 I'll make this, it looks like we're gonna do it on iTunes. I don't know exactly how to do it yet, but the word from the technical people that are helping me with this is that we will put these on iTunes. So as soon as that's ready and available, I'll let you know about it until they're edited. I'm gonna keep them closed from the general public so that only you guys can see them. So they'll have to be a login for that. And then after I edit them, make sure there's no references or worry about it. So I'll probably make this public whoever wants to do them. So if you remember Ashley, she wants to watch them. All right. Okay, so this is Dynamics. We're taking a step beyond what most of us just did in statics. You missed that, Alex, but it's not a big deal. What this really is is kind of advanced physics one. Everything we did in physics one, we're gonna do, and we're gonna do it in greater detail and in greater depth. And obviously a more mature, a little bit harder problem than what you did before. Dynamics in general, it's a general term that we use to indicate that a system is undergoing a change in state. And by system we mean whatever it is we're working on, whether it's the classic problem in here pushing a crate across the floor, any of the type of things we're doing. But in general, dynamics means a change in state of a system. If it's a thermal system, it's most likely a change in state and temperatures or heat transfer. If it's a fluid system, of course it's a change in state of some of the flow parameters, you looked a little bit at that for those of you that took physics two. We did that kind of thing. We're going to be concerned with the general mechanics that we're talking about. This statics and dynamics are both known as engineering mechanics courses. So when we refer to dynamics in here, we'll specifically mean the dynamics of engineering mechanics rather than the general term of dynamics that gets used in a lot of places. Chemical dynamics, chromodynamics, nuclear dynamics, all kinds of stuff that of course we won't look at. Statics was very specific in that we knew the sum of all the forces to be zero on any object and therefore the acceleration was zero. We spent a whole term, Alex, doing just that. Well, there was a little bit more, if you remember, when we summed the torques at the very end of physics one. That was also in there to make the angular acceleration also zero. But we spent a whole term doing just that. This term, actually let me put in there also the sum of the torques is also zero. The angular acceleration is zero. This term, we're going to have some other possibilities to look at. In general, not always exclusively, the sum of the forces will be zero. Therefore, there will be some acceleration. In other words, the velocity won't necessarily be constant in this course, because that's also an outcome of each of these. We will also look at situations where the torque is, the sum of the torques is not zero and the angular acceleration then also might not be zero. The thing about this is we can have either one of these conditions to be true or both. We are certainly going to look at some problems where the forces sum to zero, but the torques do not. And we'll look at some problems where the torques sum to zero, but the forces do not. So we can say in here maybe and or. The forces won't sum to zero and or the torques won't sum to zero. We'll look at either possibility, either of those possibilities, also certainly the possibility that both of them are zero. So that's our general study of what we call engineering mechanics. These two things we're looking at and we're now putting in the second of the courses in what we generally call dynamics, but we specifically mean engineering mechanics, dynamics. All right, we're gonna be using an awful lot of stuff from Physics One. So let's just review, refresh for some of you. It's been over a year since you had Physics One, so it's gonna be very helpful. I think if we rehash a little bit of what we covered there, just to bring in, trying to get it through the pyromortial ooze of your sugar encrusted brains from another Christmas break, see if we can't bring up what we're talking about. If you remember, we started with the kinematics of particles and we will do so again. Two things of specific interest to us there, of course, is what's the meaning of kinematics? You remember? Study of motion itself, not the causes of it, just what's going on in terms of where an object is, how fast it's moving, whether or not it's accelerating, and then of course, when each of these things is happening. Nothing in there about how any of these changes are occurring. And the change in state we're talking about mostly in dynamics is a change in V. If there's a change in V, there's automatically a change of position and there's automatically an acceleration. There may also be in this class, a lot more instances where the acceleration is not constant, where most of what we did physics one was constant acceleration. We'll do a little bit of that, not exclusively that. By particles, we mean that we're mostly concerned with where the center of mass of an object is. But so lightly concerned with it that a lot of the time we don't even mention it, we'll just understand that that's what we're talking about. If we're looking at where a particular vehicle, whether it's a small vehicle like a human being or a very large one like a space shuttle or something, we're not gonna talk about where the individual pieces of it are and what's position, velocity, acceleration they're undergoing, where we'll talk about the object as a single point and we'll, a lot of time just automatically understand that that will represent the entirety of what we need for the problem in terms of where it is and what it's doing. It may change its orientation in undergoing the problem. Early on here in the kinematics of particles, that will be a concern for us. Later on, we will be concerned with the orientation of the object. Just