Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory
Question Statement:
An ideal glider has drag characteristics given by
0D D D i
C C C=+ , where
2
DLi
C KC=
is the induced drag coefficient,
LC is the lift coefficient, and K is a
constant. For maximum range of the glider, the ratio
0DD i
CC is
Options:
(A) 1
(B) 1
3
(C) 3
(D) 3
2
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.17)**
**1. Intuitive Physical Setup & Given Parameters**:
An ideal unpowered glider operates with parabolic drag polar $C_D = C_{D0} + K C_L^2$.
We are asked to find the condition for maximum glide range (minimum glide angle $\theta_{\min}$).
**2. Step-by-Step Mathematical Derivation**:
1. **Glide Geometry**:
In steady unpowered gliding flight, the glide angle $\theta$ relates lift and drag by equilibrium:
$\tan\theta = \frac{D}{L} = \frac{C_D}{C_L}$
The glide distance is $R = \frac{h}{\tan\theta} = h \left(\frac{L}{D}\right)$.
To maximize glide distance from a given altitude $h$, the glide angle $\theta$ must be minimized, which requires maximizing the lift-to-drag ratio $(L/D)_{\max}$.
2. **Maximizing Lift-to-Drag Ratio**:
$\frac{C_D}{C_L} = \frac{C_{D0}}{C_L} + K C_L$
Differentiating with respect to $C_L$ and setting to zero:
$\frac{d}{dC_L}\left(\frac{C_D}{C_L}\right) = -\frac{C_{D0}}{C_L^2} + K = 0 \implies K = \frac{C_{D0}}{C_L^2} \implies C_{D0} = K C_L^2 = C_{Di}$
3. **Conclusion**:
Maximum glide range occurs when zero-lift parasite drag equals induced drag ($C_{D0} = C_{Di}$), so total drag is $C_D = 2C_{D0}$, and the optimum lift coefficient is $C_L = \sqrt{C_{D0}/K}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) C_D0 = C_Di**: ✅ **CORRECT**. Maximum range for an unpowered glider occurs at $(L/D)_{\max}$, where parasite drag equals induced drag ($C_{D0} = K C_L^2$).
- **(B) C_D0 = 3 C_Di**: ❌ **INCORRECT**. This condition maximizes $(C_L^{3/2}/C_D)$, which corresponds to minimum sink rate (maximum endurance).
- **(C) C_D0 = 1/3 C_Di**: ❌ **INCORRECT**. Inverts the factor of 3.
- **(D) C_Di = 0**: ❌ **INCORRECT**. Impossible for a lifting wing ($C_L > 0$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: For gliders: Maximum Glide Range requires $(L/D)_{\max} \implies C_{D0} = C_{Di}$. Minimum Sink Rate (Maximum Glide Endurance) requires $(C_L^{3/2}/C_D)_{\max} \implies C_{D0} = \frac{1}{3}C_{Di}$.
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