Wave Motion 4 Question 4

4. A stationary source emits sound waves of frequency $500 Hz$. Two observers moving along a line passing through the source detect sound to be of frequencies $480 Hz$ and $530 Hz$. Their respective speeds are in $ms^{-1}$,

(Take, speed of sound $=300 m / s$ )

(2019 Main, 10 April I)

(a) 12,16

(b) 12,18

(c) 16,14

(d) 8,18

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Answer:

Correct Answer: 4. (b)

Solution:

  1. Given,

Frequency of sound source $\left(f _0\right)=500 Hz$ Apparent frequency heard by observer $1, f _1=480 Hz$ and apparent frequency heard by observer $2, f _2=530 Hz$.

Let $v _0$ be the speed of sound.

When observer moves away from the source, Apparent

frequency, $f _1=f _0\left(\frac{v-v _0^{\prime}}{v}\right)\cdots(i)$

When observer moves towards the source,

Apparent frequency, $f _2=f _0\left(\frac{v+v _0^{\prime \prime}}{v}\right)\cdots(ii)$

Substituting values in Eq. (i), we get

$$ 480=500\left(\frac{300-v _0^{\prime}}{300}\right) $$

$$ \begin{aligned} \Rightarrow & 96 \times 3 & =300-v _0^{\prime} \\ \Rightarrow & v _0^{\prime} & =12 m / s \end{aligned} $$

Substituting values in Eq. (ii), we get

$$ \begin{aligned} 530 & =500\left(\frac{330+v^{\prime \prime}}{300}\right) \\ \Rightarrow \quad 106 \times 3 & =300+v^{\prime \prime} \\ \Rightarrow \quad v _0^{\prime \prime} & =18 m / s \end{aligned} $$

Thus, their respective speeds (in $m / s$ ) is 12 and 18 .



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