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 $V_g = \sqrt{\frac{2(W/S)}{\rho C_{L,(L/D)_{\text{max}}}}} \propto \sqrt{\frac{W/S}{\rho}}$.