A general aviation airplane is gliding with a speed Vg at minimum glide angle.
Which of the following statements is/are true?
Options:
(A) Vg is equal to the speed corresponding to the maximum lift to drag ratio of the
airplane.
(B) Vg increases with decreasing wing loading when all other parameters remain
constant.
(C) Vg increases with decreasing altitude when all other parameters remain constant.
(D) Vg increases with increasing altitude when all other parameters remain constant.
Official Answer Key:
A, D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.39)**
**1. Intuitive Physical Setup & Given Parameters**:
An airplane glides with airspeed $V_g$ at the **minimum glide angle** $\gamma_{\text{min}}$ (unpowered equilibrium glide).
We need to identify all true statements regarding $V_g$ and its dependence on flight parameters (MSQ).
**2. Step-by-Step Flight Mechanics Derivation**:
1. **Minimum Glide Angle Condition**:
- In unpowered glide: $\tan\gamma = \frac{D}{L} = \frac{1}{(L/D)}$.
- The glide angle $\gamma$ is minimized when the lift-to-drag ratio $(L/D)$ is **maximized**:
$\gamma_{\text{min}} \iff (L/D)_{\text{max}}$
- Therefore, the glide speed $V_g$ corresponding to minimum glide angle is identical to the speed for maximum lift-to-drag ratio ($V_{(L/D)_{\text{max}}}$).
- Statement A is **TRUE**.
2. **Gliding Airspeed Formula**:
- In shallow glide, lift equals weight: $L \approx W = \frac{1}{2}\rho V_g^2 S C_{L,(L/D)_{\text{max}}}$.
- For a parabolic drag polar $C_D = C_{D0} + K C_L^2$, maximum $L/D$ occurs at:
$C_{L,(L/D)_{\text{max}}} = \sqrt{\frac{C_{D0}}{K}}$
- The corresponding true airspeed is:
$V_g = \sqrt{\frac{2 W}{\rho S C_{L,(L/D)_{\text{max}}}}} = \sqrt{\frac{2}{\rho \sqrt{C_{D0}/K}}} \sqrt{\frac{W}{S}}$
3. **Parametric Dependencies of $V_g$**:
- **Wing Loading ($W/S$)**: $V_g \propto \sqrt{W/S}$. If wing loading decreases, $V_g$ **decreases**. Statement B is **FALSE**.
- **Altitude ($h$)**: As altitude increases, air density $\rho$ decreases. Since $V_g \propto 1/\sqrt{\rho}$, true airspeed $V_g$ **increases with increasing altitude**.
- Statement D is **TRUE**, Statement C is **FALSE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Vg is equal to the speed corresponding to the maximum lift to drag ratio of the airplane**: ✅ **CORRECT**. Directly follows from $\tan\gamma = 1/(L/D)$.
- **(B) Vg increases with decreasing wing loading when all other parameters remain constant**: ❌ **INCORRECT**. $V_g \propto \sqrt{W/S}$, so lighter aircraft glide slower.
- **(C) Vg increases with decreasing altitude when all other parameters remain constant**: ❌ **INCORRECT**. Decreasing altitude increases density, lowering TAS.
- **(D) Vg increases with increasing altitude when all other parameters remain constant**: ✅ **CORRECT**. Density decreases with altitude, requiring higher TAS to sustain lift.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, D**
- **Exam Takeaway**: Minimum glide angle depends only on $(L/D)_{\text{max}}$, but the speed to achieve it scales as $V_g = \sqrt{\frac{2(W/S)}{\rho C_{L,(L/D)_{\text{max}}}}} \propto \sqrt{\frac{W/S}{\rho}}$.
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