Question

A police car moving at $40m/s$ chases a thief running away at speed of $30m/s.$ The track is perpendicular to a stiff cliff as shown. The policeman blows a horn at $40Hz.$ If sound has a speed of $340m/s,$ what is the beat frequency (in $Hz$) heard by the thief?

Answer 1

Given, velocity of source (policeman),

${\upsilon }_s=40m/s$

Frequency of the source (policeman),

$n=40Hz$

Velocity of the object (thief),

${\upsilon }_0=30m/s$

Velocity of sound, ${\upsilon }_0=340m/s$

Beat frequency heard by thief $=?$

When the sound is approaching,

$\Rightarrow n_1=n\left(\frac{\upsilon -{\upsilon }_0}{\upsilon -{\upsilon }_s}\right)$

$\Rightarrow n_1=40\left(\frac{340-30}{340-40}\right)$

$\Rightarrow n_1=40\left(\frac{310}{300}\right)\dots \left(i\right)$

When the sound is receding the observer and heard from the cliff,

$\Rightarrow n_2=n\left(\frac{\upsilon +{\upsilon }_0}{\upsilon -{\upsilon }_s}\right)$

$\Rightarrow n_2=40\left(\frac{340+30}{340-40}\right)$

$\Rightarrow n_2=40\left(\frac{370}{300}\right)\dots \left(ii\right)$

$\Rightarrow n_{\text{beat}}=n_2-n_1$

$\Rightarrow n_{\text{beat}}=40\left(\frac{370}{300}\right)-40\left(\frac{310}{300}\right)$

$\Rightarrow n_{\text{beat}}=40\left(\frac{370-310}{300}\right)$

$\Rightarrow n_{\text{beat}}=40\times \left(\frac{60}{300}\right)$

$\Rightarrow n_{\text{beat}}=8$