Space Dynamics & Orbital Mechanics

### **Official Solution & Step-by-Step Derivation (GATE AE 2026 Q.51)** **1. Governing Concept & Theory**: Governed by principles of **Space Dynamics & Orbital Mechanics** (Aircraft Performance & Stability, 2 Mark, NAT). **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 **1.5**. **3. Official Answer Verification**: - **Official Result**: **1.5** (Acceptable Range: `[1.5, 1.5]`) - **Evaluation Status**: Matches official GATE 2026 master key with 100% accuracy.