Question

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

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

(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: \[f_1 = 168.182 \mathrm{~Hz}\]

(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 \(168.182 \mathrm{~Hz}\).

(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 \(n \approx 3\).

(Answer)

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

Key concepts

Beat Frequency

In physics, beat frequency is an important concept that explains sound variations perceived when two similar frequencies interact. When two sound waves of slightly different frequencies overlap, they produce an alternating pattern of loud and soft sounds. These variations are called "beats," and the frequency at which they occur is known as the beat frequency.

The beat frequency (\(n\)) is determined by the absolute difference between the two frequencies involved. Formulaically, it is given by:

• \[ n = |f_1 - f_2| \]

This means that if you have two sound waves with frequencies \(f_1\) and \(f_2\), the number of beats per second heard will be the difference between those frequencies.

In our exercise, understanding the concept of beat frequency helps us calculate how the passengers on the bus would perceive the sounds from the horn and its reflection off the wall. As a result, they perceive three distinct beats per second, aligning with the calculated and rounded value of 3 Hz.

Acoustics

Acoustics is the science that studies sound, including its production, transmission, and effects. This aspect of physics examines how sound is generated, how it travels through mediums like air and water, and how it is perceived. Understanding acoustics is key to analyzing scenarios where sound changes as it interacts with objects and environments.

In the original exercise, acoustics comes into play when we consider how the horn's sound travels towards the wall and is reflected back to the bus. This interaction involves the sound being partially absorbed by and partially reflected off a surface. The passengers inside the bus hear both the original sound and the reflected sound, leading to the beating pattern. Acoustics helps us understand how these sounds merge and how variations like beat frequencies occur in natural environments.

Velocity of Sound

The velocity of sound is a measure of how fast sound waves travel through a medium. In air, this speed is typically around 335 m/s, though it can vary based on environmental conditions like temperature, humidity, and pressure.

Understanding the velocity of sound is crucial in applications such as the Doppler effect, which describes changes in the frequency of waves as the source and observer move relative to each other. The speed of sound affects how frequencies are perceived, especially when objects like the bus and the wall in our exercise are involved.

In the given scenario, the velocity of sound is a key factor in determining the apparent frequencies perceived by the passengers. It is part of the Doppler effect formula used to calculate how the frequency of the sound from the horn changes as it moves towards and reflects back from the wall. Thus, knowledge of the velocity of sound enables accurate calculations of both the apparent frequency when the sound approaches the wall and the resultant beat frequency heard by the observers.