Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Like

Report

No Related Subtopics

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

00:38

Amy J.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:02

Anshu R.

0:00

Felicia S.

Dungarsinh P.

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

Create your own quiz

Okay, so here's our example. It involves something for physics. So I'm going to just walk you through all of this. I'm actually going Thio not right out any of the details. I'm just going to show you kind of the step by step process of how you would solve this problem. So first of all, note that maybe you know this maybe you don't. But Newton's second law for an object moving in two dimensions is given by these two differential equations. Okay? And now these air second order equations, okay? Because the, um now I guess I should say that in general, there there not second order because, you know, the forces and a direction could depend on a higher derivative. But in simple cases, the forces in the X in my direction, they're just going to be fixed. Or maybe they only depend on the velocity or something like that. So in the examples we're going to look at these will be second order differential equations, Okay. And then here M is the mass of the object. And so this is like mass. The second derivative of position is acceleration. So this is you know, sometimes you see this in the form F equals in a I mean, what it really is is net Force is equal to m A. And here the acceleration is just the second derivative. Okay, so typically, it's a second order differential equation. Okay, so under ideal conditions, and now this is going to be ignoring non conservative forces such as drag. Well, actually consider drag in just a minute. Okay? But first of all, let's just do the ideal case. So assuming that there's no drag force, this, of course, is not realistic. When a ball is thrown, the net force. So the right hand side, this differential equation is zero. So that's really nice. And then the net force in the UAE or vertical direction, is minus mg. Now, that's just gravity. Okay, so that means that in the vertical direction, there's always a force pushing me down. So here we have minus Mm. Gene. And so these these differential equations become very easy to solve, but we do need to consider the initial data. Okay, So the initial position and the initial velocity, so notice that the ball is going to be thrown with an initial speed V a trajectory, angle theta will just say, with respect to the horizontal and from ah Hei h off the ground. So let me just draw a quick picture of what's going on here is so maybe I'm a person and the ball is initially here in my hands. So this is a judge and then I'm going to throw it, and now I throw it at an angle theta. That's my trajectory angle, okay? And it's gonna land over here somewhere. So my trajectory angle is data, and then I have some initial velocity V, so I throw it at some speed. That's pretty, pretty straightforward. What's going on here? Very simple physical application. But what we're going to see is that calculus actually really allows us to dig into ah lot of different questions about what could be going on here. So the point is we get an initial value problem. We actually get to initial value problems. We get y double prime is equal to minus G. So that's just comes from canceling these factors of them. In the next double prime of tea is zero. That just comes from the fact that I can divide by zero so The second derivative is zero in the X direction. The second derivative is minus G in the Y direction and then here our initial conditions. So I'm saying that the ball starts at the 0.0 comma age. So my initial position in the X zero my initial velocity in the X is well, this is just some trigonometry, right? So I just bring my right triangle down. The X component of initial velocity is just We could send data. It's the adjacent side y of zero. Like we said, his h and the initial velocity. And why is the opposite side of this right triangle? So be cited. And then we just solve those differential equations. Okay, so we're just finding anti derivative. It's really easy to do because this is just a constant in that zero and we get the equations of motion. So the X, as you know, so t equals zero over here. But then any time later, my exposition is just to be because identity and then my position and why is just negative 16 t squared, plus the signs of T plus h. Okay, so nothing magical there. It's just solving too relatively simple second order differential equations with initial conditions. And that's what we get. And now you may recognize these. If you've taken physics. These are the Kinnah Matic equations emotion. But they're coming from solving Newton's second law, which is a different language. Okay, so hopefully that makes sense. Now, let's just kind of explore what we can do with this information that we got from solving these initial value problems. Okay, so the first thing we want to do is just determined the range and now the range if I go back to my picture is nothing more than the horizontal distance did the ball travels right again. So we have the equations of motion given initial velocity, hide off the ground and the trajectory angle. And so this distance here is what we call the range. Okay, so we can actually find when this range occurs. Okay, so that's of course, when why is equal to zero. So we just set y equal to zero, and we get this. Of course, we also need that t is bigger than zero because, of course, we're also going to get a negative solution that doesn't really make sense. It's just kind of an auxiliary or extraneous solution. It doesn't make sense for the problem. Okay, so you want the positive solution, and that's what it iss as a function of data and age. And then if we just plug in So let's call this actually, like t star. And then if we just plug in x of T star, well, that gives us a function of data, Right? So this just depends on the and, I mean depends on BIA's well, but I can think about that just being a function of the trajectory angle. Okay, so the range, Like I said, it's the X coordinate when t is equal to t star. So in other words, when it hits the ground and this is what we have And so if you try to