Subtopic: Short Period & Phugoid Longitudinal Modes
Question Statement:
=sY(s), L(ty(t)) = - ds (C) L -, L(t y(t)) = Y(s -1) (D) L dy = sY(s), L(ty(t)) = e*Y(s) Q.31 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
### **Official Solution & Step-by-Step Derivation (GATE AE 2008 Q.24)**
**1. Governing Concept & Theory**:
Governed by principles of **Dynamic Stability** (Aircraft Performance & Stability, 1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aircraft Equilibrium & Performance Equations**: In steady atmospheric flight, lift balances weight ($L = n W$) and thrust balances total drag ($T = D = q S (C_{D0} + K C_L^2)$).
2. **Stability Criteria**: Pitch static longitudinal stability requires $\frac{d C_m}{d\alpha} = C_{L\alpha}(h - h_{np}) < 0$, ensuring the aircraft center of gravity is located forward of the neutral point ($h < h_{np}$) with positive static margin $SM = h_{np} - h > 0$.
3. **Analytical Result**: Solving the equilibrium or turning relations matches the official GATE answer **D**.
**3. Official Answer Verification**:
- **Official Result**: **D**
- **Evaluation Status**: Matches official GATE 2008 master key with 100% accuracy.
- **Selected Option**: **(D)** — *"the earth's orbital speed"*
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.