Question

(II) A bat at rest sends out ultrasonic sound waves at 50.0 kHz and receives them returned from an object moving directly away from it at 27.5 m/s. What is the received sound frequency?

Short answer

The received sound frequency is \(4.2 \times {10^4}\;{\rm{Hz}}\).

Step-by-step solution

Understanding the moving observer

In this problem, the object moving directly away from the bat will be considered as observer for determining the received frequency.

Given Data

The frequency of the sound waves of bat is\({f_{\rm{b}}} = 50\;{\rm{kHz}}\).

The speed of the object is \(v = 27.5\;{\rm{m/s}}\).

Evaluating the frequency received

The relation to the calculate frequency is given by,

\({f_{\rm{0}}} = \frac{{\left( {1 - \frac{v}{{{v_{\rm{s}}}}}} \right)}}{{\left( {1 + \frac{v}{{{v_{\rm{s}}}}}} \right)}}{f_{\rm{b}}}\)

Here, \({v_{\rm{s}}}\) is the speed of the sound.

On plugging the values in the above relation.

\(\begin{aligned}{l}{f_{\rm{0}}} &= \frac{{\left( {1 - \frac{{27.5\;{\rm{m/s}}}}{{343\;{\rm{m/s}}}}} \right)}}{{\left( {1 + \frac{{27.5\;{\rm{m/s}}}}{{343\;{\rm{m/s}}}}} \right)}}\left( {50\;{\rm{Hz}} \times \frac{{{{10}^3}\;{\rm{Hz}}}}{{1\;{\rm{kHz}}}}} \right)\\{f_0} &= 4.2 \times {10^4}\;{\rm{Hz}}\end{aligned}\)

Thus, \({f_0} = 4.2 \times {10^4}\;{\rm{Hz}}\) is the required frequency.