Subtopic: Longitudinal Static Stability & Neutral Point
Question Statement:
A rocket is to be launched from the bottom of a very deep crater on Mars for earth return. The
specific impulse of the rocket, measured in seconds, is to be normalized by the acceleration due to
gravity at
Options:
(A) the bottom of the crater on Mars.
(B) Mars standard "sea level".
(C) earth's standard sea level.
(D) the same depth of the crater on earth.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.23)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Static Stability & Control (Longitudinal Static Stability & Neutral Point)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> A rocket is to be launched from the bottom of a very deep crater on Mars for earth return. The specific impulse of the rocket, measured in seconds, is to be normalized by the acceleration due to gravity at
**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 **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"the bottom of the crater on Mars."*: ❌ **INCORRECT**.
- **(B)** *"Mars standard "sea level"."*: ❌ **INCORRECT**.
- **(C)** *"earth's standard sea level."*: ✅ **CORRECT**.
- **(D)** *"the same depth of the crater on earth."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.
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