Problem Statement
A projectile travels in air of pressure 8.829 N/cm² at -10°C at a speed of 1200 km/hour. Find the Mach number and the Mach angle. Take k = 1.4 and R = 287 J/kg K.
Given Data & Constants
- Speed of projectile, \(V = 1200 \, \text{km/hr}\)
- Air temperature, \(T = -10^\circ\text{C}\)
- Adiabatic index, \(k = 1.4\)
- Gas constant, \(R = 287 \, \text{J/kg K}\)
- Pressure, \(P = 8.829 \, \text{N/cm}^2\) (Note: Not required for calculation)
Solution
1. Convert Units to SI
First, we convert the projectile's speed to m/s and the temperature to Kelvin.
2. Calculate the Local 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 projectile's speed to the local speed of sound.
4. Calculate the Mach Angle (\(\mu\))
The Mach angle is related to the Mach number by the formula \(\sin(\mu) = 1/M\).
The Mach number is approximately 1.025.
The Mach angle is approximately 77.32°.
Explanation of Mach Number and Mach Angle
Mach Number (M): A dimensionless quantity representing the ratio of an object's speed to the local speed of sound. Since M > 1, the projectile is flying at a supersonic speed.
Mach Angle (\(\mu\)): When an object travels faster than sound, it creates a conical shock wave. The Mach angle is the half-angle of this cone. It's a direct measure of the Mach number; a higher Mach number results in a narrower, more acute Mach angle.
