An aircraft is to be designed to ensure that it has enough excess power for climb. If the engine produces 500 kW of power and the power required for steady level flight at speed $V$ is 350 kW for an aircraft weight of 12.5 kN, the maximum rate of climb (in m/s) at this speed is _________ (round off to 1 decimal place).
Official Answer Key:
12.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.43)**
**1. Intuitive Physical Setup & Given Parameters**:
- Power available from engine/propeller: $P_A = 500\text{ kW} = 500 \times 10^3\text{ W}$.
- Power required for steady level flight: $P_R = 350\text{ kW} = 350 \times 10^3\text{ W}$.
- Aircraft weight: $W = 12.5\text{ kN} = 12500\text{ N}$.
**2. Step-by-Step Mathematical Derivation**:
1. **Rate of Climb and Excess Power Relation**:
From energy principles, rate of climb ($R/C$ or $V_v$) is equal to excess power divided by aircraft weight:
$R/C = \frac{P_{excess}}{W} = \frac{P_A - P_R}{W}$
2. **Substituting Given Numbers**:
$P_{excess} = 500\text{ kW} - 350\text{ kW} = 150\text{ kW} = 150000\text{ W}$
$R/C = \frac{150000\text{ W}}{12500\text{ N}} = \frac{1500}{125} = 12.0\text{ m/s}$
The aircraft climbs at exactly **12.0 m/s**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Evaluation**: **12.0 m/s** (Acceptable Range: [11.5, 12.5]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **12.0**
- **Key Formula**: $R/C = \frac{P_A - P_R}{W}$.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.