Question

A subsonic airplane is flying at a \(5000-\mathrm{m}\) altitude where the atmospheric conditions are \(54 \mathrm{kPa}\) and \(256 \mathrm{K}\). A Pitot static probe measures the difference between the static and stagnation pressures to be 16 kPa. Calculate the speed of the airplane and the flight Mach number.

Short answer

Answer: The speed of the airplane is approximately 195.1 m/s and the flight Mach number is approximately 0.616.

Step-by-step solution

1. Calculate the speed of sound

To determine the speed of sound, we can use the following equation: \( a = \sqrt{\gamma R T} \) Where \(a\) is the speed of sound, \(\gamma\) is the ratio of specific heats (1.4 for air), \(R\) is the specific gas constant for air (287 J/kg·K), and \(T\) is the temperature in Kelvin. Given the temperature, T = 256 K: \( a = \sqrt{1.4 \cdot 287 \cdot 256} \approx 316.3 \;\text{m/s} \)

2. Calculate the static pressure at the altitude

We are given the atmospheric conditions at the altitude - the pressure is 54 kPa. Thus, the static pressure \(P_{static} = 54\;\text{kPa}\)

3. Determine the stagnation pressure

We are given the difference between the static and stagnation pressures as 16 kPa. Thus, the stagnation pressure \(P_{stag}\) can be found by adding this difference to the static pressure: \(P_{stag} = P_{static} + \Delta P = 54\;\text{kPa} + 16\;\text{kPa} = 70\;\text{kPa}\)

4. Calculate the airplane's speed

We can now use the Bernoulli equation relating static and stagnation pressures to find the airplane's speed, \(V\). The Bernoulli equation in this case is: \(P_{stag} = P_{static} + \frac{1}{2} \rho V^2\) First, we need to find the air density (\(\rho\)) at the given altitude using the ideal gas law: \(\rho = \frac{P_{static}}{R T} = \frac{54\times 10^3\;\text{Pa}}{287\;\text{J/kg·K}\cdot 256\;\text{K}} \approx 0.73\;\text{kg/m^3}\) Now, we can use the Bernoulli equation and solve for \(V\): \(V = \sqrt{\frac{2(P_{stag} - P_{static})}{\rho}} = \sqrt{\frac{2(70\times 10^3\;\text{Pa} - 54\times 10^3\;\text{Pa})}{0.73\;\text{kg/m^3}}} \approx 195.1\;\text{m/s}\)

5. Calculate the flight Mach number

Finally, we calculate the flight Mach number using the airplane's speed and the speed of sound found in step 1: \(M = \frac{V}{a} = \frac{195.1\;\text{m/s}}{316.3\;\text{m/s}} \approx 0.616\) So, the speed of the airplane is approximately 195.1 m/s and the flight Mach number is approximately 0.616.