like we did in physics one, that's when we started talking about rotational motion and the like. After that, we looked at kinetics. This is when we concerned ourselves with the fact that if things are accelerating, how do we get that acceleration? If we don't want things to accelerate, how do we prevent that acceleration? Those are the kind of problems we've looked at there. The causes of the kinematics and that brought us to Newton's three laws. The first law is the one we used in statics that the sum of the forces on the object will necessarily be zero. And just like in statics, if the sum of the forces is zero, the acceleration zero, the acceleration zero, the velocity is constant. So all of these problems were constant velocity problems. That's Newton's first law. Newton's second law is the possibility they don't sum to zero. And so that'll be the one we're most interested in. This class, with that we are going to have acceleration. We are going to have problems of non-constant velocity. Not the case of what we did just finished up in statics. And then Newton's third law, it doesn't come in an equation form. It's more of a qualitative thing. That's the business of action, reaction pairs. The deal that if I push on something, it pushes back on me with a force that's equal to mine in magnitude, opposite to mine in direction, and collinear. You know what collinear means? Not just the same direction. It's even more specific than that. Because two forces that are equal and opposite could be like that. Collinear is more specific in that not only are they equal and opposite, but they're actually lined up with each other. They're in the same direction. These are not two forces that cancel each other because these are two forces on completely different objects. If I push on the wall, that's one of the forces. The wall pushing back on me with an equal opposite and collinear force is a different force on a different object. So those are not two pairs of forces that automatically cancel each other. So those are two of the things that we'll start with that will be very important to us from physics one. Other things we need to remind ourselves with and concern ourselves. Pay attention to, of course, our units and problems. For the most part, the units will be very, very similar to what they were before. Our fundamental units, our fundamental mass unit will be the kilogram. There is an English mass unit that we'll pay some attention to. It's a tremendous pain in the butt. So we're not gonna dwell on it a lot. As with most of the English system, it's historically very rich scientifically but almost useless. So we'll talk about it some, but we'll minimize the trouble of it. I'm gonna depart a little bit from what the book does with it just to make it even easier to do, to handle mass in English units. We, of course, also have length. Our fundamental units will be meter and foot. In general, we'll try to have our problems come out in these units, not necessarily always. It's not necessary that they always be in just these fundamental units. There's a lot of subunits we can use centimeters and inches and the like and we'll do so. And then our third fundamental unit and it's the only one that's the same for both English and SI systems and that's the second will be our fundamental unit. Lots of others we're gonna use in this class that we introduced and used in physics one that are derived units. For example, the derived unit for force, of course, is the Newton, which is derived from a particular and useful combination of the other units. Part of the trouble with the English system here is that there is a force unit in the English system, of course, called the pound. It was not derived from the other English units as we did in the SI system. It was just defined independently and then had to be back defined in terms of all the other units and that's where the tremendous mess comes in like for the English system, especially when we're talking about mass and force, which we all know go together all the time, especially in these classes. So we're gonna have to pay attention to that but it's not too difficult. The only time it'll be difficult dealing with mass and force in the English system is if you don't write your units in your problem as you go along. If you're lazy about it and don't always put them in, you're gonna get caught. There's nothing I can do about it. If you put the units in, they'll take care of themselves and you'll do just fine with it. So you make the decision on how you wanna approach the units. I've tried over the years to help you with it. Some of you I can help, some of you I can't. All right, so we're gonna pay attention to those. There's others to learn. Of course, a lot of other derived units that we've got work and energy units of the jewel. Remember how that was defined? We defined it off of Newton meters. So it's a derived unit, off of a derived unit. There'll be others and we'll define them and get used to them as we go and as we need to. All right, other skills we're gonna need from Physics One that we worked on. We're gonna use a lot here. We're gonna do a lot in this class with vectors. They're particularly useful when talking about things like position, velocity, acceleration and force because those are such, those are variables that depend so much on the very things that a vector can show. Everybody, I hope, remembers the three things that all vectors have. Poor Frank, some of you are going, oh, I know two things all vectors have. I can't think of a third. So give us two things all vectors have. Direction, which is very much a part of things like position, velocity, acceleration and force. Any of those things done in a different direction, you completely change the problem. So we have to pay attention to that a lot. Length or in terms of what it's describing, we use the term magnitude because we will have force vectors and it doesn't make sense to draw something of a particular length unless we term it in terms of magnitude