Problem

The figure shows a beam of length $L$ embedded in…

Problem 46 Easy Difficulty

In the theory of relativity, the energy of a particle is
$$E=\sqrt{m_{0}^{2} c^{4}+h^{2} c^{2} / \lambda^{2}}$$
where $m_{0}$ is the rest mass of the particle, $\lambda$ is its wave
length, and $h$ is Planck's constant. Sketch the graph of $E$ as
a function of $\lambda .$ What does the graph say about the energy?

Answer

$E=\sqrt{m_{0}^{2} c^{4}+h^{2} c^{2} / \lambda^{2}}$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

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Video Transcript

So here in this equation we have f is a function of, um Lando. All right, so this is wave link. Um, the mass of the particle at rest. Speed of light. Planck's constant and wavelength, so really are variable is buidling Orland. Okay, so the domain here, um, let's make this a little smaller to fit everything. The domain here is gonna be X has to be positive. So X is gonna have to be greater than zero. And why is that? Because wavelength has to be positive because it's, you know, measuring wavelengths. Okay, so there are no points of intersection, but we do have asked him to. It's so vertical. Ask them to, um we can see because of the Lambda here. Dividing by zero. Can't do that. So, vertical ass in two we're gonna see at Lambda equals zero. Okay? And now we're gonna find the horizontal Lassen totes by looking as the limit. Right? Says limit as X approaches infinity over function. So off of Lambda. Let me see that it's gonna turn out to be. And that wrist see, C squared. All right, so I'm sorry. My bed. So here is not exits. Not our, um, variable. It's Linda. Great. So here we have a horizontal ask himto e equals and at rest, see X squared. All right. And now we know that half of Lando is going to be decreasing. Unstow ming, so it's decreasing. All right, so now if we look at the graph, this is Lando, and this is energy or f of lambda. We know that we haven't asked himto a x equal Lambda equals zero, which is here. And we know we have another ask himto at e equals e equals and initial C squared. And now we don't know what that is. What this is exactly. Because, for example, we know that, um, c squared is a constant speed of light squared. Um, but, um ah, and the mess initially, we're gonna have to, you know, gain that in another way. But let's just say that it's somewhere around here. So let's say that this is e equals and that wrist C squared. All right, so it's somewhere there, and we know that it's always decreasing or function is always decreasing over are doing. And it'll proto been on touch asthma, toots. So we can see that this is an inverse relationship. It's Enver Energy and, um, wavelength are inversely proportional meaning as energy as well as energy increases are as energy energy decreases as wavelength increases.

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