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Chapter 16: Problem 62

(I) A bat at rest sends out ultrasonic sound waves at 50.0 \(\mathrm{kHz}\) and receives them returned from an object moving directly away from it at 30.0 \(\mathrm{m} / \mathrm{s} .\) What is the received sound frequency?

Short Answer

Expert verified
The received sound frequency is approximately 45,990 Hz.

Step by step solution

01

Understand the Doppler Effect Formula

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. For sound waves, the formula for the observed frequency \( f' \) when the source and observer are moving directly towards or away from each other is given by:\[ f' = f \left( \frac{v + v_0}{v + v_s} \right)\]Where:- \( f \) is the source frequency.- \( v \) is the speed of sound in air (approximately \(343 \text{ m/s} \) at room temperature).- \( v_0 \) is the speed of the observer (positive if moving towards the source).- \( v_s \) is the speed of the source (positive if moving away from the observer).
02

Identify the Given Values

From the exercise, we are given:- Source frequency, \( f = 50.0 \text{ kHz} = 50,000 \text{ Hz} \).- Speed of the object (observer) moving away, \( v_0 = 30.0 \text{ m/s} \).- The bat (source) is at rest, so \( v_s = 0 \text{ m/s} \).- Speed of sound in air, \( v \approx 343 \text{ m/s} \).
03

Apply the Doppler Effect Formula

Substitute the given values into the Doppler Effect formula:\[ f' = 50,000 \left( \frac{343 + 0}{343 + 30} \right)\]The formula simplifies to:\[ f' = 50,000 \times \left( \frac{343}{373} \right)\]
04

Calculate the Received Sound Frequency

Compute the calculation:\[ f' = 50,000 \times 0.9198 \approx 45,990 \text{ Hz} \]Thus, the frequency of the sound waves the bat receives is approximately \( 45,990 \text{ Hz} \).
05

Check the Result for Consistency

Since the object is moving away from the bat, the received frequency should be lower than the emitted frequency. The computed frequency of \( 45,990 \text{ Hz} \) is indeed less than \( 50,000 \text{ Hz} \), confirming that the Doppler effect was applied correctly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Shift
A key part of understanding the Doppler Effect involves the notion of frequency shift. In simple terms, the frequency shift refers to the change in the pitch or tone of a sound that occurs when the source of the sound and the observer are moving relative to each other.
- When they move towards each other, the frequency appears to increase, sounding higher.
- Conversely, when they move apart, the frequency appears to decrease, sounding lower.
This change in frequency, or frequency shift, is caused by the compression or stretching of the sound wave depending on the direction of movement.
Understanding this concept is crucial in applications such as radar and sonar, where waves are reflected off objects and analyzed based on their frequency shifts to determine the speed and direction of the object. In our given problem, the bat experiences a decrease in frequency, a classic scenario of a frequency shift due to the object moving away.
Ultrasonic Waves
Ultrasonic waves are sound waves with frequencies higher than the upper audible limit of human hearing. Most humans can hear up to 20 kHz, while ultrasonic waves typically start from 20 kHz onwards.
- These waves are used in various fields due to their high frequency and ability to reflect off small objects.
- Bats, for example, use ultrasonic waves for echolocation; they emit ultrasonic pulses and listen to the echoes that bounce back off objects. This helps them navigate and hunt even in complete darkness.
The exercise involves a bat emitting ultrasonic waves at 50 kHz, utilizing this method to detect an object. The object reflects these waves back, allowing the bat to perceive the environment through the changes in the frequency of the returned ultrasonic waves.
Velocity of Sound
The velocity of sound is the speed at which sound waves travel through a medium. In air, this speed is approximately 343 m/s at room temperature. However, the velocity can change based on several factors:
- **Temperature:** Warmer air results in faster sound transmission.
- **Medium:** Sound travels faster in liquids and even faster in solids, thanks to closely packed molecules.
In our problem, the given velocity of sound is used to determine the frequency shift experienced by the bat. The formula for the Doppler Effect relies heavily on the precise value of the velocity of sound to accurately calculate the observed frequency shift. Thus, knowing the correct velocity of sound is essential for such calculations and practical applications.

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