Chapter 15, Problem 85A

(a)

To determine

To calculate: The velocity of the car.

Answer to Problem 85A

Velocity of the car is $68.6\text{ m/s}$ .

Explanation of Solution

Given:

Velocity of sound in air at ${20}^{\text{o}}\text{C}$ ( $v$ ) = $+343\text{ m/s}$

Velocity of detector ( $v_d$ ) = $0\text{ m/s}$ (at rest)

The source sounds one major third above the stationary trumpet.

Formula used:

Doppler Effect equation:

  $f_d=f_s\left(\frac{v-v_d}{v-v_s}\right)$

Where $f_d$ represents frequency perceived by the detector, $f_s$ represents wave’s frequency, $v$ represents speed of sound in air at ${20}^{\text{o}}\text{C}$ , $v_s$ represents velocity of source and $v_d$ represents velocity of detector.

Calculation:

Velocity of sound in air at ${20}^{\text{o}}\text{C}$ ( $v$ ) = $+343\text{ m/s}$

Velocity of detector ( $v_d$ ) = $0\text{ m/s}$ (at rest)

The frequency of source is one major third ratio above the detector’s frequency.

One major third ratio = $\frac{5}{4}$

Ratio of $\frac{f_s}{f_d}=\frac{4}{5}$

Velocity of car (source) ( $v_s$ ) = $24.6\text{ m/s}$

To calculate the velocity of source rearrange the formula

  $\begin{array}{l}{f}_d=f_s\left(\frac{v-v_d}{v-v_s}\right)\\ v_s=v-\frac{(v-v_d)f_s}{f_d}\end{array}$

Substitute the above given values

  $\begin{array}{l}{v}_s=343\text{ m/s}-\frac{(343\text{ m/s}-0\text{ m/s})4}{5}\\ v_s=343\text{ m/s}-274.4\text{ m/s}=68.6\text{ m/s}\end{array}$

Conclusion:

Velocity of the moving car is $68.6\text{ m/s}$ .

(b)

To determine

Whether the experiment must be tried or not.

Answer to Problem 85A

The car is moving dangerously fast, $153.5\text{ mph}$ , hence the experiment must not be tried.

Explanation of Solution

Given:

Velocity of the moving car is $68.6\text{ m/s}$

Calculation:

As per problem,

Convert the velocity of car from $\text{m/s to mph}$ .

  $\begin{array}{l}1\text{ mi= 1609 m}\\ \text{1 h= 3600 s}\end{array}$

  $\begin{array}{l}{v}_s=68.6\text{ m/s}\\ v_s=68.6\cdot \left(\frac{3600\text{ s}}{1\text{ h}}\right)\cdot \left(\frac{1\text{ mi}}{1609\text{ m}}\right)\\ v_s=153.5\text{ mph}\end{array}$

Conclusion:

The car is moving dangerously fast, $153.5\text{ mph}$ , hence the experiment must not be tried.