Chapter 12, Problem 60P

(a)

To determine

To Calculate: The shockwave that makes an angle with the direction of the airplane’s motion.

Answer to Problem 60P

The shockwave that makes an angle with the direction of the airplane’s motion is $25.77°$ .

Explanation of Solution

Given:

Speed of sound $\left(C\right)$ $=310\text{ m/s}$

Mach number $\left(M\right)$ $=2.3$

Formula used:

  $\sin \theta =\frac{C}{v}\text{ or }\frac{1}{M}$

  $C=$ Velocity of sound $=310\text{ m/s}$

  $v=$ The velocity of an object, m/s

  $M=$ Mach number

Calculation:

The sine of the angle is equal to the ratio of the velocity of sound to the velocity of an object and it is the reciprocal of Mach number.

Mathematically,

  $\sin \theta =\frac{C}{v}\text{ or }\frac{1}{M}$

Where,

  $C=$ Velocity of sound $=310\text{ m/s,}$

  $v=$ The velocity of an object, m/s

  $M=$ Mach number

  $\begin{array}{l} v=CM\text{ m/s}\\ v=310\times 2.3\end{array}$

  $v=713\text{ m/s}.$

Now, substitute the values in the equation,

  $\sin \theta =\frac{\text{310 m/s}}{\text{713 m/s}}$

  $\theta ={\sin }^{-1}(0.43)$

  $\theta =25.77°$

Alternative solution,

  $\sin \theta =\frac{1}{M}$

  $\theta ={\sin }^{-1}\left(\frac{1}{M}\right)$

  $\theta ={\sin }^{-1}\left(\frac{1}{2.3}\right)$

  $\theta =25.77°$

The shock wave angle is $25.77°$ .

Conclusion:

The shockwave that makes an am angle with the direction of the airplane’s motion is ${25.77}^{\circ }$ .

(b)

To determine

To Calculate: The time that a person on the ground can hear the shock wave after an airplane directly overhead.

Answer to Problem 60P

The time that a person can hear a shock wave after an airplane flying directly overhead is $20.62\text{ s}.$

Explanation of Solution

Given:

Height of airplane $\left(h\right)$ $=7100\text{ m}\text{.}$

The velocity of the airplane, $\left(v\right)$ $=713\text{ m/s}$ .

Shock wave angle $\left(\theta \right)$ $=25.77°$

Formula used:

  $\tan \theta =\frac{h}{v.t}$

  $t=\left(\frac{h}{v\cdot \tan \theta }\right)$ s.

  $t=$ a time that the person can hear the shock wave.

Calculation:

To find the time that a person can hear a shock wave, use the following equation.

  $\tan \theta =\frac{h}{v\cdot t}$

  $t=\left(\frac{h}{v\cdot \tan \theta }\right)$ s

  $t=\left(\frac{7100\text{ m/s}}{713\text{ m/s}\times \tan (25.77)}\right)\text{ s}$

  $t=20.62\text{ s}$

Conclusion:

The time that a person can hear a shock wave after an aeroplane flying directly overhead is $20.62\text{ s}.$