Problem Statement
Calculate the Mach number at a point on a jet propelled aircraft which is flying at 900 km/hour at sea-level where air temperature is 15°C. Take k = 1.4 and R = 287 J/kg K.
Given Data & Constants
- Speed of aircraft, \(V = 900 \, \text{km/hr}\)
- Air temperature, \(T = 15^\circ\text{C}\)
- Adiabatic index, \(k = 1.4\)
- Gas constant, \(R = 287 \, \text{J/kg K}\)
Solution
1. Convert Units to SI
First, we convert the aircraft speed to m/s and the temperature to Kelvin.
2. Calculate the Speed of Sound (c)
The speed of sound in the air is calculated using the formula \(c = \sqrt{kRT}\).
3. Calculate the Mach Number (M)
The Mach number is the ratio of the aircraft's speed to the local speed of sound.
The Mach number of the aircraft is approximately 0.735.
Explanation of Mach Number
The Mach number (M) is a dimensionless quantity representing the ratio of an object's speed through a fluid to the local speed of sound in that fluid. It's a crucial parameter in aerodynamics and fluid dynamics.
- Subsonic (M < 1): The object is moving slower than the speed of sound. This is the case for our aircraft (M ≈ 0.735).
- Transonic (M ≈ 1): The object is moving at or near the speed of sound.
- Supersonic (M > 1): The object is moving faster than the speed of sound, creating a shock wave (sonic boom).
The speed of sound is not constant; it depends on the temperature of the medium. That's why we must first calculate the local speed of sound at 15°C before we can determine the aircraft's Mach number.
