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Chapter 2: Problem 9

The Doppler effect formula involves two speeds, \(v\) and c. The Doppler formula for sound involves three speeds (source, listener, and sound). Why a different number?

Short Answer

Expert verified
The Doppler formula for sound involves three speeds (sound, source, listener) because the sound's speed varies with the motion of both the source and the listener. Contrarily, the Doppler formula for light involves only two speeds (source and light), because the speed of light remains constant regardless of the motion of source or observer.

Step by step solution

01

Understanding the Doppler Effect

The Doppler Effect is the change in frequency or wavelength of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. It is the difference in sound or light wave frequency as perceived by an observer due to the motion of the source, the observer, or both.
02

Defining the Parameters: Speed of Source, Observer, and Wave

In the Doppler formula for sound, three speeds are involved: speed of the source (the object creating the sound), speed of the observer (the one perceiving the sound), and speed of sound (the speed at which sound travels in a certain medium). If either the source or the observer is moving, the frequency of the sound differs between them due to the difference in their movement speeds.
03

Understanding the Doppler Effect for Light

When it comes to light, there are only two speeds involved in the Doppler formula: \(v\) (the relative speed of the source and the observer) and \(c\) (the speed of light). This is because, according to the theory of relativity, the speed of light \(c\) is always constant and doesn't change with the motion of source or observer. It only depends on the medium it is traveling through.

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Key Concepts

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

Wave Frequency
Wave frequency refers to the number of waves that pass a fixed point in a given amount of time, and it is typically measured in Hertz (Hz). This concept is crucial when discussing the Doppler Effect, as the phenomenon directly affects perceived frequency. For instance, when an ambulance approaches an observer, the sound waves compress, leading to higher frequency and pitch. Conversely, as it moves away, the waves stretch, decreasing frequency and pitch.

It's important to comprehend that the original frequency emitted by the source doesn't change; it is the relative motion between the source and the observer that causes a variation in the observed frequency. Therefore, if the source is moving towards the observer, the observer will detect a higher frequency than what is being emitted. If the source moves away, the observer will detect a lower frequency.
Sound Speed
Sound speed, or the speed of sound, is the rate at which sound waves propagate through a medium. It is a fundamental element in the Doppler Effect equations for sound. Different factors, such as the medium's density, temperature, and composition, can affect this speed. For example, sound travels faster in water than in air and even faster in solids.

In air at room temperature, sound travels at approximately 343 meters per second (m/s). When discussing the Doppler Effect for sound, it鈥檚 crucial to consider the actual speed of sound in the specific medium, as it's the speed at which the sound waves are moving between the source and the observer and plays a role in how much the observed frequency is shifted.
Speed of Light
The speed of light, symbolized as 'c', is a universal physical constant important in many areas of physics, including the Doppler Effect for light waves. In a vacuum, it is precisely 299,792,458 meters per second and is used as the base speed at which all electromagnetic waves, including light, travel.

When considering the Doppler Effect for light, only the relative speed between the source and the observer ('v') is relevant along with the constant speed of light, because the speed of light remains unchanged regardless of the motion of either the source or the observer. This seeming independence of the speed of light's constancy leads into one of the most groundbreaking theories in physics: the theory of relativity.
Theory of Relativity
The theory of relativity, developed by Albert Einstein, fundamentally changed our understanding of space, time, and gravity. It consists of two parts: special relativity and general relativity. Special relativity introduced the idea that the laws of physics are the same in all inertial frames of reference and the speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source.

This has direct implications for the Doppler Effect concerning light. Unlike sound, where the medium's properties can affect the wave's speed, light in a vacuum always travels at 'c', making it invariant and not dependent on the movement of the source or observer. This core principle explains why the Doppler Effect for light only considers the relative velocity between the source and the observer and not the speed at which the light itself travels, as it remains constant.

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Most popular questions from this chapter

Classically, the net work done on an initially stationary object equals the final kinetic energy of the object. Verify that this also holds relativistically. Consider only one-dimensional motion. It will be helpful to use the expression for \(p\) as a function of \(u\) in the following: \(W=\int F d x=\int \frac{d p}{d t} d x=\int \frac{d x}{d t} d p=\int u d p\)

A particle of mass \(m_{0}\) moves through the lab at \(0.6 c\). Suddenly if explodes into two fragments. Fragment I. mass \(0.66 m_{0}\), moves at \(0.8 c\) in the same direction the original particle had been moving. Determine the velocity (magnitude and direction) and mass of fragment \(2 .\)

In the frame in which they are at rest, the number of muons at tiroe \(r\) is given by $$ N=N_{0} e^{-\nu / \tau} $$ where \(N_{0}\) is the number at \(r=0\) and \(\tau\) is the mean lifetime 2.2 \mus. (a) If muons are produced at a height of \(4.0 \mathrm{~km}\), beading toward the ground at \(0.93 \mathrm{c}\). what fraction will survive to reach the ground? (b) What fraction would reach the ground if classical mechanics were valid?

For reasons having to do with quantum mechanics, a given kind of atom can emit only certain wavelengths of light. These spectral lines serve as a "fingerprint." For instance, hydrogen's only visible spectral lines are \(656,486.434,\) and \(410 \mathrm{nm} .\) If spectral lines were of absolutely precise wavelength, they would be very difficult to discern. Fortunately, two factors broaden them: the uncertainty principle (discussed in Chapter 4 ) and Doppler broadening. Atoms in a gas are in motion, so some light will arrive that was emitted by atoms moving toward the observer and some from atoms moving away. Thus, the light reaching the observer will cover 8 range of wavelengths. (a) Making the assumption that atoms move no faster than their rms speed-given by \(v_{\mathrm{nns}}=\sqrt{2 k_{\mathrm{B}} T / m},\) where \(k_{\mathrm{B}}\) is the Boltanann constant obtain a formula for the range of wavelengths in terms of the wavelength \(\lambda\) of the spectral line, the atomic mass \(m,\) and the temperature \(T\). (Note: \(\left.v_{\text {rms }} \ll c .\right)\) (b) Evaluate this range for the 656 nm hydrogen spectral line, assuming a temperature of \(5 \times 10^{4} \mathrm{~K}\).

Consider the collisions of two identical particles. each of mass \(m_{0}\). In experiment \(A\), a particle moving at \(0.9 c\) strikes a stationary particle. (a) What is the total kinetic energy before the collision? (b) In experiment \(B\), both particles are moving at a speed \(u\) (relative to the lab), directly roward one another. If the total kinetic energy bef ore the collision in experiment \(B\) is the same as that in experiment A, what is \(u\) ? c) In both experiments, the particles stick together. Find the mass of the resulting single particle in each experiment. In which is more of the initial kinetic energy converted to mass?
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