Question:
A policeman on duty detects a drop of $15 \%$ in the pitch of the horn of a motor car as it crosses him. If the velocity of sound is $330 \, m \, s^{-1}$ , calculate the speed of the car.
A policeman on duty detects a drop of $15 \%$ in the pitch of the horn of a motor car as it crosses him. If the velocity of sound is $330 \, m \, s^{-1}$ , calculate the speed of the car.
Updated On: May 11, 2024
- $26.7 \, m \, s^{-1}$
- $27.6 \, m \, s^{-1}$
- $53.4 \, m \, s^{-1}$
- $54.3 \, m \, s^{-1}$
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The Correct Option is A
Solution and Explanation
Here, velocity of sound, $v = 330 \, m \, s^{-1}$
Frequency of the horn = $ \upsilon$
Let $\upsilon_c$ be the speed of the car.
The frequency of the horn of the car heard by the policeman before it crosses him is,
$ \upsilon' = \upsilon \left( \frac{v}{v - v_c} \right) $ ....(i)
and after it crosses him is
$ \upsilon" = \upsilon \left( \frac{v}{v + v_c} \right) $ ....(ii)
Divide (ii) by (i), we get
$\frac{\upsilon"}{\upsilon'} = \frac{\nu -\nu_{c}}{\nu+\nu_{c} } = \frac{330 - \nu_{c}}{330 + \nu_{c}}$
As the pitch (i.e. frequency) drops by $15 \%$ so
$ \frac{ \upsilon"}{\upsilon'} = \frac{85}{100} = \frac{17}{20} , Thus , \frac{17}{20} = \frac{330 - \nu _{c}}{330 + \nu _{c}}$
$ 5610 + 17 \upsilon_{c} = 6600 - 20\upsilon_{c} $
$37 \upsilon_{c}=990 \, \, \therefore \, \, \nu_{c} = \frac{990}{37} = 26.7 m s^{-1}$
Frequency of the horn = $ \upsilon$
Let $\upsilon_c$ be the speed of the car.
The frequency of the horn of the car heard by the policeman before it crosses him is,
$ \upsilon' = \upsilon \left( \frac{v}{v - v_c} \right) $ ....(i)
and after it crosses him is
$ \upsilon" = \upsilon \left( \frac{v}{v + v_c} \right) $ ....(ii)
Divide (ii) by (i), we get
$\frac{\upsilon"}{\upsilon'} = \frac{\nu -\nu_{c}}{\nu+\nu_{c} } = \frac{330 - \nu_{c}}{330 + \nu_{c}}$
As the pitch (i.e. frequency) drops by $15 \%$ so
$ \frac{ \upsilon"}{\upsilon'} = \frac{85}{100} = \frac{17}{20} , Thus , \frac{17}{20} = \frac{330 - \nu _{c}}{330 + \nu _{c}}$
$ 5610 + 17 \upsilon_{c} = 6600 - 20\upsilon_{c} $
$37 \upsilon_{c}=990 \, \, \therefore \, \, \nu_{c} = \frac{990}{37} = 26.7 m s^{-1}$
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Waves are a disturbance through which the energy travels from one point to another. Most acquainted are surface waves that tour on the water, but sound, mild, and the movement of subatomic particles all exhibit wavelike properties. inside the most effective waves, the disturbance oscillates periodically (see periodic movement) with a set frequency and wavelength.
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