Question

A siren mounted on a police car (A) chasing car (B) and in the same direction emits sound of frequency \[180\,{\text{Hz}}\] . A thief is sitting on car (B) and velocity of both the cars is equal to \[30\,{\text{m}}\,{{\text{s}}^{ - 1}}\] . The wavelength of sound received by observer is (velocity of sound \[360\,{\text{m}}\,{{\text{s}}^{ - 1}}\] )

A. \[1.82\,{\text{m}}\]
B. \[2\,{\text{m}}\]
C. \[2.4\,{\text{m}}\]
D. \[0.28\,{\text{m}}\]

Answer 1

Hint:First of all, we will use the Doppler’s effect to find the apparent frequency of the sound as received by the observer. We will find the relative velocity of the sound with respect to the observer. Then we will use the relation between wavelength, frequency and velocity to find the wavelength.

Formula used:
The formula which is used to calculate the apparent frequency of the sound as received by the observer, when both the source and the listener are moving in the same direction:
\[n' = \left( {\dfrac{{v - {v_{\text{L}}}}}{{v - {v_{\text{S}}}}}} \right) \times n\] …… (1)
Where,
\[n'\] indicates the apparent frequency as received by the observer.
\[v\] indicates the velocity of the sound.
\[{v_{\text{L}}}\] indicates the velocity of the thief’s car i.e. the listener.
\[{v_{\text{S}}}\] indicates the velocity of the source i.e. the siren.
\[n\] indicates the frequency of the siren.
Again, the formula which gives the wavelength of the sound wave.
\[{\lambda _{{\text{observed}}}} = \dfrac{{{v_{{\text{apparent}}}}}}{{n'}}\] …… (2)
Where,
\[{\lambda _{{\text{observed}}}}\] indicates the observed wavelength of the sound wave.
\[{v_{{\text{apparent}}}}\] indicates the apparent velocity of the sound.
\[n'\] indicates the apparent frequency of the sound.

Complete step by step answer:
In the given question, we are supplied with the following data:
There is a police car with a siren, who is chasing a thief. The frequency of the siren is \[180\,{\text{Hz}}\] . The velocity of both the cars is \[30\,{\text{m}}\,{{\text{s}}^{ - 1}}\] .The velocity of the sound is given as \[360\,{\text{m}}\,{{\text{s}}^{ - 1}}\] .

Let us proceed to solve the numerical.
We will substitute the required values in the equation (1) and we get:
$n' = \left( {\dfrac{{v - {v_{\text{L}}}}}{{v - {v_{\text{S}}}}}} \right) \times n \\
\Rightarrow n' = \left( {\dfrac{{360 - 30}}{{360 - 30}}} \right) \times 180 \\
\Rightarrow n' = 180\,{\text{Hz}} $
Therefore, the apparent frequency of the sound is found out to be \[180\,{\text{Hz}}\] .

Since, both the cars is moving in the same direction, then the apparent velocity of sound will be:
${v_{{\text{apparent}}}} = 360 - 30 \\
\Rightarrow {v_{{\text{apparent}}}} = 330\,{\text{m}}\,{{\text{s}}^{ - 1}} \\$
Now, we use the formula (2) and we get:
${\lambda _{{\text{observed}}}} = \dfrac{{{v_{{\text{apparent}}}}}}{{n'}} \\
\Rightarrow {\lambda _{{\text{observed}}}} = \dfrac{{330}}{{180}} \\
\therefore {\lambda _{{\text{observed}}}} = 1.82\,{\text{m}} \\$
Hence, the observed wavelength of the sound wave is found out to be \[1.82\,{\text{m}}\] .

The correct option is A.

Note: While solving this problem, most of the students seem to have confusion regarding the correct use of the formula. In this case, both the observer and the source are moving in the same direction, hence relative velocity of the sound with respect to the observer is less than the actual velocity of the sound. If the observer and the source move towards each other, then the relative velocity is more than the actual velocity.