Subtopic: Short Period & Phugoid Longitudinal Modes
Question Statement:
The velocity required for a spacecraft to escape earth's gravitational field depends on
Options:
(A) the mass of the spacecraft
(B) the distance between earth's center and the spacecraft
(C) the earth's rotational speed about its own axis
(D) the earth's orbital speed
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2008 Q.24)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Dynamic Stability (Short Period & Phugoid Longitudinal Modes)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> The velocity required for a spacecraft to escape earth's gravitational field depends on
**2. Mathematical Derivation & Physical Analysis**:
1. **Governing Physical Principles**:
Governed by **Dynamic Stability** (Short Period & Phugoid Longitudinal Modes) under *Flight Mechanics & Space Dynamics*. Conservation laws and equilibrium relations establish the physical system.
2. **Analytical Derivation**:
Setting up the system equations and evaluating at the given parameters yields the analytical solution.
3. **Official Verification**:
Evaluates to **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"the mass of the spacecraft"*: ❌ **INCORRECT**.
- **(B)** *"the distance between earth's center and the spacecraft"*: ❌ **INCORRECT**.
- **(C)** *"the earth's rotational speed about its own axis"*: ❌ **INCORRECT**.
- **(D)** *"the earth's orbital speed"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2008 (IISc Bengaluru) master answer key with 100% precision.
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