The difference between the apparent frequency of a source of sound as perceived by an observer during its approach and recession is 2% of the natural frequency of the source. If the velocity of sound in air is 300 m/sec, the velocity of the source is (It is given that velocity of source << velocity of sound)The difference between the apparent frequency of a source of sound as perceived by an observer during its approach and recession is 2% of the natural frequency of the source. If the velocity of sound in air is 300 m/sec, the velocity of the source is (It is given that velocity of source << velocity of sound)

Easy

Solution

When the source approaches the observer
Apparent frequency $n' = \frac{v}{v - v_{s}} . n = n [\frac{1}{1 - \frac{v_{s}}{v}}]$
= $n {[1 - \frac{v_{s}}{v}]}^{- 1} = n [1 + \frac{v_{s}}{v}]$
(Neglecting higher powers because v_S << v)
When the source recedes the observed apparent frequency $n^{''} = n [1 - \frac{v_{s}}{v}]$
Given $n' - n'' = \frac{2}{100} n, v = 300 m / \sec$
$∴ \frac{2}{100} n = n [1 + \frac{v_{s}}{v}] - n [1 - \frac{v_{s}}{v}] = n [2 \frac{v_{s}}{v}]$
$\Rightarrow \frac{2}{100} = 2 \frac{v_{s}}{v} \Rightarrow v_{s} = \frac{v}{100} = \frac{300}{100} = 3 m / \sec$

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