do this by I'm going to tell you, I use the computer to solve these equations into the calculations. If you try to do this by hand, you're really going to see that the hardest part of any of this were already done with calculus. Essentially, we'll do a little bit more calculus, but the hardest part of all this is algebra. Okay, so we get this really cool formula. And so, you know, if you just make some observations right off the bat, Well, you know, if you plug in, Thatta is equal to Pi over two. Okay, then notice that cosine of pi over two is going to be zero. And all of this is not even going to matter. Right? But what is that saying? What's saying that if I should throw the ball straight up, it's going to go up? It is going to come right back down. So the range will be zero. And that's exactly what we expect. Okay, so this this equation is really cool. I mean, it's really complicated. It looks crazy, but there's so much packed into this equation is extremely powerful. Okay, is what I'm trying to say. Okay, so let's let's think about some of the questions that we can ask given this range equation as a function of theta. Okay, So recall that we have this crazy expression for the range is a function of fate, and now we've already seen that if they despise over to if I throw the ball straight up, the range will be zero. Also, if there, uh, data is minus pirate, too. The range will be zero. And so, actually, is a simple application of the extreme value theory. Um, we know that there is a an angle that will maximize the wrench. Right. And this is extremely useful. You know, I have a ball, and I'm trying to see how far I can throw it. I know I The hardest I can throw it is V maybe 100 MPH. I'm not really worried about that. But the question is, what angle do I want to throw it out? And now remember, the tricky thing is, my height depends on how tall I am, right? And so I think a lot of people would guess that the angle is pirate or for the optimal trajectory angle. But that actually depends on how tall you are. It depends on how far are how high you are, above where the ball is gonna link. So how do we find the trajectory angle? Okay, that maximizes the range. This time is a function of my height or how far I am above the ground, right? Throw the ball and where the ball is gonna land. So I do this by just well, finding a critical point. So I take the derivative, set it equal to zero, and then maybe do a little bit of work. And, you know, Please don't take any of this. You know too seriously. This is just algebra, okay? I don't want you to think like, Oh, my gosh, How would you even do this? Well, it's not that important. I mean, this is a good bit of algebra, and in fact, I would be extremely impressed if you could do this because I used to computer. So the advantage of computers is that they could do these, you know, extremely tedious and just really non enlightening computations for you. Okay. But you know, the calculus tells you what to do. At least you set the derivative equal to zero. So you find the critical point and we can actually see from the context that there should only be one critical point in that angle is here. So notice that the angle depends, of course, on the velocity and the height, we're assuming the velocity is fixed. And we now have the optimal trajectory angle as a function of H. Now, notice that if h is zero. So I just plug in zero for H. I get okay so square to be squared over square to be square. So our co sign of squared of one half that's pyre or four. So what that means is, if I throw the ball from the ground and I want the optimal trajectory angle, it will be pi over four. But it's not going to be pi over four if I throw the ball from a height significantly above where the ball is going to land. Okay, so that's kind of an interesting side note about this problem is that they're definitely more factors here thing. You may think it really depends on things like height and velocity and notice that we're not even considering drag force yet. Okay, so we're going to consider drag force, and that's going to throw even more into the mix. But, you know, let's just take it step by step. Okay, so here is the optimal trajectory angle. Okay, So, as I just said, and you can confirm, the data of zero is pie before, and what that's saying is that if I throw the ball from the ground and it's going toe land at the same height, then the optimal trajectory angle is pi over four. But if I throw the ball from way up here, course you would expect that I can throw the ball further. What is the optimal trajectory? Angle as H increase. Okay, so So in other words, if I am taller, do I want to throw the ball at an angle higher than higher before equal to pyre before or lower than pyre before? Well, we can again answer this question using calculus because we have a function that represents the optimal trajectory angle as a function of the height. So if I just take the derivative, I see that as a function of H, the derivative of the optimal trajectory angle is negative. So what does that mean? Think about the derivative is a rate of change. So that means if H increases Okay, right here. So is H increases. The optimal trajectory decreases. So the taller I am, the more horizontal I want to throw the ball to get an a maximum ranch, and we can even take a limit. His h goes to infinity or just think about his h. You know, being very, very tall Or in other words, if I'm on the top of a cliff and I'm throwing a ball and trying to maximize the range, I basically want to through the angle horizontally so that the directory angle optimal trajectory angle is going to zero is h goes to infinity. Okay, so that should also make sense. You know, if you think about if you're on the top of a cliff throwing the ball higher, you're already so far above the ground. You sort of want to put most of the velocity into the horizontal because it's going to stay off the ground a significant amount of time. Okay, so these air just really good conceptual questions to ask yourself and to, you know, I'm I'm actually verifying them using calculus, but, you know, to see if these types of things make sense. Okay. And this is exactly what you do is a scientist, right? You ask questions, you try to maybe get some intuition. You try