Question

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

• A. 33 m/s
• B. 22 m/s
• C. zero
• D. 11 m/s

Answer 1

Answer: (B)

22 m/s

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,

$n^{\prime }=n_{car}\left(\frac{v-v_m}{v-v_s}\right)$ ...................eq1 .

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,

$n^{\prime \prime }=n_{siren}\left(\frac{v+v_m}{v}\right)$ ...................eq2 ,.

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, $176\left(\frac{v-v_m}{330-22}\right)=165\left(\frac{v+v_m}{330}\right)$ ,

or $\left(\frac{v-v_m}{v+v_m}\right)=\frac{165}{176}\times \frac{308}{330}=7/8$ ,

or $15v_m=v$,

Speed of sound in air $v=330m/s$

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