Question

A police van, moving at $22 \mathrm{~m} / \mathrm{s},$ chases a motor- cyclist. The policeman sounds his horn at $176 \mathrm{~Hz}$, while both of them move towards a stationary siren of frequency 165 Hz as shown in the figure. If the motor-cyclist does not observe any beats, his speed must be: (take the speed of sound=330 m/s)

Answer 1

Calculate obvious sound frequency and the apparent frequency of sound heard by motorcyclist, according to the question no beats are observed by the motorcyclist. Therefore, Take the difference of obvious sound frequency and the apparent frequency and equate it to zero. Then calculate speed.

Formula used:
$\quad \mathrm{n}_{\text {car }}\left(\dfrac{\mathrm{v}-\mathrm{v}_{\mathrm{m}}}{\mathrm{v}-\mathrm{v}_{\mathrm{s}}}\right)=\mathrm{n}_{\text {siren }}\left(\dfrac{\mathrm{v}+\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}\right)$

Complete solution Step-by-Step:
Frequency describes the number of waves in a given amount of time that pass through a fixed place. So, if the time it takes is 1/2 second for a wave to pass. The measurement of hertz, abbreviated as Hz, is the number of waves that pass per second.
A police car (sound source) with a speed of $\text{vs}$ is approaching a motorcycle in the first situation.
Observer), which moves at a speed of $\text{vm}$ away from the police car.
Therefore, the obvious sound frequency heard by motorcyclists,
$\mathrm{n}^{\prime}=\mathrm{n}_{\mathrm{car}}\left(\dfrac{\mathrm{v}-\mathrm{v}_{\mathrm{m}}}{\mathrm{v}-\mathrm{v}_{\mathrm{s}}}\right)$
In second situation, motorcyclist (observer) is approaching a stationary siren (source), with a
speed of $\mathrm{v}_{\mathrm{m}}$
Therefore, apparent frequency of sound heard by motorcyclist,
$\mathrm{n}^{\prime \prime}=\mathrm{n}_{\mathrm{siren}}\left(\dfrac{\mathrm{v}+\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}\right)$
This is only possible when the difference in frequencies heard by the motorcyclist is zero, since no beats are observed by the motorcyclist.
i.e. $\mathrm{n}^{\prime}-\mathrm{n}^{\prime \prime}=0$
or $\mathrm{n}^{\prime}=\mathrm{n}^{\prime \prime}$
or $\quad \mathrm{n}_{\text {car }}\left(\dfrac{\mathrm{v}-\mathrm{v}_{\mathrm{m}}}{\mathrm{v}-\mathrm{v}_{\mathrm{s}}}\right)=\mathrm{n}_{\text {siren }}\left(\dfrac{\mathrm{v}+\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}\right)$
Given $\mathrm{n}^{\prime}=176 \mathrm{~Hz}, \mathrm{n}^{\prime \prime}=165 \mathrm{~Hz}, \mathrm{v}=330 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{s}}=22 \mathrm{~m} / \mathrm{s}, \mathrm{n}_{\mathrm{car}}=176 \mathrm{~Hz}$ and $\mathrm{n}_{\mathrm{siren}}=165 \mathrm{~Hz}$
Hence, $176\left(\dfrac{\mathrm{v}-\mathrm{v}_{\mathrm{m}}}{330-22}\right)=165\left(\dfrac{\mathrm{v}+\mathrm{v}_{\mathrm{m}}}{330}\right)$
or $\left( \dfrac{\text{v}-{{\text{v}}_{\text{m}}}}{\text{v}+{{\text{v}}_{\text{m}}}} \right)=\dfrac{165}{176}\times \dfrac{308}{330}=\dfrac{7}{8}$
$\therefore 15{{\text{v}}_{\text{m}}}=\text{v}$
$\therefore$ Speed of sound in air $\mathrm{v}=330 \mathrm{~m} / \mathrm{s}$
${{\text{v}}_{\text{m}}}=\dfrac{330}{15}=22~\text{m}/\text{s}$

Note:
Frequency is the number of occurrences per unit of time of a repeating event. It is also referred to as temporal frequency, where the contrast between spatial frequency and angular frequency is emphasized. The frequency is measured in hertz (Hz) units, which is equal to one repeated event occurrence per second.