to see what you can get. Just from from the equations from the calculus. See how things were changing and then, you know, see if things make sense. Okay. See if maybe something surprises you. Uh, so, yeah, it's a really cool process. Let's keep moving on. Okay, so we're gonna go all the way back to the equations of motion. So if you need Thio, rewind to a little bit and grab him. That's fine. So going back, we want to determine the maximum height. So the highest point the ball reaches as a function of state. And then, of course, we want toe Think about the angle that maximizes the site. This question is, I think is a little bit simpler. So the maximum height I think we've seen this before occurs when the velocity in the Y direction is zero. That's because the velocity is positive. We're assuming to begin with, and then in order for the ball to start falling back down the velocity need just needs to reach zero and then become negative. Okay, So when it reaches zero, the ball is at its maximum hype. And that's really easy to see that that happens when t is V signed data over 32. And so the maximum height is given by this equation. Okay, so then it's just a matter of finding okay, We can kind of cut off this this interval or this domain of this function for data between, say, zero and pie. So it's a continuous function on a closed interval. It has an absolute maximum. It's going to occur in a critical point. It's going to occur the only critical point. And so we see pretty quickly. I mean, I say immediately that well, the maximum occurs in Data's pirate or two. Okay. And this this should be no surprise at all that if I want to maximize the height, then I want to throw the ball straight up. Okay, that's pretty silly. But, I mean, it is important that, you know, you'd be able to do that from just the equations. Just the calculus. Okay. You should verify what your intuition is telling you. Okay? Things that air. Obviously true. You should be able to verify. Okay, so the maximum height occurs when data aspire to. But you see that? You know, even if I change the angle, Aiken, see the maximum height. So maybe somehow I want toe balance the maximum height in the maximum range. So that would be another interesting question. Maybe I want thio instead of just maximize the height or maximize the range. Maybe I want to maximize the some of the height in the range, the maximum height in the range so you can do things like that in certain situations. You know, this is the example that comes to mind is in American football when you are punting. Okay, so you definitely want the the ball to go very far. So you want your range to be very, very long. But you don't want to make the range so long at the expense of what's called out kicking your coverage. So you you want your teammates to be able to run down and make a tackle about the time the ball is being caught. So if you kick the ball too far, actually, it's not a good thing, because then the player that catches the ball has time. Thio kind of see everybody running at him. What you really want is you want the ball to go as far as possible and also let the ball go high enough so that your teammates air there to make the tackle. Okay, so that's just, you know, one of the situations you can think of where you're trying, Thio. I'm not going to say maximize or minimize, but optimize h of data and our data. So this is kind of a you know, you could spend, you know, the whole class, you know, riding out different situations of how can you know one of the best ways to kind of optimize this h and R. Maybe you want to use, like, an arithmetic mean a geometric mean? I don't know, but I mean, there's all different types of things you could dio, but I'm gonna move on because again, I just want this to be kind of an introductory project so you can see how powerful calculus is in a very simple situation. There are a lot of questions that calculus can answer. Okay, so there's a very simple way to determine how high a ball will go when it's thrown straight up. And so what we do in this case is we just set theta to be pyre or two because we know that's going to be how the maximum it occurs and then actually solve for H is a function of t the time it will take the ball to hit the ground and The reason I want to solve for T is because in practical purposes, the easiest thing to find in an experiment involving projectile motion is the time I can always time how long it takes for something to hit the graft after I threw it. Or if I drop something how far it takes to hit the ground. So what we do is again just more algebra. You see the steps here you just saw for H as a function of velocity and then we want to solve for V A t using live t equals zero. So when it hits the ground again and we get this equipped So this is the maximum height Onley based on how long the ball is in the air. Okay. This, I think, is one of the most powerful equations to know and particularly the case when you're you're dropping a ball from rest. Okay, so what? This actually allows you to what? This what this can tell you So suppose that you are on the top of the cliff or something and you want to know how high the cliff is. All you dio is you just drop the object. Maybe a small pebble and your time. Okay. And if you get the time it takes to hit the ground, you can actually compute very accurately how tall the cliff is. Okay, this was one of the first things that really got me interested in calculus. I had a teacher that we went on a field trip and we were on top of a bridge, and he showed us how to use this equation to calculate how high we work above the water on this bridge. And it's it's just really fascinating that you could do this. So, again, this is a cool equation on. You know, I put a little example here, So if you're 6 ft tall and you throw a ball straight up and it takes two seconds to hit the ground, then the ball reaches a maximum height of about 19.14 ft. Okay? And the point is, is that you can measure time pretty accurately. Okay. You can, you know, within. You know, a few 10th of a second, you know, may only alter you buy a few feet. You know, if you're just looking for an estimate, this is extremely useful. So again, you know, we just did the simple thing of solving an initial value problem, actually