A source of sound and a detector are placed at the same place on ground. At $t = 0$ , the source $S$ is projected towards reflector with velocity $v_{0}$ in vertical upward direction and reflector starts moving down with constant velocity $v_{0}$ . At $t = 0$ , the vertical separation between the reflector and source is $H (> \frac{v_{0}^{2}}{2 g})$ . The speed of sound in air is $v (>> v_{0})$ . Take $f_{0}$ as the frequency of sound waves emitted by source. Find the frequency of sound received by detector after being reflected by reflector at $t = \frac{v_{0}}{2 g}$ .

f_{2} = f_{0} (\frac{v^{2}}{(v - v_{0}) (2 v - v_{0})})

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f_{2} = 2 f_{0} (\frac{v^{2}}{(v - v_{0}) (2 v - v_{0})})

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f_{2} = f_{0} (\frac{v (v + v_{0})}{(v - v_{0}) (2 v - v_{0})})

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f_{2} = 2 f_{0} (\frac{v (v + v_{0})}{(v - v_{0}) (2 v - v_{0})})

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Solution

The correct option is D $f_{2} = 2 f_{0} (\frac{v (v + v_{0})}{(v - v_{0}) (2 v - v_{0})})$
From the data given in the question,

Initially source is projected with velocity $v_{0}$ upwards, the velocity of the source after a time $t = \frac{v_{0}}{2 g}$ is

$v_{s} = v_{0} - \frac{v_{0}}{2 g} \times g = \frac{v_{0}}{2}$

Number of waves reaching the reflector per second.

$f_{1} = ⎛ ⎜ ⎜ ⎝ \frac{v + v_{0}}{v - \frac{v_{0}}{2}} ⎞ ⎟ ⎟ ⎠ f_{0}$

Now, the reflector behaves as a source of sound of frequency $f_{1}$ moving towards the detector at rest.

$f_{2} = \frac{v}{v - v_{0}} f_{1}$

$\Rightarrow \frac{v}{v - v_{0}} \times ⎛ ⎜ ⎜ ⎝ \frac{v + v_{0}}{v - \frac{v_{0}}{2}} ⎞ ⎟ ⎟ ⎠ \times f_{0}$

Number of waves reaching the detector per second is

$f_{2} = 2 f_{0} (\frac{v (v + v_{0})}{(v - v_{0}) (2 v - v_{0})})$

Hence, option (d) is the correct answer.

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Similar questions

Q. A source of sound and detector are placed at the same place on ground. At t = 0, the source S is projected towards reflector with velocity

V_{0}

in vertically upward direction and reflector starts moving down with constant velocity

v_{0}

. At t = 0, the vertical separation between the reflector and source is

H (> \frac{v_{0}^{2})}{2 g})

. The speed of sound in air is

v (≫ v_{0})

. Take f_0 as frequency of sound waves emitted by source. Based on above information answer the following questions.

Wavelength of sound waves as received by detector before reflection at

t = \frac{V_{0}}{2 g}

,is

A source of sound and a detector are placed at the same place on ground. At t = 0, the source S is projected towards reflector with velocity $V_{0}$ in vertically upward direction and reflector starts moving down with constant velocity $V_{0}$ . At t = 0, the vertical separation the reflector and source is $H (> \frac{v_{0}^{2}}{2 g})$ . The speed of sound in air is v(>> $v_{0}$ ). Take f₀ as the frequency of sound waves emitted by source. Based on above information answer the following questions.