Question

A policeman is doing his duty and is found to be detecting a drop of 10% in the pitch of the horn of a car in motion as it passes him. When the velocity of the sound is $330m{{s}^{-1}}$, then what will be the speed of the car?
$\begin{aligned} & A.36.7m{{s}^{-1}} \\\ & B.17.3m{{s}^{-1}} \\\ & C.25m{{s}^{-1}} \\\ & D.27m{{s}^{-1}} \\\ \end{aligned}$

Answer 1

The apparent velocity of the sound will be the product of the actual velocity and the ratio of the actual velocity to the sum of the actual velocity and the speed of the car. Find out the apparent velocity in terms of actual velocity and substitute this in the equation. This will help you in answering this question.

Complete step by step solution:
As it is mentioned in the question that the 10% of pitch of the horn is lost. Therefore he will be detecting 90% of it.
Let us assume that the original velocity of the car be $V$. As he is detecting 90% only, then the detectable velocity be,
$V\times \dfrac{90}{100}=0.9V$
This can be sometimes called the apparent velocity also.
The apparent velocity of the car with respect to the policeman can be found by the equation,
${V}'=V\left( \dfrac{V}{V+{{V}_{S}}} \right)$
Where ${{V}_{S}}$ be the speed of the car and ${V}'$ be the apparent velocity of the car. Let us substitute the values in it will give,
$0.9V=V\left( \dfrac{V}{V+{{V}_{S}}} \right)$
Let us rearrange the equation in terms of the speed of the car. That is,
$0.9V+0.9{{V}_{S}}=V$
That is,
$\begin{aligned} & 0.9{{V}_{S}}=0.1V \\\ & \Rightarrow {{V}_{S}}=\dfrac{V}{9} \\\ \end{aligned}$
The general velocity of sound in air can be shown as,
$V=330m{{s}^{-1}}$
Substituting the values in it will give,
${{V}_{S}}=\dfrac{330}{9}=36.7m{{s}^{-1}}$
Therefore the speed of the car has been calculated as $36.7m{{s}^{-1}}$.

The answer has been given as option A.

Note:
The Doppler Effect is defined as the variation in frequency of a wave with respect to an observer who is in motion relative to the source of the wave. A general example of the Doppler shift will be the variation of the pitch heard when a car sounding a horn approaches and goes away from an observer.