Gross weight of an airplane is 7000 N, wing area is 16 m$^2$, and the maximum lift coefficient is 2.0. Assuming air density at the altitude as 1.23 kg/m$^3$, the stall speed (in m/s) of the aircraft is _________ (accurate to two decimal places).
Official Answer Key:
18.85
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2018 Q.58 / AE Q.48)**
**1. Intuitive Physical Setup & Given Parameters**:
- Aircraft Weight: $W = 7000\text{ N}$
- Wing Area: $S = 16\text{ m}^2$
- Maximum Lift Coefficient: $C_{L,\max} = 2.0$
- Air Density: $\rho = 1.23\text{ kg/m}^3$
**2. Step-by-Step Mathematical Derivation**:
1. **Steady Level Flight Lift Equation**:
At stall speed $V_{stall}$, the airplane produces lift equal to weight at its maximum lift coefficient $C_{L,\max}$:
L = W = \frac{1}{2} \rho V_{stall}^2 S C_{L,\max}
2. **Solving for Stall Speed $V_{stall}$**:
V_{stall} = \sqrt{\frac{2W}{\rho S C_{L,\max}}}
3. **Substituting Numerical Values**:
V_{stall} = \sqrt{\frac{2 \times 7000}{1.23 \times 16 \times 2.0}} = \sqrt{\frac{14000}{39.36}} = \sqrt{355.691} \approx 18.8597\text{ m/s}
Rounding to two decimal places:
V_{stall} \approx 18.86\text{ m/s}
This falls exactly in the official interval **[18.80, 18.90]**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **18.80 to 18.90 m/s**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **18.85** (Acceptable Range: **18.80 to 18.90**)
- **Key Formula**: $V_{stall} = \sqrt{\frac{2(W/S)}{\rho C_{L,\max}}}$.