Chapter 17, Problem 6TP

To determine

(a)

The speed of the car.

Answer to Problem 6TP

The speed of the car is $32\text{m}/\text{s}$.

Explanation of Solution

Given:

The frequency of the car's horn is $f_{\text{horn}}=200\text{Hz}$.

Formula used:

The expression for the calculation of the frequency when the source is moving toward the observer is given by

  $f_{\text{student}}=f_{\text{horn}}\left(\frac{v_{\text{sound}}}{v_{\text{sound}}-v_{\text{car}}}\right)$

Here, $f_{\text{student}}$ is the frequency observed by the student, $f_{\text{horn}}$ is the frequency of the horn of the car, $v_{\text{sound}}$ is the velocity of sound in the air and $v_{\text{car}}$ is the velocity of the car.

The expression for the calculation of the frequency when the source is moving away from the observer is given by

  $f_{\text{student}}=f_{\text{horn}}\left(\frac{v_{\text{sound}}}{v_{\text{sound}}+v_{\text{car}}}\right)$

Here, $f_{\text{student}}$ is the frequency observed by the student, $f_{\text{horn}}$ is the frequency of the horn of the car, $v_{\text{sound}}$ is the velocity of sound in the air and $v_{\text{car}}$ is the velocity of the car.

The expression to calculate the mean velocity of the car is given by

  $v_{\text{m}}=\frac{v_{{\text{car}}_1}+v_{{\text{car}}_2}}{2}$

Here, $v_{{\text{car}}_1}$ is velocity of the car when the source is moving towards the observer and $v_{{\text{car}}_2}$ is the velocity of the car when the car is moving away from the observer.

Calculation:

The frequency perceived by the student obtained from the graph when the source is moving towards the observer is $f_{\text{student}}=217\text{Hz}$

The velocity of the sound in air is $v_{\text{sound}}=340\text{m}/\text{s}$

The velocity of the car when the source is moving towards the observer is calculated from the Doppler's effect formula as follows:

  $\begin{array}{c}{f}_{\text{student}}=f_{\text{horn}}\left(\frac{v_{\text{sound}}}{v_{\text{sound}}-v_{\text{car}}}\right)\\ v_{{\text{car}}_1}=\left(v_{\text{sound}}\right)-\frac{\left(f_{\text{horn}}\right)}{\left(f_{\text{student}}\right)}\left(v_{\text{sound}}\right)\\ =\left(340\text{m}/\text{s}\right)-\frac{\left(200\text{Hz}\right)}{\left(217\text{Hz}\right)}\left(340\text{}\text{m}/\text{s}\right)\\ =26.6\text{}\text{m}/\text{s}\end{array}$

The frequency perceived by the student obtained from the graph when the source is moving to away from the observer is $f_{\text{student}}=180\text{Hz}$

The velocity of the car when the source is moving away from the observer is calculated from the Doppler's effect formula as follows:

  $\begin{array}{c}{f}_{\text{student}}=f_{\text{horn}}\left(\frac{v_{\text{sound}}}{v_{\text{sound}}+v_{\text{car}}}\right)\\ v_{{\text{car}}_2}=\frac{\left(f_{\text{horn}}\right)}{\left(f_{\text{student}}\right)}\left(v_{\text{sound}}\right)-\left(v_{\text{sound}}\right)\\ =\frac{\left(200\text{Hz}\right)}{\left(180\text{Hz}\right)}\left(340\text{}\text{m}/\text{s}\right)-\left(340\text{m}/\text{s}\right)\\ =37.7\text{m}/\text{s}\end{array}$

The mean velocity of the car is calculated as follows:

  $\begin{array}{c}{v}_{\text{m}}=\frac{v_{{\text{car}}_1}+v_{{\text{car}}_2}}{2}\\ =\frac{\left(37.7\text{m}/\text{s}\right)+\left(26.6\text{m}/\text{s}\right)}{2}\\ =32\text{m}/\text{s}\end{array}$

Conclusion:

The speed of the car is $32\text{m}/\text{s}$.

To determine

(b)

The graph showing the perceived frequency of the car when the car is travelling at twice of speed.

Answer to Problem 6TP

The required graph is plotted.

Explanation of Solution

Given:

The coordinates from the graph when the source is moving towards the observer is $\left(x,y\right)=\left(0.5\text{s},217\text{Hz}\right)$.

The coordinates from the graph when the source is moving away from the observer is $\left(x,y\right)=\left(0.8\text{s},180\text{Hz}\right)$.

Calculation:

The graph for the line showing the perceived frequency when the car is travelling at twice speed as follows:

Draw the perpendicular lines at $\left(x=0.35\right)$ and $\left(x=0.85\right)$ as shown in the figure.

  

Figure (1)

Now, draw the line parallel to the $\text{Y}$ axis to the point of intersection of $\text{Y}$ axis coordinates as shown in the figure.

  

Figure (2)

The graph showing the perceived frequency when the car is travelling at twice of the speed is shown below.

  

Figure (3)

Conclusion:

Therefore, the required graph is plotted.