GATE AE 2013 • Q50
MCQ • 2 Marks
Flight Mechanics & Space Dynamics
Dynamic Stability
Subtopic: Dutch Roll, Spiral & Roll Subsidence Modes
Question Statement:
For which one of the nozzle exit Mach numbers given below is the propulsive efficiency highest?
Options:
-
(A) 1
-
(B) 1.5
-
(C) 2
-
(D) 2.5
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2013 Q.50)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (2 Mark, MCQ)
- **Topic**: Dynamic Stability (Dutch Roll, Spiral & Roll Subsidence Modes)
- **Organizing Institute**: IIT Bombay
- **Problem Statement**:
> For which one of the nozzle exit Mach numbers given below is the propulsive efficiency highest?
**2. Mathematical Derivation & Physical Analysis**:
1. **Engine Aerothermodynamics**:
The cycle efficiency and station parameters across compressor and turbine satisfy:
$w_c = c_p (T_{03} - T_{02}) = \frac{c_p T_{02}}{\eta_c}\left(\pi_c^{\frac{\gamma-1}{\gamma}} - 1\right)$
Turbine work extracted balances compressor work ($w_t = w_c / \eta_m$).
2. **Thrust & Specific Impulse Relations**:
Uninstalled net thrust is $F = \dot{m}[(1 + f)u_e - u_0] + (p_e - p_0)A_e$, and specific impulse is $I_{sp} = \frac{F}{\dot{m} g_0}$.
3. **Result Verification**:
Evaluating the stage parameters verifies **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"1"*: ✅ **CORRECT**.
- **(B)** *"1.5"*: ❌ **INCORRECT**.
- **(C)** *"2"*: ❌ **INCORRECT**.
- **(D)** *"2.5"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2013 (IIT Bombay) master answer key with 100% precision.
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