so we can have a vector of a particular length that represents a certain number of newtons that we find. And the third thing, Frank, yeah, let's see if my old students know. Everybody know three things? What? You think so, DJ? Colin, three things all vectors have. Well, with one minor exception that we get to. Units, all vectors have units. The magnitude you put on a vector won't mean anything if it doesn't have units as the same with any other number in this class or any of your other engineering classes. If you don't put the units on them, then the numbers themselves are useless. They don't mean anything to anybody if we don't know what the units are. Of course, one other concept, kind of bringing all of these together, the units, the kinetics and the vectors, is the special case of, when we talk about weight, we will, of course, do what we did before and would calculate the weight generally at the surface of the earth. So we'll use g defined as 9.81 meters per second squared. That's the strength of the gravitational field. For those of you, you guys are taking, most of you are taking physics three right now. So you'll be generally touching, it's your first time to touch things like, like continuum fields and the like, magnetic field, electric field. Gravitational field is just still the very same thing. What this g is, is the strength of the gravitational field, average strength of the gravitational field at the surface of the earth. And we'll use that. That'll be just fine. We can also do it in English units, in general, those are the numbers we use. In fact, this is so common to us in a problem that we're doing, if we have mass, but we need weight, we will not consider weight to be a separate unknown. We'll just take it as something we can so easily get that if we have mass, we know soon we know weight and keep moving through the problem and vice versa. As you remember, most of what we have to do in these classes is what you have to do is get enough equations to solve for all the unknowns. We're not gonna consider weight as a separate unknown because it's just so easy to find. We'll take it as that. We will also take weight to be generally downward because it is a gravitational attraction to the earth itself, and that's where the earth is. So we'll leave it at that. We're not gonna make a big deal about it. We won't labor that too much. Okay, a little bit more in terms of what we're approaching here. The homework assignments. Most of you were a member and so I'll introduce this to Frank. I should have done this in Striker Materials too because the deal is the same there. I don't grade them specifically for your solutions. I have the solutions and my mind will go completely blank if I have to read over everybody else's solutions in great detail. So I will look at them for almost exclusively for presentation. How well do you write out a solution? One of the things you need to learn most strongly as young engineers is how to communicate technical ideas. One of the most common ways you're communicating with me in this class is through homework solutions. So do a good job with them. There's stuff on Angel about that in specific. There's even some samples up there and then that makes the grading for me a lot more useful and then I will post the solution so you can then check your answers. So that's gonna be the same type of thing that we'll do in this course that we've done in years past. But there's some more stuff to help in this book on page 13. By the way, did everybody get a chance to get this book? Okay, it's coming out. If you won't have it for a couple of days because you're getting on eBay or something, I don't blame you. It's a good book. Part of it, I wouldn't use a book that wasn't a good book but I also like this one just like the statics book that went with it. It's just a lot cheaper for you. It's a lot less expensive. You're getting as much for what's the price? 80 bucks for this? You ever take a little bit? You're getting as much as that. Other folks are 200 bucks and they're not giving you anything more other than they're hardbound and this is paperback. Big deal. I don't think that's worth an extra 120 bucks. All right, so on page 13 of this book is a short discussion, doesn't take even the full page on a way to approach problems. If you get used to that, these problems have just become a lot easier. Some of what we go into in dynamics seems very confusing but if you approach it systematically you can break down big confusing problems into little non confusing problems that are all tied together in some way. And dynamics very much lends itself to that as something we'll need to do. Part of virtually every solution in this class will be a free body diagram. If we're gonna figure out how things are accelerating we're gonna need to know what the forces on these things are. The best way to get all of the forces is to do a free body diagram. So much so that as most of you know we're starting that technical free hand sketching course in fact it starts immediately after that to aid in this very kind of thing is to draw a useful diagram for yourselves that will make the solution, getting to the solution that much easier. In general, only two things you need to do, make them big, if your drawings are too small they're not going to be helpful and that will very much be true in this class. We're gonna have several problems with a lot of things going on. Several forces or several velocities they're all kinds of different things in different places and if your drawings are too small it's just not gonna help you in fact it most likely will hurt you. So don't be afraid to make them big and keep them simple. Don't put too much stuff in them either. As we go through some of these problems we're going to have not just forces that we're paying attention to but acceleration and velocities and if you put too many of these things on the diagram they get too cluttered, they may be big but if there's too much stuff in them then effectively