Question

A bullet is fired with a speed of $\mathbf{685}\mathbf{\text{\hspace{0.17em}m}}\mathbf{/}\mathbf{\text{s}}$ . Find the angle made by the shock cone with the line of motion of the bullet.

Short answer

Angle made by shock cone is$30°$

Step-by-step solution

The given data

Bullet fired with speed of ${\mathrm{v}}_{\mathrm{b}}=685\text{\hspace{0.17em}m}/\text{s}$.

Understanding the concept of shock wave 

Here we have to use the concept of Mach cone angle in which the ratio of speeds is equal to the sine of the angle. Angle made by the shock cone can be found by finding the sine of the angle between the speed of sound and speed of moving body. Here, the speed of a moving body is the speed of a bullet, that is, $\mathbf{685}\mathbf{\text{\hspace{0.17em}m}}\mathbf{/}\mathbf{\text{s}}$

Formula:

The speed of the sound using equation of shock wave (or the Mach cone angle),

$\mathbf{sin\theta }\mathbf{=}\frac{\mathbf{v}}{{\mathbf{v}}_{\mathbf{b}}}$ …(i)

Calculation of the angle made by shock wave cone 

Here we are given the speed of bullet, and we know the speed of sound as $\mathrm{v}=343\text{\hspace{0.17em}m}/\text{s}$.

We can use the following formula of equation (i) to find angle of the shock wave as:

$\begin{array}{c}\mathrm{sin\theta }=\frac{343\text{\hspace{0.17em}m}/\text{s}}{685\text{\hspace{0.17em}m}/\text{s}}\\ \mathrm{\theta }=30°\end{array}$

Hence, the angle made by the shock wave is $30°$.