to initial value problems. But then, just with some manipulation using some calculus, etcetera, there's just so much reading talk about. I mean, we could just keep going on and on and on. But I do want to get to a very simple model where we account for drag force and just show how that actually effects the differential equation that we need to solve before we do all this analysis. Okay, so don't get overwhelmed. I mean, this is a paragraph. I'm just going to summarize it for you. So we want a slightly more realistic model. So what we can assume is that instead of the net force in the horizontal direction being zero, we can assume that it's proportional to the negative square of the velocity. So why am I assuming that? Okay, why is that justified? So think about when you're underwater and you move your hand through the water. If I move my hand very slowly, I don't feel a lot of what's called the Retarding force or the force that's pushing against me as I move my hand in the water, But the faster I try to move my hand in the water, the stronger the retarding forces. Okay, so the more, uh, push back I get from the water. So that being said, there is sort of kind of this, uh, square law that you use in these situations when it comes to the area. Okay, so my hand has a specific area anyway, So what we do is we just assume that, you know, it's it's pretty good to assume that the net force in the X direction is proportional to the negative square the velocity. So just take my word for it if you didn't follow my explanation. So the proportionality constant then is this number C, which is called the drag coefficient. Now the drag coefficient is going to depend on a lot of things. Okay, it's going to depend on the shape. So how aerodynamic the ball is. So is it a beach ball? You know, that's not very aerodynamic. Or is it? You know, a dart that's very aerodynamic. And then it's going to also depend on properties of the air that you are throwing in, so maybe wind speed or humidity, okay. And then, you know, we could also make a similar assumption in the vertical direction. But for simplicity, we're just going to consider horizontal drag. And the reason is probably because in the situation, we're trying to maximize range. And so most of the velocity we're assuming is going to be in the X direction anyway. So the why why drag is gonna not play as big of a role. Okay, so we're just making one step into more realistic model. So I'm going to skip all the steps you just saw for the new equations of motion here. You still have a second order separable differential equation that you can solve. You just do it. Okay? You solve this initial value problem and of course, why if he is going to be the same, But exit T becomes this alright, because signed data over K times natural lot of one plus Katie. And here K is CV coastline data over em. So K really depends on the velocity and trajectory angle depends on the mass of the object, and it depends on the drag coefficient. Okay, so all I really want to get out of this is to show you that. You know, just by making you know those assumptions, you have to, of course, justify your assumptions. Say why there reasonable and maybe say what their limitations are. You can start studying this situation in more detail. And so what I wanna do is actually just show you a graph. Okay, so a few sample trajectories, they're shown. I just did hh zero. The initial velocity is 100 ft per second. And then I just used various values of C and data. Okay, so let's see what some of these trajectories look like. And try to understand, at least on an initial snapshot, how things like the optimal trajectory angle change when you have to account for drag boards. Okay. So, as I said, this is a graph of various trajectory angles with drag coefficients. Okay, so in the blue kind of radiant we have data is equal to 45 degrees, and you see that when see a zero, that's the skinny blue line that's going to be the longest range. And that makes sense because, well, when there's no drag force, we already said, if I throw the ball from the ground, the optimal trajectory angle will be pi over four or 45 degrees. Okay. And you see that if I introduce some drag and throw the ball at that same trajectory angle So here C is equal to point to that might mean, you know, less aerodynamic ball. Or maybe I'm throwing into the wind. I land here and then, you know, and even stronger force. Ah, land here. And what you see is that for the same values of C C equals 0.20 point four. If I switch to 40 degrees now for the optimal a case, A C equals zero or the ideal case, of course, the ball is not going to go is far. But when there's drag, I actually see that 40 degrees the ball goes further than 45 degrees. Okay, so what that tells me is that the optimal trajectory angle is getting smaller when I have a stronger drag force or I'm you know, there's a couple ways you can think about that if I have a less aerodynamic ball, um, or, uh, this kiss or dense fluid, or if I'm throwing into the wind so any of those things can play a big role, and that's what we're seeing here. Okay, you can also think about, um, negative values of scene. I didn't do that here, but that's actually really interesting because, you know, you may say, Well, why would I think about negative values of C? Well, this last case, if I'm turning into the wind, there's definitely a drag force that's resisting the ball moving forward. But what if I throw into the wind? If I throw into the wind, then actually, there's a track force behind the ball pushing the ball forward, so that's as sort of like a negative drag force, Okay, and in that case, you know you can do the same announced, so I didn't do it here. But that's another interesting question you can ask. So hopefully that gives you a snapshot into what you can do with calculus. Like I said, this is an introductory to differential equations. But, you know, really, the the simplest concepts in differential equations just involved basic calculus finding anti derivatives. So that's that. Well, this is the end of the course. I hope you enjoyed it. Hope you learned a lot and good luck with the rest of your studies.

Area Between Curves

Volume

Arc Length and Surface Area

Integration Techniques

Trig Integrals