Question

Source and observer start moving simultaneously along x and y-axis respectively. The speed of source is twice the speed of observer, $V_0$. If the ratio of observed frequency to frequency of the source is $0.75$, find the velocity of sound

• A. $\frac{11}{\sqrt{5}}{v}_0$
• B. $\frac{17}{\sqrt{5}}{v}_0$
• C. $\frac{16}{\sqrt{5}}{v}_0$
• D. $\frac{19}{\sqrt{5}}{v}_0$

Answer 1

The correct option is D $\frac{19}{\sqrt{5}}{v}_0$

Distance moved by observer in time t: $y=v_{0}t$

Distance moved by source m in time t: $x=2v_{0}t$

$f_{apparent}=f_0\left(\frac{v+v_0}{v-v_s}\right)$

$\frac{f_{apparent}}{f_0}=\left(\frac{v+v_0}{v-v_s}\right)$

We have $v_0=-2v_{0}cos\theta$ and $v_s=-v_{0}sin\theta$

$tan\theta =\frac{v_{0}t}{2v_{0}t}=\frac{1}{2}$

$0.75=\frac{v-2v_{0}cos\theta }{v-(-v_{0}sin\theta )}=\frac{v-2v_{0}cos\theta }{v+v_{0}sin\theta }$

$\frac{3}{4}=\frac{v-2v_0\frac{2}{\sqrt{5}}}{v+v_0\frac{1}{\sqrt{5}}}$

$3v+\frac{3v_0}{\sqrt{5}}=4v-\frac{16v_0}{\sqrt{5}}$

$v=\frac{19v_0}{\sqrt{5}}$

Hence, option $\left(D\right)$ is correct answer.