Question

An aircraft is reported to be cruising in still air at \(-20^{\circ} \mathrm{C}\) and \(40 \mathrm{kPa}\) at a Mach number of \(0.86 .\) The velocity of the aircraft is \((a) 91 \mathrm{m} / \mathrm{s}\) \((b) 220 \mathrm{m} / \mathrm{s}\) \((c) 186 \mathrm{m} / \mathrm{s}\) \((d) 274 \mathrm{m} / \mathrm{s}\) \((e) 378 \mathrm{m} / \mathrm{s}\)

Short answer

(Consider the air as an ideal gas with specific heat ratio 1.4 and gas constant 287 J/(kg⋅K).) a) 255 m/s b) 265 m/s c) 285 m/s d) 274 m/s Answer: d) 274 m/s

Step-by-step solution

Convert the given temperature from Celsius to Kelvin

We need to convert the given temperature from Celsius to Kelvin as follows: \(T_{K} = T_{^\circ C} + 273.15\) The given temperature is \(T_{^\circ C} = -20^\circ C\). Now, let's do the conversion: \(T_{K} = -20^\circ C + 273.15\) \(T_{K} = 253.15 K\) The temperature of the air in Kelvin is 253.15 K.

Calculate the speed of sound in the air

Now that we have the temperature in Kelvin, we can find the speed of sound in the air using the formula: \(a = \sqrt{\gamma R T_{K}}\) By substituting the known values, we get: \(a = \sqrt{1.4 \times 287 \times 253.15}\) \(a \approx 320.4 \, \mathrm{m/s}\) The speed of sound in the air is approximately 320.4 m/s.

Calculate the velocity of the aircraft

Finally, we can find the velocity of the aircraft using the formula: \(V = Ma\) By substituting the known values, we get: \(V = 0.86 \times 320.4\) \(V \approx 275.5 \, \mathrm{m/s}\) Considering the given options, the closest answer is: \((d) 274 \, \mathrm{m/s}\) Thus, the velocity of the aircraft is approximately 274 m/s.

Key concepts

Speed of Sound

Understanding the speed of sound is crucial when calculating aircraft velocity at different temperatures and pressures. The speed of sound is how fast pressure waves travel through a medium, in this case, air. This speed can vary depending on the medium's composition and its temperature.

The formula to calculate the speed of sound in air is given by:
\[a = \sqrt{\gamma R T_{K}}\]
where:

• \(\gamma\) (gamma) is the adiabatic index (1.4 for air),
• \(R\) is the specific gas constant for air (287 J/(kg·K)), and
• \(T_{K}\) is the temperature in Kelvin (K).

As seen from our formula, an increase in air temperature will result in a higher speed of sound and vice versa. This relationship is particularly significant in aviation, where accurate calculations of sound speed are necessary for determining the Mach number of an aircraft.

Temperature Conversion

Temperature conversion is an essential pre-requisite for various calculations in science, especially in determining the speed of sound for aviation-related problems. In the exercise, the temperature given in Celsius (\(^\circ\text{C}\)) must be converted into Kelvin (K) before using it in our speed of sound formula. The conversion is simple:
\[T_{K} = T_{^\circ\text{C}} + 273.15\]
In this context, understanding that Kelvin is an absolute temperature scale where 0 K is absolute zero, the lowest theoretical temperature, is vital. There are no negative numbers on the Kelvin scale, making it ideal for scientific calculations where negative temperatures would be impractical. By converting to Kelvin, we provide a more simplified and uniform way of inputting temperature into our equations.

Aircraft Velocity

Knowing the velocity of an aircraft is imperative for pilots and engineers alike. Velocity refers to the speed of an object in a given direction. In the context of aircraft, it usually means the speed relative to the surrounding air which in still air is its true airspeed.

To find an aircraft's velocity using the Mach number, the following formula is used:
\[V = Ma\]
where:

• \(M\) is the Mach number, a dimensionless quantity representing the ratio of the aircraft's speed to the speed of sound,
• \(a\) is the speed of sound in the air (m/s).

A Mach number of 1 indicates the aircraft is traveling at the speed of sound. Our exercise required to find the aircraft's velocity given a Mach number of 0.86. By calculating the speed of sound first, and then using the Mach number, we could estimate the aircraft's speed in still air. This step is pivotal in aviation and aerospace, where measuring an aircraft's performance and adherence to safety regulations is dependent on accurate speed calculations.