathematical Derivation**:
1. **Parabolic Drag Polar Formulation**:
For a subsonic aircraft at low to moderate angles of attack, the total drag coefficient is expressed as:
C_D = C_{D0} + K C_L^2 = C_{D0} + \frac{C_L^2}{\pi e AR}
where $C_{D0}$ is the zero-lift parasite drag coefficient, and $K C_L^2$ is the induced (lift-dependent) drag coefficient.
2. **Geometric Properties of the Curve**:
- When $C_L = 0$, $C_D = C_{D0} > 0$ (positive intercept on the horizontal $C_D$ axis).
- The curve is a parabola symmetric about the horizontal axis (or slightly shifted upward for cambered wings: $C_D = C_{D,\min} + K(C_L - C_{L0})^2$).
- $C_D$ increases quadratically with $C_L$.
3. **Graph Identification**:
The plot of $C_L$ (vertical axis) versus $C_D$ (horizontal axis) is a parabola opening to the right, originating from $C_D = C_{D0}$ at $C_L = 0$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)**: ❌ **INCORRECT**. Shows a straight line ($C_D \propto C_L$).
- **(B)**: ❌ **INCORRECT**. Passes through the origin $(0, 0)$, neglecting parasite drag ($C_{D0} = 0$).
- **(C)**: ❌ **INCORRECT**. Shows negative curvature ($d^2C_D/dC_L^2 < 0$).
- **(D)**: ✅ **CORRECT**. Correct parabolic shape opening to the right with non-zero positive intercept $C_{D0}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Drag polar is parabolic: $C_D = C_{D0} + K C_L^2$. A tangent drawn from the origin to the drag polar gives the point of maximum aerodynamic efficiency $(L/D)_{\max}$.
