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A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant $k$ . Set up the differential equation and solve for the velocity.

$\mathbf{v}(t)=\sqrt{\frac{m g}{k}} \frac{\exp \left(-2 \sqrt{\frac{k g}{m}} t\right)-1}{1+\exp \left(-2 \sqrt{\frac{k g}{m}} t\right)} \hat{j}$Note that it's a tanh function

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

Campbell University

Harvey Mudd College

Boston College

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in this problem, we want to derive the linear drag formula, which is X equal to be not over K one minus e to the negative. Katie. Coastline Alfa. And why equal to be not over K one minus e to the negative. Katie sign Alfa plus G over K squared one minus K t minus. Key to the negative. Katie. So we want to find this from the initial value problem D squared are over e t squared equals negative G j minus k d r d t. So that's our Cody. And then we have initial conditions are 00 and d r d t zero is the not the vector, which is gonna be so v not co sign Alfa I Plus we not sign out, J Okay, so let's go ahead and just break up this into two problems. So one for X and one for white. So we'll have X double Prime is equal to you. Negative K X prize. We can We read that as X double prime plus k ex prime. His equals zero and so one of the initial conditions being so we'll have x of zero is zero and ex prime of zero is going to be be not co sign off and then we'll do the same for wife. So have why Double prime is equal. Teoh. Why? Double crying is equal the negative g minus k y five where we can write that as wide open prime plus k y is equal to negative and the initial conditions. So at y zero screw and why prime of zero is gonna be v not sign Alva. So that's our other problems. So notice that X is a homogeneous problem. And why has the non homogeneous from eso We'll start with X Because we're gonna do similar steps when we do. Why? Because why walk to split up in a homogeneous particular solutions? Okay, so for X right, we have, uh so we want to find the characteristics so we'll have lambda squared. Plus K lambda is equal to zero. So when we saw the Lambda will get landed, people zero and they get Okay, Spike, Factoring outlined, you get landed times. Landry plus K. All right, so that means that X looks like so, except he is gonna look like some constant times e to the negative. Katie. Plus some other constant times either The zero t which is gonna be one so upset that constantly So then we know that X zero is equal to zero is going to be equal Teoh C one plus C two. So that means that see, 18 bullets negative C to whatever that maybe we'll find out a second so than ex prime. So if we take the derivative of this Yugi, we would get, uh, negative k c one e to the negative. Katie close here. So we know that ex prime of zero is equal to be not close eye in Alfa. And that's gonna be a so click click and zero into this equation will just get negative. K C one. So that means that C one is equal. Teoh Negative V not over k coastline Alfa. All right, so what is X so X? Uh, t is gonna be a So we have Zi Juan or Sorry. See one so negatively not over. Okay. Coastline alfa Times e to the negative, Katie. And then we have plus C two So c two is negative C one So I have plus B not over k faux sign Alfa and we can rewrite the sole factor out the so we can factor out the Veena over K times, coastline, Alfa and then tidy it up of it. We have that in both terms, so we'll have to be not over k co sign Alfa time. So here's the positive one, and this will be a negative e. So they get Katie. All right. It looks pretty good. That's what we have here. So X makes sense. Why is not as much fun, but well, I mean, they just find if you like it. But see, Variant, copy this whole thing. Here we go. Come here. So, similarly, the X we're gonna have the same character. Six. So we're gonna have why? Homogeneous is going to be some constant e to the negative, Katie. Plus some other constant. The particular solution, though. So why particular? If we look here, we're gonna get Why, particularly double Prime Plus Kyi prime is equal to your negative g so that the particular solution is gonna look something like So what do you see? Three t squared, plus seat. Uh, for T plus C five. You never run out of enough sees until you get to like, too many numbers. Okay, so what are we doing here? All right, so then what happens when we take the derivative? So then why particular? Private E? It's gonna equal to C three t plus C four. And why a particular double Private e is gonna B two c three. That's it. So now we want to just plug all this into their and then, you know, we can get some of these constants to fall out. So we have to see three. That's my white. A little time plus K times to see three t plus C four is equal to negative, Gene. All right, so we have tombs three plus Xu K C three t plus K c for is equal to negative G R. K. So look over here. There are no t terms, so that means that si three equals zero. So now we get Oops. That's the ugliest cake. We have k c for as equal to your negative G. That means that C four is gonna be negative. G over k. All right. What do we do now? So we found What does this even find us? We have Ah. Oh, yeah. so note that we don't need the c five because we already have a constant term in my homogeneous solution. So when I add those together, they're gonna combined in the same concept. So I didn't really need that there. Um, OK, so that tells me that why particular of t is equal to negative G Over K. So now So why the solution is going to be the sum of its homogeneous solution. And it's particular solution. So that's gonna be C one e to the negative k T plus C two plus hoops minus G over K t. Okay, that's pretty good. So we have two unknowns and we have to initial values left, so that should get us what we want. So first we'll look at wives era because it's gonna get through some stuff. So we have zero is equal. Teoh C one plus C two. So that means that see one, you close negative season. Break it. So let's look at what is swiper. So we take the derivative will have negative K c one you to the negative. Katie minus, uh G over k. Really good. So now what we do. So we want by prime of zero is going to be. You could do Veena, sign Alfa. That's gonna equal negative K C one. So each of the zeros one, my ass G over. Okay, okay, So what is C one? So C one is going to be We're gonna have V not sign Alfa plus G over K Times one over. Negative K. What a mess. OK, and then so let's Heidi that up a bit. So that's gonna be be not over. Okay, they get sorry. Negative. You not over K sign. Alfa plus very fine US G over K squared. That means that C two is going to be be not over K sign Alfa plus G over Case word. All right, what is next? So now we need to plug everything back in. So we have gonna copy this because it's too much to keep track of Chelsea. We'll be OK. So I have negative one over. Okay? Oh, yeah. What's just how are you? This Okay, so I have negative. We not over K sign. Alfa Finest. She over case weird e to the negative. Katie, Right? Plus C two. So c two is v not over k sign Alfa plus P over K squared and that minus G over K T. Okay, so are we just need to combine, like terms. So let's put everything with sign off. Um, first, because that's how they did it in the formula. So be not over. K. Sign off. So we have this piece and this piece, so it'll be times one remind us. He to the negative. Katie. Okay, so next let's grab everything with G in it. So we have this piece this'll piece and this piece, and we'll do it like the book did it before they factored factored out of G over K square. So, uh, we'll try to do it in the order. So we have one from here, and then we have negative Katie here, and then the last piece is gonna be minus E to the negative. Katie, that was equal to y all right. Looks good. That matches what we started with. So we're going to go. We did it

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