Question

A police car moving at $22m/s$, chases a motorcyclist. The policeman sounds his horn at $176Hz$ , while both of them move towards a stationary siren of frequency $165Hz$. Calculate the speed (in $m/s$) of the motorcycle, if it is given that he does not observe any beats. Take speed of the sound, $v=330m/s$.

Answer 1

Given,
Speed of the police car, $v_s=22m/s$
Frequency of the police car, $n=176Hz$
We know that speed of the sound, $v=330m/s$

Speed of the observer (motorcycle), $v_0=v$

Frequency of the police car horn heard by motorcyclist,

$\Rightarrow n_1=n\left(\frac{v-v_0}{v-v_s}\right)$

$\Rightarrow n_1=176\times \left(\frac{330-v}{330-22}\right)\dots \left(i\right)$

Here, the siren is stationary, $v_s=0m/s$
Frequency of the siren, $n=165Hz$
We know that speed of the sound,
$v=330m/s$
Speed of the observer (motorcycle), $v_0=v$
Frequency of the siren heard by motorcyclist,

$\Rightarrow n_2=n\left(\frac{v+v_0}{v-v_s}\right)$

$\Rightarrow n_2=165\times \left(\frac{330+v}{330}\right)\dots \left(ii\right)$

$\Rightarrow n_1-n_2$

$\Rightarrow 176\times \left(\frac{330-v}{330-22}\right)-165\times \left(\frac{330+v}{330}\right)$

Since the motorcyclist does not observe any beats,

$\Rightarrow n_1-n_2=0$

$\Rightarrow 176\times \left(\frac{330-v}{308}\right)=165\times \left(\frac{330+v}{330}\right)$

$\Rightarrow \left(\frac{330-v}{330+v}\right)=\frac{165}{176}\times \frac{308}{330}$

$\Rightarrow \left(\frac{330-v}{330+v}\right)=\frac{7}{8}$

$\Rightarrow (8\times 330)-8v=(7\times 330)+7v$

$\Rightarrow 15v=330$

$\Rightarrow v=\frac{330}{15}$

$\Rightarrow v=22m/s$

Final answer: $22m/s$