Chapter MPS, Problem 14.7MPS

To determine

Find the match number of Jet if a sonic boom created by a cone with a vertex angle of $75°.$

Answer to Problem 14.7MPS

  $\boxed{1.6427}.$

Explanation of Solution

Given information:

When a jet travels at speeds greater than the speed of sound, a sonic boom is created by the sound waves forming a cone behind the jet. If $\theta$ is the measure of the angle at the vertex of the

Cone, then the Mach number $M$ can be determined using the formula sin $\frac{\theta }{2}=\frac{1}{M}.$

Calculation:

Then, the mach number $M$ can be determined using the formula,

  $\sin \frac{\theta }{2}=\frac{1}{M}\mathrm{.......}\left(1\right)$

It is enough to find the number $M$ when $\theta =75°$

Substitute $\theta =75°$ in the given equation $\left(1\right)$ to get,

  $\begin{array}{l}\sin \frac{75°}{2}=\frac{1}{M}\\ M=\frac{1}{\sin \frac{75°}{2}}\end{array}$  $\mathrm{.........}\left(2\right)$

Need to find the value of $\sin \frac{75°}{2}.$

Use half- angle formula,

  $\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}$

  $\sin \frac{75°}{2}=\sqrt{\frac{1-\cos 75°}{2}}$

  $=\sqrt{1-\frac{\sqrt{6-\sqrt{2}}}{\frac{4}{2}}}$  $\mathrm{Sin}ce\cos 75°=\frac{\sqrt{6-\sqrt{2}}}{4}$

  $=\sqrt{4-\frac{\sqrt{6+\sqrt{2}}}{8}}$

Use this value in equation $\left(2\right)$ to get,

  $\begin{array}{l}M=\frac{1}{\sin \frac{75°}{2}}\\ =\frac{1}{\left(\sqrt{\frac{4-\sqrt{6}+\sqrt{2}}{8}}\right)}\\ =\sqrt{\frac{8}{4-\sqrt{6+\sqrt{2}}}}\\ \approx 1.6427\end{array}$

Hence, the Mach number of jet is $\boxed{1.6427}.$