in-the-theory-of-relativity-the-mass-of-a-particle-with-speed-v-is-mfvfracm_0sqrt1-v2-c2-where-m_0-2

Question

In the theory of relativity, the mass of a particle with speed $v$ is
$$m=f(v)=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}$$
where $m_{0}$ is the rest mass of the particle and $c$ is the speed of light in a vacuum. Find the inverse function of $f$ and explain its meaning.

Carlos Pinilla · Accepted Answer

So you have the bullying problem. Mom must off. I&#x27;m object. Moving bullion sp You know, relativity is a model by this question where not from us Erased particle of interests must sees the velocity of light. Thank you. So, uh, with his extreme verse, So, uh mm. It&#x27;s a function off. The finding a numbers means that we want to fine fortune. The U S. Unfortunate for us. So that gives is there with us? So, in other words, we said, Well, muss are if he is gonna be ableto that expression and over square with one minus this quote over she squared. I don&#x27;t want to find, uh, our velocity a PSA fortune on the b expressing terms of unless terms off. Okay, so your mom&#x27;s too the soul for G desecration on this world on me, cancer is not heart will be easier, is there? So simply five, uh, so, uh, well began move. Usually there the Yeah, right. Multiplying right square with on both sides. So that&#x27;s gonna give us quote one. My nose be squirt over. See, squirt is equal to and not for him. Right? So at this point, you can when it&#x27;s squared up square both sides of the equation, so well, since this choir room always e this, uh, yeah, that&#x27;s Mrs was eating number. Should people. So is gonna give us one lying us? Yeah. Article waas must be traveling at a smaller rescue than the speed of light. So that&#x27;s gonna be pushing. So you have these, um, I&#x27;m going on at this point. What if Louise Well, that guy there, these run there. Uh, bye. So striking that both sides in the question not over him. Word on being RD on both sides of the question. The square. Where is he? Square. So, doing all that? We&#x27;re gonna get one minus these. So it&#x27;s obstructing that one. Minus him, Not over. Mmm. Or the squared should be equal to these. You&#x27;re excited, too. Gets up. Sign the square. We&#x27;re square. I&#x27;m going. At this point, you can take school. Exclude of that, they take square it. Well, the velocities are positive. So we&#x27;re gonna get Oh, only the busty, this quote squirrel one. Oh, uh, I&#x27;m not divided by but one way. This I&#x27;m not. If I am, It&#x27;s a word vehicle too. Over a seat. I&#x27;ve been at this point or move this year they&#x27;re moving bloody both sides with you. So you gonna get that these velocity vehicle to see Time&#x27;s up. So see, one mine does not over. So these didn&#x27;t burst fortune. Oh, so orange is that if you plug into these your ma&#x27;s, it gives you back the velocity. So, uh, what does it mean? That so it means, uh, well, so if you&#x27;re going to these So to get, um, us, you have a must look, some us which velocity you should you should be. So if universal them, we&#x27;ll give you but but which velocity? You should be traveling two. But being that must, it gives you that of us on there.

Linda Hand · Answer

Okay. So M is the mass of an object and M sub not. Is the initial mass How much it way before he started moving around? Okay. V is the velocity of the mass and C. Is the speed of light? So the question is, what happens to this mass? This mass as your as its velocity approaches the speed of light? I put a little minus sign there because I mean I&#x27;m approaching it from numbers less than it. Because we have to slowly get up. We have to come from a slower speed to get to see. Okay, so as we get close to see then what happens to V squared over C squared? Well B and C are almost the same then V squared over C squared is almost one. Okay, so we have um sub zero over the square to one minus one nipple. Little minus sign here and buy this minus. I mean the same thing. It&#x27;s a little bit less than one. Okay, so the bottom is one minus something. A tiny little bit less than it. So that&#x27;s zero. Okay, but it&#x27;s positive zero. It&#x27;s a little bit bigger than zero because this is a little bit smaller than one. So it&#x27;s square root is a little bit bigger than zero. So what happens if you have a number and you divide it by a tiny tiny tiny little thing. That&#x27;s almost zero. Well it goes to infinity infinite mass. As its velocity approaches the speed of light, its mass becomes infinite

