Subtopic: Short Period & Phugoid Longitudinal Modes
Question Statement:
A general aviation airplane is flying at an altitude of 5000 m. The indicated
airspeed is 250 km/h. Assume that there are no instrument errors and position
errors. Neglecting compressibility effects, which one of the following options is
FALSE?
Options:
(A) The true airspeed is greater than 250 km/h.
(B) The calibrated airspeed is 250 km/h.
(C) The true airspeed is 250 km/h.
(D) The equivalent airspeed is 250 km/h.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.24)**
**1. Intuitive Physical Setup & Given Parameters**:
- Airplane flying at altitude: $h = 5000\text{ m}$.
- Indicated airspeed: $V_{\text{IAS}} = 250\text{ km/h}$.
- No instrument or position errors (so $V_{\text{CAS}} = V_{\text{IAS}}$).
- Compressibility effects neglected (so $V_{\text{EAS}} = V_{\text{CAS}}$).
We are asked to identify which of the four statements is **FALSE**.
**2. Step-by-Step Flight Mechanics Derivation**:
1. **Airspeed Hierarchy and Relationships**:
- **Calibrated Airspeed ($V_{\text{CAS}}$)**:
$V_{\text{CAS}} = V_{\text{IAS}} - \Delta V_{\text{inst}} - \Delta V_{\text{pos}}$
Since instrument and position errors are zero:
$V_{\text{CAS}} = 250\text{ km/h}$
Statement B is **TRUE**.
- **Equivalent Airspeed ($V_{\text{EAS}}$)**:
At low speeds or neglecting compressibility:
$V_{\text{EAS}} = V_{\text{CAS}} = 250\text{ km/h}$
Statement D is **TRUE**.
- **True Airspeed ($V_{\text{TAS}}$)**:
True airspeed accounts for the decrease in ambient air density with altitude:
$V_{\text{TAS}} = \frac{V_{\text{EAS}}}{\sqrt{\sigma}} = \frac{V_{\text{EAS}}}{\sqrt{\rho / \rho_0}}$
At $h = 5000\text{ m}$, the density ratio is:
$\sigma = \frac{\rho}{\rho_0} \approx 0.601 < 1$
$\sqrt{\sigma} \approx \sqrt{0.601} \approx 0.775$
$V_{\text{TAS}} = \frac{250}{0.775} \approx 322.5\text{ km/h} > 250\text{ km/h}$
Statement A is **TRUE**.
2. **Evaluation of Statement C**:
- Statement C asserts: *"The true airspeed is 250 km/h."*
- As derived above, because density at 5000 m is substantially lower than sea level ($\rho < \rho_0$), $V_{\text{TAS}}$ must be strictly greater than 250 km/h.
- Therefore, Statement C is **FALSE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) The true airspeed is greater than 250 km/h**: ❌ True statement. At 5000 m, $\rho < \rho_0 \implies V_{\text{TAS}} > V_{\text{IAS}}$.
- **(B) The calibrated airspeed is 250 km/h**: ❌ True statement. Zero instrument and position errors mean $V_{\text{CAS}} = V_{\text{IAS}}$.
- **(C) The true airspeed is 250 km/h**: ✅ **CORRECT (FALSE STATEMENT)**. Contradicts $V_{\text{TAS}} = V_{\text{IAS}} / \sqrt{\sigma}$.
- **(D) The equivalent airspeed is 250 km/h**: ❌ True statement. Incompressible assumption equates $V_{\text{EAS}}$ to $V_{\text{CAS}}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Remember the airspeed conversion chain: $\text{IAS} \xrightarrow{\text{err}} \text{CAS} \xrightarrow{\text{comp}} \text{EAS} \xrightarrow{\rho/\rho_0} \text{TAS}$. At altitude ($\rho < \rho_0$), $V_{\text{TAS}}$ is always strictly greater than $V_{\text{EAS}}$.
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