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Question 47
Question Statement:
An aircraft with a turbojet engine, having an inlet area of 1 m$^2$, is flying at a speed of 250 m/s at an altitude where air density is 0.4 kg/m$^3$. The mass flow rate of fuel added is negligible. If the jet exhaust speed is 600 m/s relative to the aircraft, the net thrust (in kN) produced by the engine is _________ (round off to the nearest integer).
Official Answer Key:
35.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.47)**
**1. Intuitive Physical Setup & Given Parameters**:
- Flight speed: $V_0 = 250\text{ m/s}$
- Air density at altitude: $\rho_0 = 0.4\text{ kg/m}^3$
- Engine inlet capture area: $A_i = 1\text{ m}^2$
- Jet exhaust velocity: $V_j = 600\text{ m/s}$
- Fuel mass flow rate is negligible ($\dot{m}_f \ll \dot{m}_a$).
**2. Step-by-Step Mathematical Derivation**:
1. **Mass Flow Rate of Air ($\dot{m}_a$)**:
$\dot{m}_a = \rho_0 A_i V_0 = 0.4 \times 1 \times 250 = 100\text{ kg/s}$
2. **Net Thrust Equation**:
For a fully expanded nozzle ($P_j = P_0$):
$F_{net} = \dot{m}_a (V_j - V_0)$
3. **Evaluating Net Thrust**:
$F_{net} = 100\text{ kg/s} \times (600 - 250)\text{ m/s} = 100 \times 350 = 35000\text{ N} = 35\text{ kN}$
The net thrust produced is exactly **35 kN**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Evaluation**: **35.0 kN** (Acceptable Range: [34, 36]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **35.0**
- **Key Formula**: $F_{net} = \dot{m}_a (V_j - V_0)$.
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