The power-off glide range for an airplane with a maximum Lift to Drag ratio of 18, when the glide starts from an altitude of 4 km above ground level, is _________ km (round off to the nearest integer).
Official Answer Key:
72.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2019 Q.27 / AE Q.17)**
**1. Intuitive Physical Setup & Given Parameters**:
- Maximum lift-to-drag ratio: $(L/D)_{\max} = 18$
- Initial glide altitude: $h = 4\text{ km} = 4000\text{ m}$ above ground level.
- Unpowered (power-off) equilibrium descent.
**2. Step-by-Step Mathematical Derivation**:
1. **Gliding Flight Equilibrium**:
In steady, power-off glide at glide angle $\gamma$:
$\tan\gamma = \frac{D}{L} = \frac{1}{L/D}$
2. **Glide Range ($R$) Relation**:
From trigonometry of the descent path:
$R = \frac{h}{\tan\gamma} = h \left( \frac{L}{D} \right)$
3. **Maximum Glide Range**:
To maximize range $R$, the airplane must fly at the maximum lift-to-drag ratio $(L/D)_{\max}$:
$R_{\max} = h \times (L/D)_{\max}$
4. **Numerical Calculation**:
$R_{\max} = 4\text{ km} \times 18 = 72\text{ km}$
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: **72.0 km** (Acceptable Range: [72, 72]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **72.0**
- **Key Formula**: $R = h \cdot (L/D)_{\max}$ for power-off glide.
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