Question

The Doppler effect (named after Christian Doppler) is the change in the pitch (frequency) of the sound from a source (s) as heard by an observer (o) when one or both are in motion. If we assume both the source and the observer are moving in the same direction, the relationship is

$f\text{'}=f_a\left(\frac{v-v_o}{v-v_s}\right)$ Where,

role="math" localid="1646148999776" $$\begin{gathered} f\text{'}=perceivedpitchbytheobserver \\ {f}_a=actualpitchofthesource \\ v=speedofsoundinair(assume772.4mph) \\ {v}_o=speedoftheobserver \\ {v}_s=speedofthesource \end{gathered}$$

Suppose that you are traveling down the road at $45$mph and you hear an ambulance (with siren) coming toward you from the rear. The actual pitch of the siren is$600$hertz (Hz).

Part (a): Write a function $f\text{'}\left(v_s\right)$that describes this scenario.

Part (b): If $f\text{'}=620Hz$, find the speed of the ambulance.

Part (c): Use a graphing utility to graph the function.

Part (d): Verify your answer from part (b).

Short answer

Part (a): The required function that describes the given scenario is $f\text{'}\left(v_s\right)=\frac{436440}{772.4-v_s}$.

Part (b): The speed of the ambulance is localid="1646150114120" $68.5mph$.

Part (c): The graph of the function is given below,

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given question,

$$\begin{gathered} f_a=600, \\ v=772.4, \\ {v}_o=45 \end{gathered}$$

Substitute the value in the given function,

localid="1646149276358" $$\begin{gathered} f\text{'}\left(v_s\right)=600\left(\frac{772.4-45}{772.4-v_s}\right) \\ f\text{'}\left(v_s\right)=\frac{436440}{772.4-v_s} \end{gathered}$$

Part (b) Step 1. Substitute the values in the function.

Consider the given question,

$f\text{'}=620$

role="math" localid="1646149399999" $f\text{'}\left(v_s\right)=\frac{436440}{772.4-v_s}......\left(i\right)$

Substitute the values in equation (i),

role="math" localid="1646149457809" $$\begin{gathered} 620=\frac{436440}{772.4-v_s} \\ 620\left(772.4-v_s\right)=436440 \\ 478888-620v_s=436440 \\ -620v_s=-42448 \\ {v}_s\approx 68.5mph \end{gathered}$$

Part (c) Step 1. Plot the function.

Consider the given function,

$f\text{'}\left(v_s\right)=\frac{436440}{772.4-v_s}$

On plotting the function, we get,

Part (d) Step 1. Verify the answer of part (b).

Consider the previous part,

$v_s\approx 68.5mph$

On plotting the value on the graph,

From the graph, we can say$f\text{'}=620$, then$v_s\approx 68.5$.