Question Statement:
The Joukowskii airfoil is studied in aerodynamics because
Options:
-
(A) it is used in many aircraft
-
(B) it is easily transformed into a circle, mathematically
-
(C) it has a simple geometry
-
(D) it has the highest lift curve slope among all airfoils
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.14)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Differential Equations (First Order ODEs)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> The Joukowskii airfoil is studied in aerodynamics because
**2. Mathematical Derivation & Physical Analysis**:
1. **Aerodynamic Theory & Finite Wing Relations**:
Lift and induced drag follow Lanchester-Prandtl lifting-line theory:
$C_L = a_0 (\alpha - \alpha_{L=0} - \alpha_i), \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Induced Drag Minimization**:
For an elliptic lift distribution ($e = 1$), downwash is constant across the span, providing minimum induced drag for a given aspect ratio.
3. **Official Verification**:
Matches **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"it is used in many aircraft"*: ✅ **CORRECT**.
- **(B)** *"it is easily transformed into a circle, mathematically"*: ❌ **INCORRECT**.
- **(C)** *"it has a simple geometry"*: ❌ **INCORRECT**.
- **(D)** *"it has the highest lift curve slope among all airfoils"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.
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