Question

A traffic policeman sounds a whistle to stop a car-driver approaching towards him. The car-driver does not stop and takes the plea in court that because of the Doppler shift, the frequency of the whistle reaching him might have gone beyond the audible limit of 25 kHz and he did not hear it. Experiments showed that the whistle emits a sound with frequency closed to 16 kHz. Assuming that the claim of the driver is true, how fast was he driving the car? Take the speed of sound in air to be 330 m s⁻¹. Is this speed practical with today's technology?

Answer 1

Given:
Frequency of whistle $f_0$ = 16 × 10³ Hz
Apparent frequency $f$ = 20 × 10³ Hz
(f is greater than that value)
Velocity of source $v_s$ = 0
Let $v_0$ be the velocity of the observer.
Apparent frequency $\left(f\right)$ is given by:

$f=\left(\frac{v+v_0}{v-v_s}\right)f_0$

On substituting the values in the above equation, we get:

$$\begin{gathered} 20\times {10}^3=\left(\frac{330+v_0}{330-0}\right)\times 16\times {10}^3 \\ \Rightarrow \left(330+v_0\right)=\frac{20\times 330}{16} \\ \Rightarrow v_0=\frac{20\times 330-16\times 330}{4} \\ =\frac{330}{4}\mathrm{m}/\mathrm{s}=297\mathrm{km}/\mathrm{h} \end{gathered}$$

(b) This speed is not practically attainable for ordinary cars.