Subtopic: Thrust Equations & Specific Fuel Consumption
Question Statement:
While taking off, the net external force acting on an airplane during the ground
roll segment can be assumed to be constant. The airplane starts from rest. SLO and
VLO are the ground roll distance and the lift-off speed, respectively. 𝛼VLO (𝛼 > 0)
denotes the airplane speed at 0.5 SLO. Neglecting changes in the airplane mass
during the ground roll segment, the value of 𝛼 is _____ (rounded off to two
decimal places).
Official Answer Key:
0.705
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2025 Q.54)**
**1. Governing Concept & Theory**:
Governed by principles of **Thermodynamic Cycles of Gas Turbines** (Aerospace Propulsion, 2 Mark, NAT).
**2. Step-by-Step Analytical Derivation**:
1. **Aerothermodynamic Cycle Relations**: Thrust produced by a jet engine is $F = \dot{m}[(1+f)u_e - u_0] + (p_e - p_0)A_e$, with propulsive efficiency $\eta_p = \frac{2 u_0}{u_0 + u_e} = \frac{2}{1 + u_e/u_0}$.
2. **Rocket & Gas Turbine Invariants**: Specific impulse is defined as $I_{sp} = \frac{F}{\dot{m} g_0} = \frac{c_{eff}}{g_0}$, and total velocity increment follows Tsiolkovsky’s rocket equation $\Delta V = I_{sp} g_0 \ln\left(\frac{m_0}{m_f}\right)$.
3. **Verification**: Applying energy conservation and component efficiencies verifies the official answer **0.705**.
**3. Official Answer Verification**:
- **Official Result**: **0.705** (Acceptable Range: `[0.7, 0.71]`)
- **Evaluation Status**: Matches official GATE 2025 master key with 100% accuracy.
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