Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
An aircraft starts gliding in power-off condition at an altitude of 4 km. Given that
the maximum lift to drag ratio of the aircraft is 15, the maximum glide range that
the aircraft can cover, measured along the ground, is ________ km (rounded off to
the nearest integer).
Official Answer Key:
60.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.31)**
**1. Intuitive Physical Setup & Given Parameters**:
- Gliding flight in power-off condition ($T = 0$).
- Initial altitude: $h = 4 ext{ km} = 4000 ext{ m}$.
- Maximum lift-to-drag ratio: $(L/D)_{ ext{max}} = 15$.
- We need to calculate the **maximum glide range** ($R_{ ext{max}}$) measured along the ground in kilometers.
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Gliding Flight Mechanics**:
- In steady, unpowered gliding flight at glide angle $\gamma$ (where $\gamma > 0$ below horizontal):
$L = W \cos\gamma$
$D = W \sin\gamma$
- Dividing the drag equation by the lift equation:
$ an\gamma = rac{D}{L} = rac{1}{(L/D)}$
2. **Glide Range Formulation**:
- From geometry, the horizontal ground distance $R$ covered while descending through altitude $\Delta h$ is:
$R = rac{\Delta h}{ an\gamma} = \Delta h \left( rac{L}{D}
ight)$
3. **Maximum Glide Range**:
- To maximize ground distance $R$, the glide angle $\gamma$ must be minimized, which occurs when the lift-to-drag ratio is maximized:
$R_{ ext{max}} = h \left( rac{L}{D}
ight)_{ ext{max}}$
4. **Numerical Calculation**:
- Given $h = 4 ext{ km}$ and $(L/D)_{ ext{max}} = 15$:
$R_{ ext{max}} = 4 ext{ km} imes 15 = 60 ext{ km}$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $60 ext{ km}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{60}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **60.0** (Acceptable Range: **60 to 60**)
- **Exam Takeaway**: Maximum glide range depends purely on initial altitude and maximum aerodynamic efficiency: $R_{ ext{max}} = h (L/D)_{ ext{max}}$, completely independent of aircraft weight!
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