To determine the equivalent airspeed of an aircraft using the total pressure measured by a Pitot tube, we need to find the air density at the altitude of 24,000 feet, convert that to appropriate units, and then calculate the equivalent airspeed.
Step 1: Understanding the Problem
The total pressure \( P_t \) measured by the Pitot tube is given as 0.5564 bar. We need to convert this pressure to Pascals for calculations, since: \[ 1 \text{ bar} = 100,000 \text{ Pa} \]
Thus: \[ P_t = 0.5564 \text{ bar} = 0.5564 \times 10^5 \text{ Pa} = 55,640 \text{ Pa} \]
Step 2: Calculate Static Pressure at 24,000 feet
At an altitude of 24,000 feet (approximately 7,315 meters), we can use the International Standard Atmosphere (ISA) model to find the static pressure \( P_s \).
Using the ISA, at 24,000 feet, the temperature \( T \approx -6.5 \text{ °C} = 266.65 \text{ K} \), and the standard pressure at this altitude \( P_s \approx 0.2924 \text{ bar} \) or \( 29,240 \text{ Pa} \).
Step 3: Calculate Air Density
The air density \( \rho \) can be found using the ideal gas law: \[ \rho = \frac{P_s}{R T} \] where:
- \( R \) (specific gas constant for dry air) \( \approx 287.05 , \text{J/(kg} \cdot \text{K)} \)
- \( P_s = 29,240 \text{ Pa} \)
- \( T = 266.65 \text{ K} \)
Substituting the values: \[ \rho = \frac{29,240}{287.05 \times 266.65} \approx \frac{29,240}{7650.2} \approx 3.82 , \text{kg/m}^3 \]
Step 4: Determine Equivalent Airspeed (EAS)
The dynamic pressure \( q \) is given by: \[ q = P_t - P_s = 55,640 \text{ Pa} - 29,240 \text{ Pa} = 26,400 \text{ Pa} \]
The equivalent airspeed (EAS) can be calculated using the formula: \[ EAS = \sqrt{\frac{2q}{\rho}} \] Substituting the values: \[ EAS = \sqrt{\frac{2 \times 26,400}{3.82}} = \sqrt{\frac{52,800}{3.82}} \approx \sqrt{13,831.64} \approx 117.03 \text{ m/s} \]
Conclusion
The equivalent airspeed of the aircraft at the altitude of 24,000 feet with the given conditions is approximately: \[ \boxed{117.03 , \text{m/s}} \]
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