Problem Statement
A projectile is travelling in air having pressure and temperature as 8.829 N/cm² and - 5°C. If the Mach angle is 30°, find the velocity of the projectile. Take k = 1.4 and R = 287 J/kg K.
Given Data & Constants
- Air temperature, \(T = -5^\circ\text{C}\)
- Mach angle, \(\mu = 30^\circ\)
- 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 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 Mach Number (M)
The Mach number is related to the Mach angle by the formula \(\sin(\mu) = 1/M\).
4. Calculate the Velocity of the Projectile (V)
The velocity of the projectile is its Mach number multiplied by the local speed of sound.
The velocity of the projectile is approximately 656.4 m/s.
Explanation of Mach Angle
When an object travels faster than the speed of sound (supersonic flight, M > 1), it creates a conical shock wave, similar to the wake of a boat. The Mach angle (\(\mu\)) is the half-angle of this cone.
This angle has a direct trigonometric relationship with the Mach number. By knowing the Mach angle, we can determine how many times faster than sound the object is traveling (its Mach number). Once the Mach number is known, we can find the object's true velocity by multiplying the Mach number by the local speed of sound, which itself depends on the air temperature.