Paul George · Answer

everybody we&#x27;re doing. Chapter two, Section three Problem 44. This problem stands out to me in this book because you deal a lot with limits and you deal a lot with calculations. And this problem tends to go just a little bit more theoretical. And I appreciate that. So if we drew out what the actual you know equation is when we talk about the parts, we can break this thing down and we can have a good understanding of what&#x27;s going on. Okay, so it talks about what all the pieces are right. M sub zero is just your initial mass. M is the mass of the particle. V is the velocity. So this is This is the thing that we got to talk about, right? It says what happens as the approaches see from the left. We need to talk about that because it&#x27;s a limit and it&#x27;s a it&#x27;s a good application. Actually, of limits is the approach. Is he from the left? Eventually, hypothetically, right. Let&#x27;s assume that the becomes C, right. Um, well, if they were to become see then that means that V squared over C squared would become one and one minus one would be zero, and you would be dividing by zero. Which means that you&#x27;d be getting a vertical ash bento, which means that this thing would be to people up or it would be going down. Um, since this is sort of a massive a particle, it&#x27;s going to be positive. So that means that when, um, you have sort of this line going right And you have, you know, see, and you have this line of things that are happening in the is going to see right. You have things that are happening, things that are happening. Things were happening. It as it goes to see it&#x27;s going to just absolutely explode. So this is the really interesting thing that happens is like, That&#x27;s what that&#x27;s what goes on is like as the imaginative use, a lot smaller than it right. Then if C is much bigger than the that, that means that C essentially, uh, kills kills this fracture. Sort of imagine. Imagine if you had to write as your V squared and you had, like, 1000 as your C squared right two is approaching 1000. It&#x27;s really far away, but it&#x27;s coming out of 1000 from from the left and it&#x27;s looking forward. Five and six and 78 It&#x27;ll get bigger, right? It&#x27;ll eventually if you&#x27;d let those numbers get bigger than V squared. If you let it get bigger, it will become she&#x27;s quite. But right now this thing is really close to zero, right? It&#x27;s one over 500. It&#x27;s really tiny. And so that&#x27;s the thing is that you come actually from this world of like being really close to zero and you get bigger and bigger and bigger and bigger until you explode. Um, yeah, so I mean, this is a theoretical problem. That&#x27;s kind of just what we&#x27;re talking about, right? We&#x27;re not getting any sort of distinct answers. It&#x27;s just we want you to be able to understand what&#x27;s going on and what I want you to question yourself and to be able to ask good questions and come up with some stuff, so I hope this helps until next time.

Yaw Asomani · Answer

you have the mass to be, uh, the initial square root off one minus velocity squared over his feet of nice squared. And you&#x27;re taking the limit as the velocity approaches the speed of light. Well, and this is gonna be the limit. As velocity approaches feed off night, it is gonna be off. This is gonna be I am not over one minus. Now, if velocities approaching, see, then this one is gonna be C squared over. C squared. So what is he squared over? C squared. It is one. Right. So this becomes I am not over screwed off one minus one. So it&#x27;s zero, right? So it becomes infinity, right?

Jamie M · Answer

were sketching our function m as a function of V. So I just called the f a V. Um, so this is our function. So first, to find her domain, we have that zero to seize our domain because it must be positive. Under the radical intercepts we have from zero to em initial, because if we set our extra zero, we just have a minute Shal over one symmetry none. Assam toots. We have a vertical ass in tow vehicle, See? And the reason for that is if he does equal, see, we have one. So one minus 10 and we cannot have our function over zero. All right, so of prime of the equals. So we&#x27;re doing the first derivative to see where it&#x27;s increasing. This is her first derivative C squared, minus B squared, cubed. All right. And we see that this is increasing over our entire domain, which is from zero to see. So there are no local men&#x27;s or Max is. And if we find a double prime Seacon cavity you see, see on initial times to V squared plus C squared over V squared minus C squared. That&#x27;s all squared times radical C squared minus B squared. And we see that&#x27;s also above zero everywhere over our our domain zero to see. So that means it&#x27;s Kong cave up over our domain. Right? And now we can graft or function. So you know it&#x27;s gonna be our X axis is gonna be V. And our Y axis is gonna be because here M is equal to F V. All right, so we haven&#x27;t intercept at zero. I am initial. We haven&#x27;t asked him to at sea, and our graph will approach but not touch, See?

August Miller · Answer

for this a problem were given a large expression and asked to isolate. See? So let&#x27;s start by multiplying by the denominator and dividing by M to bring this entire expression over to the other side of the problem. Now we have a radical one minus b squared over C squared is equal toe. I&#x27;m not over em. This makes things easy for us because we can comfortably square both sides of the equation. So we now have one minus b squared over C squared is equal toe m not squared over m squared. Now we subtract. I&#x27;m not squared over m square and add v squared over c squared. This might be a little confusing, but we&#x27;re basically just switching these two, bringing everything over to the side We wanted to be on. So we have the square over Xi squared is equal to one minus that I m not square over em. Score. Next. We&#x27;re gonna take this entire term of one minus. I&#x27;m not squared over m squared and divide by it. Then we&#x27;re gonna multiplied by C squared. So again, we&#x27;re basically switching this denominator c squared with the entire expression on the other side. This makes things really easy for us. We simply have C squared by itself on one side of the equation and this entire term of V squared over one minus m not squared over m squared so we can simply put a square root over each of these and get C is finally equal to V&#x27;s radical V squared over one minus. I&#x27;m not swear over em a long process, but it gives us a pretty smooth final solution.

Problem

Find a formula for the inverse of the function. …

Problem 20 Easy Difficulty

In the theory of relativity, the mass of a particle with speed $v$ is
$$m=f(v)=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}$$
where $m_{0}$ is the rest mass of the particle and $c$ is the speed of light in a vacuum. Find the inverse function of $f$ and explain its meaning.

Answer

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James Stewart

Biocalculus Calculus for the Life Sciences

Anna Marie V.

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Precalculus Review - Intro

Functions on the Real Line - Overview

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See answer

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

James Stewart

Biocalculus Calculus for the Life Sciences