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Chapter 10: Problem 1484

A bus is moving with a velocity of \(5 \mathrm{~m} / \mathrm{s}\) towards a huge wall. The driver sounds a horn off frequency \(165 \mathrm{~Hz}\). If the speed of sound in air is \(335 \mathrm{~m} / \mathrm{s}\), the number of beats heard per second by the passengers in the bus will be........ (A) 3 (B) 4 (C) 5 (D) 6 A vehicle with a horn of frequency \(\mathrm{n}\) is moving with a velocity

Short Answer

Expert verified
The number of beats heard per second by the passengers in the bus is approximately 3, which corresponds to option (A).

Step by step solution

01

(Step 1: Calculate the apparent frequency as the bus approaches the wall)

First, let's calculate the apparent frequency (\(f_1\)) of the horn due to the Doppler effect as the bus moves towards the wall. The Doppler effect formula for a moving source and stationary observer is given by: \[f_1 = f_0 \frac{v}{v - v_s}\] Where: - \(f_0\) is the original frequency of the source (165 Hz in this case) - \(v\) is the speed of sound in air (335 m/s) - \(v_s\) is the velocity of the source, i.e., the bus (5 m/s) Now let's plug the given values into the formula: \[f_1 = 165 \frac{335}{335 - 5} = 165 \frac{335}{330}\] Solving, we get: \[f_1 = 168.182 \mathrm{~Hz}\]
02

(Step 2: Calculate the apparent frequency as the sound is reflected off the wall)

Since the wall is stationary and behaves like an observer, we can use Doppler effect formula again to calculate the apparent frequency (\(f_2\)) heard after the sound is reflected off the wall and comes back to the bus. This time, the wall acts as a moving source and the bus as a stationary observer: \[f_2 = f_1 \frac{v + v_w}{v}\] Where: - \(v_w\) is the velocity of the wall But since the wall is stationary, \(v_w = 0\), and the formula becomes: \[f_2 = f_1 \frac{v}{v}\] So, \(f_2 = f_1\), which is \(168.182 \mathrm{~Hz}\).
03

(Step 3: Calculate the number of beats heard per second)

Now that we have both the original frequency (\(f_0\)) and the apparent frequency after reflection (\(f_2\)), the number of beats per second can be calculated by finding the absolute difference between them: \[n = |f_2 - f_0|\] Plugging in the values we found, we get: \[n = |168.182 - 165|\] Solving, we get: \[n = 3.182\] Rounding it to the nearest whole number, we get \(n \approx 3\).
04

(Answer)

The number of beats heard per second by the passengers in the bus is approximately 3, which corresponds to option (A).

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