Question

 Mach angle α and Mach number M are related as

1. \(M = {\sin ^{ - 1}}\left( {\frac{1}{\alpha }} \right)\)
2. \(\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)\)
3. \(\tan \alpha \left( {\sqrt {{M^2} - 1} } \right)\)
4. \(\alpha = cose{c^{ - 1}}\left( {\frac{1}{M}} \right)\)

Answer 1

Correct Answer - Option 2 : \(\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)\)

Explanation:
Mach number:

Mach number has been defined as the square root of the ratio of the inertia force of a flowing fluid to the elastic force.

Mach Number (M) = \(\sqrt{\frac{Inertia\;Force}{Elastic\;Force}}=\sqrt{\frac{\rho AV^2}{KA}}=\frac{V}{\sqrt{K/\rho}}=\frac{V}{C}\) 

Mach Angle:

It is defined as the half of the angle of the Mach cone. It is given by 

\(\sinα=\frac{C}{V}=\frac{1}{M}\)

cos²α = 1 - sin²α

\(\cos^2\alpha=1-\frac{1}{M^2}\)

\(\cos^2\alpha=\frac{M^2-1}{M^2}\)

\(\cos\alpha=\sqrt{\frac{M^2-1}{M^2}}\)

\(\alpha = {\cos ^{ - 1}}\left( {\sqrt {\frac{{{M^2} - 1}}{{{M^2}}}} } \right)\)

Flow can be classified on the basis of Mach number:

Flow	Mach number
Subsonic	M < 1
Sonic	M = 1
Supersonic	1 < M < 3
Hypersonic	M > 3