Question

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

Answer 1

In first situation, a police car (source of sound) with a speed $v_s$, is approaching a motorcycle ( observer), which is moving away from police car with a speed of $v_m$. Therefore apparent frequency of sound heard by motorcyclist,
$n^{\prime }=n_{car}\left(\frac{v-v_m}{v-v_s}\right)$

In second situation, motorcyclist (observer) is approaching a stationary siren (source), with a speed of $v_m$. Therefore apparent frequency of sound heard by motorcyclist, $n^{\prime \prime }=n_{siren}\left(\frac{v+v_m}{v}\right)$

Now as no beats are observed by motorcyclist, this is possible only when difference in frequencies heard by motorcyclist is zero, i.e. $n^{\prime }-n^{\prime \prime }=0$, or $n^{\prime }=n^{\prime \prime }$,
or $n_{car}\left(\frac{v-v_m}{v-v_s}\right)=n_{siren}\left(\frac{v+v_m}{v}\right)$

Given $n^{\prime }=176Hz,n^{\prime \prime }=165Hz,v=330m/s,v_s=22m/s,n_{car}=176Hz$ and $n_{siren}=165Hz$

Hence,
$176\left(\frac{v-v_m}{330-22}\right)=165\left(\frac{v+v_m}{330}\right)$
$\left(\frac{v-v_m}{v+v_m}\right)=\frac{165}{176}\times \frac{308}{330}=7/8$
$15v_m=v$
Speed of sound in air $v=330m/s$

$v_m=330/15=22m/s$