Question

A state trooper chases a speeder along a straight road; both vehicles move at 160km/h. The siren on the trooper’s vehicle produces sound at a frequency of 500Hz. What is the Doppler shift in the frequency heard by the speeder?

Short answer

The Doppler shift in the frequency heard by the speeder is, 0.

Step-by-step solution

The given data

1. Frequency of the siren is, f = 500Hz
2. Speed of the trooper is, v_s = 160km/h or$44.44\frac{m}{s}$
3. Speed of the speeder is, v_D = 160km/h or$44.44\frac{m}{s}$

Understanding the concept of the Doppler Effect

Inserting the given values in the formula obtained from Doppler’s Effect, we can find the frequency heard by the speeder. Using this and the frequency of the siren, we can find the Doppler shift in the frequency heard by the speeder.

Formula:

The frequency received by the observer according to the Doppler Effect, (since the motion of the speeder is away from the source.)

$\mathit{f}\mathbf{\text{'}}\mathbf{=}\mathit{f}\frac{\left(v-v_D\right)}{\left(v-v_s\right)}$ …(i)

Calculation of the Doppler shift in frequency

We know that the speed of sound is,

Using equation (i) andsubstitute all the value in this equation, the frequency received by the speeder is given as:

$$\begin{gathered} f\text{'}=500Hz\times \frac{\left(343m/s-44.44m/s\right)}{\left(343m/s-44.44m/s\right)} \\ f\text{'}=500Hz \end{gathered}$$

Hence, net shift in the frequency due to the two vehicles of same speed is given as:

$\begin{array}{rcl}∆f& =& f\text{'}-f\\ & =& 500Hz-500Hz\\ & =& 0\end{array}$

Therefore, Doppler shift in the frequency heard by the speeder is 0.