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Chapter 16: Problem 99

You are riding your bicycle directly away from a stationary source of sound and hear a frequency that is \(1.0 \%\) lower than the emitted frequency. The speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What is your speed?

Short Answer

Expert verified
Your speed is approximately 3.5 m/s.

Step by step solution

01

Understand the Doppler Effect

The Doppler effect explains the change in frequency of a wave relative to an observer moving relative to the source of the wave. In this case, the observer (you on the bicycle) is moving away from the sound source, resulting in hearing a lower frequency than what is emitted.
02

Use the Doppler Effect Formula for Sound

The formula for the frequency heard by an observer moving away from a stationary source is given by: \[ f' = f \frac{v}{v + v_o} \] where \( f' \) is the observed frequency, \( f \) is the emitted frequency, \( v \) is the speed of sound, and \( v_o \) is the speed of the observer.
03

Relate Observed Frequency to Emitted Frequency

We know the observed frequency \( f' \) is \( 1.0 ext{\%} \) lower than the emitted frequency \( f \), meaning: \( f' = 0.99f \). Substitute this into the Doppler effect formula to set up the equation as follows: \[ 0.99f = f \frac{343}{343 + v_o} \].
04

Solve for Observer's Speed

Cancel \( f \) from both sides of the equation: \[ 0.99 = \frac{343}{343 + v_o} \]. Solve for \( v_o \):\[ 343 + v_o = \frac{343}{0.99} \]. Calculate \( 343 / 0.99 \) to find that the result is approximately \( 346.4646... \). Thus, \( v_o = 346.4646 - 343 \approx 3.4646 \).
05

Round Off to Reasonable Precision

Since the problem asks for the speed, rounding to one decimal place is appropriate: \( v_o \approx 3.5 \text{ m/s} \).

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Key Concepts

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

Understanding Frequency
Frequency refers to the number of waves that pass a point in a certain amount of time, typically measured in Hertz (Hz). It determines the pitch of the sound we hear. When observing sound waves, the frequency can seem different from its actual value due to motion. This effect is known as the Doppler Effect. In our scenario, when you ride a bicycle away from a stationary sound source, the sound waves are stretched. This stretching leads to a decrease in the frequency compared to the original sound emitted. Therefore, you hear a lower pitch.
Nature of Sound Waves
Sound waves are longitudinal waves that travel through air, water, or solids, carrying sound energy from one place to another. These waves consist of compressions and rarefactions, which travel through the medium. For sound to occur, there needs to be a medium to travel through, as sound cannot propagate in a vacuum. As you move away from the sound source, the spacing between the waves increases, altering the frequency that you perceive.
Speed of Sound Explained
The speed of sound is the velocity at which sound waves travel through a medium. In dry air at room temperature, this speed is approximately 343 meters per second (m/s). However, the speed can vary depending on factors like medium, temperature, and atmospheric pressure. In our problem, the speed of sound remains constant, enabling us to calculate how your motion affects the frequency. It forms a baseline for understanding how your movement alters what you hear.
Role of Relative Motion
Relative motion is the idea that motion is always described relative to something else. In the context of sound, it's crucial because the changes in frequency you perceive depend on your movement relative to the sound source. If the observer and source are stationary, the frequency remains unchanged. Here, because you are moving away from the sound source, relative motion causes the frequency of the sound to decrease.
Observer's Velocity Impact
The observer's velocity is the speed at which the person is moving concerning the source of the sound. This motion affects the frequency through the Doppler Effect. By moving away, you increase the distance the sound waves must travel, which increases the time between receiving consecutive waves. This makes the frequency appear lower. We can calculate this speed using the Doppler formula, given the observed 1% decrease, resulting in the observer's speed of approximately 3.5 m/s.

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Most popular questions from this chapter

A car is accelerating while its horn is sounding. Just after the car passes a stationary person, the person hears a frequency of \(966.0 \mathrm{Hz}\) Fourteen seconds later, the frequency heard by the person has decreased to \(912.0 \mathrm{Hz}\). When the car is stationary, its horn emits a sound whose frequency is \(1.00 \times 10^{3} \mathrm{Hz} .\) The speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What is the acceleration of the car?

An observer stands \(25 \mathrm{m}\) behind a marksman practicing at a rifle range. The marksman fires the rifle horizontally, the speed of the bullets is \(840 \mathrm{m} / \mathrm{s},\) and the air temperature is \(20^{\circ} \mathrm{C} .\) How far does each bullet travel before the observer hears the report of the rifle? Assume that the bullets encounter no obstacles during this interval, and ignore both air resistance and the vertical component of the bullets' motion.

A typical adult ear has a surface area of \(2.1 \times 10^{-3} \mathrm{m}^{2}\). The sound intensity during a normal conversation is about \(3.2 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) at the listener's ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?

When Gloria wears her hearing aid, the sound intensity level increases by \(30.0 \mathrm{dB}\). By what factor does the sound intensity increase?

A water-skier is moving at a speed of \(12.0 \mathrm{m} / \mathrm{s}\). When she skis in the same direction as a traveling wave, she springs upward every \(0.600 \mathrm{s}\) because of the wave crests. When she skis in the direction opposite to the direction in which the wave moves, she springs upward every 0.500 s in response to the crests. The speed of the skier is greater than the speed of the wave. Determine (a) the speed and (b) the wavelength of the wave.
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