Problem Statement
An aeroplane is flying at an height of 20 km, where the temperature is - 40°C. The speed of the plane is corresponding to M = 1.8. Assuming k = 1.4 and R = 287 J/kg K, find the speed of the plane.
Given Data & Constants
- Mach number, \(M = 1.8\)
- Air temperature, \(T = -40^\circ\text{C}\)
- Adiabatic index, \(k = 1.4\)
- Gas constant, \(R = 287 \, \text{J/kg K}\)
Solution
1. Convert Temperature to Absolute Scale (Kelvin)
The formula for the speed of sound requires the temperature to be in Kelvin.
2. Calculate the Local Speed of Sound (c)
The speed of sound in the air at the given altitude is calculated using the formula \(c = \sqrt{kRT}\).
3. Calculate the Speed of the Plane (V)
The speed of the plane is its Mach number multiplied by the local speed of sound.
The speed of the plane is approximately 550.87 m/s.
Explanation of the Calculation
The speed of an aircraft is often described by its Mach number (M), which is the ratio of its speed to the speed of sound in the surrounding air. A key point is that the speed of sound is not constant; it changes with the temperature of the air.
Therefore, to find the actual speed of the plane in meters per second, we must first calculate the local speed of sound at its cruising altitude (where the temperature is -40°C). Once we have the local speed of sound, we can multiply it by the given Mach number to find the plane's true speed.
