Chapter 12, Problem 64P

(a)

To determine

To calculate:The angle of the shock cone $\theta$ .

Answer to Problem 64P

The shock wave angle is $48.54°$ .

Explanation of Solution

Given:

Height of plane above ground $\left(h\right)$ $=1.5\text{ km }=1500\text{ m}$ .

The plane travelled horizontal distance $\left(x\right)$ $=2\text{ km }=2000\text{ m}$ .

Formula used:$$

  $\sin \theta =\frac{c}{v}$

  $c=$ velocity of sound, 343 m/s

  $v=$ velocity of object, m/s

Calculation:

To calculate v , determine the time first.

  $t=\frac{h}{c}\text{ s}$

  $t=\frac{1500}{343}\text{ s}$

  $t=4.37\text{ s}$

To determine the $v$

  $v=\frac{\text{distance}}{\text{time}}\text{ m/s}$

  $v=\frac{2000}{4.37}\text{ m/s}$

  $v=457.66\text{ m/s}$

Now, calculate the shock wave angle,

  $\sin \theta =\frac{c}{v}$

  $\theta ={\sin }^{-1}\frac{343\text{ m/s}}{457.66\text{ m/s}}$

  $\theta =48.54°$

Conclusion:

The shock wave angle is $48.54°$ .

(b)

To determine

To calculate:The velocity of anaeroplane.

Answer to Problem 64P

The velocity of an aeroplane is $440.34\text{ m/s}.$

Explanation of Solution

Given:

Height of plane above ground $\left(h\right)$ $=1.5\text{ km }=1500\text{ m}$ .

The plane travelled horizontal distance $\left(x\right)$ $=2\text{ km }=2000\text{ m}$

velocity of sound $\left(v\right)$ $=330\text{ m/s}$ .

Formula used:

  $v=\frac{c}{\sin \theta }\text{ m/s}$

  $v=$ Speed of the sonic boom

  $c=$ velocity of sound = 343 m/s

Calculation:

Horizontal distance travelled by the plane $=2\text{ km }\left(2000\text{ m}\right).$

  $v=\frac{c}{\sin \theta }\text{ m/s}$

  $v=\frac{330}{\sin 48.54}\text{ m/s}$

  $v=440.34\text{ m/s}$

Conclusion:

The velocity of an aeroplane is $440.34\text{ m/s}.$