Problem 3.14

Air at a temperature of ^oC is flowing with a velocity of m/s. A projectile is fired into the air stream with a velocity of m/s in the opposite direction to that of the airflow. Calculate the angle that the Mach waves from the projectile make to the direction of motion.

Given:
\( T = \) 25^oC = 298 K,
\( V_{a} = \) 180 m/s,
\( V_{p} = \) 800 m/s.

To calculate: \( \mu=? \)

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

By placing the frame of reference on the projectile, we get a configuration, as shown on the right side of Fig.1. In this configuration, the Mach wave is stationary with respect to the frame of reference. The Mach number for the flow can be calculated as,

\[ M=\frac{V_{a}}{a}=\frac{V_{a}}{\sqrt{\gamma\,R\,T}}=\frac{980}{\sqrt{1.4\times287\times298}}\ . \]

The Mach angle can be calculated using the formula,

\[ \sin\mu=\frac{1}{M} \]

\[ \mu=\sin^{-1}\left(\frac{\sqrt{1.4\times287\times298}}{980}\right) \]

\[ \boxed{\mu=20.6765^{\circ}}\ . \]

Problem 3.14:

Solution: