Question

(II) Two automobiles are equipped with the same single frequency horn. When one is at rest and the other is moving toward the first at 18 m/s, the driver at rest hears a beat frequency of 4.5 Hz. What is the frequency the horns emit? Assume\(T = 20^\circ {\rm{C}}\).

Short answer

The frequency of horn is \(81.2\;{\rm{Hz}}\).

Step-by-step solution

Understanding the frequency of the automobile

In this problem, the frequency emitted by the automobile will be smaller than the frequency received by the automobile.

Given Data

The beat frequency is\({f_{\rm{b}}} = 4.5\;{\rm{Hz}}\).

The speed of the automobile is\(v = 18\;{\rm{m/s}}\).

The temperature is\(T = 20^\circ {\rm{C}}\).

Evaluating the frequency of the horn

The relation to the calculate frequency is given by,

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

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

On plugging the values in the above relation.

\(\begin{aligned}{l}{f_{\rm{h}}} &= \left( {\frac{{343\;{\rm{m/s}}}}{{18\;{\rm{m/s}}}} - 1} \right)\left( {4.5\;{\rm{Hz}}} \right)\\{f_{\rm{h}}} &= 81.2\;{\rm{Hz}}\end{aligned}\)

Thus, \({f_{\rm{h}}} = 81.2\;{\rm{Hz}}\) is the required frequency emitted by the horn.