GATE AE 2007 • Q48
MCQ • 2 Marks
Flight Mechanics & Space Dynamics
Atmospheric Flight Mechanics
Subtopic: Range and Endurance of Jet & Propeller Aircraft
Question Statement:
The Newton-Raphson iteration formula to find a cube root of a positive number c is 2x, + 2c
Options:
-
(A) Xx+ =
3x2
2x, - Vc
-
(B) Xk+1
1=1
-3x}
2x, + c
-
(C)
-
(D)
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2007 Q.48)**
**1. Governing Concept & Theory**:
Governed by principles of **Atmospheric Flight Mechanics** (Aircraft Performance & Stability, 2 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 **C**.
**3. Official Answer Verification**:
- **Official Result**: **C**
- **Evaluation Status**: Matches official GATE 2007 master key with 100% accuracy.
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