/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 243 A bus is moving towards a huge w... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 243

A bus is moving towards a huge wall with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). The driver sounds a horn of frequency \(200 \mathrm{~Hz}\). What is the frequency of beats heard by a passenger of the bus, if the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\).

Short Answer

Expert verified
The beat frequency heard by a passenger on the bus is approximately \(2.02\,\text{Hz}\).

Step by step solution

01

Analyze the given data

In this problem, we have: - Velocity of the bus (source), \(v_s = 5 \,\text{m/s}\) - Frequency of the horn, \(f_s = 200\,\text{Hz}\) - Speed of sound in air, \(v = 330 \,\text{m/s}\) The aim is to find the frequency of beats heard by the passenger on the bus.
02

Apply Doppler effect formula for the moving source and stationary listener

The Doppler effect formula for a moving source and a stationary listener is given by: \[f_{obs} = \frac{v}{v - v_s} * f_s\] where \(f_{obs}\) is the frequency observed by a stationary listener (in this case, the wall). Plug in the given values to find the frequency observed by the wall: \[f_{obs} = \frac{330}{330 - 5} * 200 = \frac{330}{325} * 200\]
03

Calculate the reflected frequency from the wall

Now we need to calculate the reflected frequency heard by the passenger on the bus. Since the wall is stationary, we need to consider the bus as the observer, and hence use the Doppler effect formula for the stationary source and moving observer: \[f_r = \frac{v + v_s}{v} * f_{obs}\] Plug in the given values (using the calculated frequency observed by the wall as \(f_{obs}\)): \[f_r = \frac{330 + 5}{330} * \frac{330}{325} * 200 = \frac{335}{330} * \frac{330}{325} * 200\]
04

Calculate the beat frequency

The beat frequency is the difference between the reflected frequency from the wall and the original frequency of the horn: \[f_{beat} = |f_r - f_s|\] Plug in the calculated reflected frequency (\(f_r\)) and the original frequency (\(f_s\)): \[f_{beat} = \left|\frac{335}{330} * \frac{330}{325} * 200 - 200\right|\]
05

Calculate the final result

Now, we can compute the final result, the beat frequency: \[f_{beat} = \left|\frac{335}{330} * \frac{330}{325} * 200 - 200\right| \approx 2.02\,\text{Hz}\] The beat frequency heard by a passenger on the bus is approximately 2.02 Hz.

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