Subtopic: Longitudinal Static Stability & Neutral Point
Question Statement:
Consider a steady, isentropic, supersonic flow (Mach number M > 1) entering a
Convergent-Divergent (CD) duct as shown in the figure. Which one of the
following options correctly describes the flow at the throat?
Options:
(A) Can only be supersonic
(B) Can only be sonic
(C) Can either be sonic or supersonic
(D) Can only be subsonic
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.23)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Static Stability & Control** (Longitudinal Static Stability & Neutral Point) within the *Flight Mechanics & Space Dynamics* curriculum (1 Mark, MCQ).
**Problem Statement**:
> Consider a steady, isentropic, supersonic flow (Mach number M > 1) entering a Convergent-Divergent (CD) duct as shown in the figure. Which one of the following options correctly describes the flow at the throat?
**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 **`C`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"Can only be supersonic"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"Can only be sonic"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"Can either be sonic or supersonic"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(D)** *"Can only be subsonic"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2024 (IISc Bengaluru) key.
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