Chapter 12, Problem 61P

(a)

To determine

To calculate: The probe’s Mach number.

Answer to Problem 61P

The Mach number of the probe is $\mathrm{119.04.}$

Explanation of Solution

Given:

Speed of sound $\left(C\right)$ $=35\text{ m/s}$

Speed of probe $\left(v_{probe}\right)$ $=15000\text{ km/h}$ = $15000\text{ kmph}\times \frac{5}{18}\text{ m/s}$ $=4166.6\text{ m/s}.$

Formula used:

  $M=\frac{v_{probe}}{C}$

  $M=$ Mach number

  $v_{probe}=$ speed of probe, m/s

  $C=$ speed of sound, m/s

Calculation:

Substitute the values in the equation,

  $M=\frac{4166.66\text{ m/s}}{\text{35 m/s}}$

  $M=119.04$

The Mach number of the probe is $\mathrm{119.04.}$ It has no units.

Conclusion:

The probe’s Mach number is $119.04,$ if its initial velocity is $15000\text{ km/h}.$

(b)

To determine

To calculate: The angle of the shock wave relative to the direction of motion.

Answer to Problem 61P

The angle of shock wave is $0.48°$ , relative to the direction of motion.

Explanation of Solution

Given:

Speed of sound $\left(C\right)$ $=35\text{ m/s}$

Speed of probe $\left(v_{probe}\right)$ $=15000\text{ km/h}$ = $15000\text{ kmph}\times \frac{5}{18}\text{ m/s}$ $=4166.6\text{ m/s}.$

Formula used:

Shock wave is defined as, it is the strong pressure wave in an elastic medium like air, water or any solid and it is produced by supersonic speeds of an object.

Mathematically the shock wave is defined as:

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

Where,

  $v_{probe}=$ speed of probe, m/s

  $C=$ speed of sound, m/s

Calculation:

Substitute the values in the equation

  $\sin \theta =\frac{1}{119.04}\text{ or }\frac{35\text{ m/s}}{4166.6\text{ m/s}}$

  $\theta =0.48°$

Conclusion:

The angle of shock wave is $0.48°$ , relative to the direction of motion.