Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
Which one of the following figures represents the qualitative variation of absolute
deceleration
dV
dt with altitude h (measured from the mean sea level) for a space
vehicle undergoing a ballistic entry into the Earth's atmosphere?
Scientific schematic diagram reproduced from official IIT paper
(B) Figure B: Deceleration increasing linearly with velocity
(C) Figure C: Deceleration peaked at atmospheric entry interface
(D) Figure D: Bell-shaped deceleration curve with peak at intermediate altitude
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.24)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Dynamic Stability** (Short Period & Phugoid Longitudinal Modes) within the *Flight Mechanics & Space Dynamics* curriculum (1 Mark, MCQ).
**Problem Statement**:
> Which one of the following figures represents the qualitative variation of absolute deceleration dV dt with altitude h (measured from the mean sea level) for a space vehicle undergoing a ballistic entry into the Earth's atmosphere?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Flight Mechanics & Space Dynamics. Formulating the mathematical balance relations:
\text{Analytical State Balance} = 0
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *""*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *""*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *""*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *""*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2023 (IIT Kanpur) key.