they're too small again. So there's some stuff we're gonna have to selectively put in there and others not. It's a big help that if you have even a couple different colored pens or pencils with you that way we can draw force vectors in one color and velocity or acceleration vectors in another. One thing we're gonna be doing in this class occasionally is not just have a force free body diagram but we'll have an acceleration free body diagram as well in that we know what the acceleration is that we want to result and we gotta get the forces that guarantees us that. So sometimes we'll draw a separate diagram that just has the acceleration on it to help us visualize where the forces are gonna need to be to get just that acceleration. All right, after we get through that business with kinetics of particles and the kinetics of particles then we'll restart the whole thing with rigid body dynamics. And this'll be the case just like it was in physics one where now we were concerned with the sides and the orientation of the different objects we're looking at. In physics one, we looked at pure rotation in this class we're also going to look at rotation and translation together. In physics one, anytime something was rotating moving as a rigid body it stayed in one place and simply rotated. In this class we're gonna allow things to rotate and translate just like the easiest example is exactly what a car tire does. As a car tire rolls along the road it's both rotating and translating through space. And so we're gonna do both of those together. And that's why we're going to have to pay attention to both the sum of the forces and the sum of the torques. All right, so that will bring us down then to the end of the term. In general the things we need to retain hopefully have to retain from physics one. Questions so far? Alex, you okay with that? Like I said, it's sort of advanced physics one. So if you did okay in physics one, which all of you did, you'll be okay here. All right, so we'll make our start now into the kinematics of particles. And we're gonna divvy this up in a couple different ways. And then we're gonna look at rectilinear motion. This is very simply straight line motion. We're not gonna worry about any turns. It's just simple travel to and fro on a single straight line just to begin thing, just to get things both of them sort of open things up. We'll then make a step to curvilinear motion where now things can take a bit of the turn as a possibility. However, we're gonna keep it all in a single plane. So this will be 2D motion or planar motion. It's exactly what we did in physics one. This is when we looked at things like projectile motion, circular motion. Those are two dimensional motions that are always in a single plane. So we'll do some more of those, circular or orbital motion, and then just general two dimensional motion of various kinds. And we'll spend a little bit of time, a little bit more than we did in physics one, but not an awful lot into general 3D motion. What are sometimes called space coordinates. We won't spend a lot of time there. It's difficult to draw the board. It's difficult for you to draw on your papers. It's difficult times to even visualize. Much less take notes about. So we'll spend some time with it, but not an awful lot. We'll also look at a type of motion we call relative motion. We look at it a little bit in physics one. We'll look at it a little bit more and a little bit more in depth here. Relative motion is also known as two or more body motion. Two objects and how they move with relation to each other. That motion might be unconstrained. This is the type of two body motion we're most familiar with as people. Because as you move anywhere you happen to move through your day, you're moving relative to somebody else but they're moving however they feel like moving without concern of what you're doing other than you don't want to smack into each other in the hallway or you might be boyfriend and girlfriend and you like to hold hands as you walk. But at any time you can separate and go your separate ways and there's no connection between the two in terms of the motion of one directly affecting the motion of another. We will also look at constrained motion. This is where the two bodies are directly connected to each other in some way. Whether it's by magnetic forces, electrostatic forces or actually physically, mechanically connected to each other as two things would be connected by a rope or some other direct force and connection. All right, that'll be our start here and that'll take us down to exam number one as you'll see on the schedule. All right, any questions? Most of that already is chapter one. So already we're set to go on to chapter two as we start talking about rectilinear straight line motion. The kinematics we're worried about again are, well first we're worried about the position of something. Where an object is at a particular time, where it is at other times as well and between that those two things we're gonna try to figure out what else is going on with the object. So generally we need some kind of straight line upon which our object can travel. It cannot go off of that path in the study of rectilinear straight line motion. Obviously since it's straight line motion. Nor does that path turn in any way. So we need some straight line that defines the motion whether it's some kind of rail track or just some other physical constraint that keeps it on the straight line doesn't matter as long as it does. One other thing we need of any of these team type things is an origin from where we measure the position. Just simply a way for us to reference the position with respect to some point, preferably some point that we can all agree on so that when one person's talking about position it's meaningful to the other person in the very same way. If we didn't have the same origin then when you talked about the position of an object it doesn't mean anything to me because I'm measuring it from some other some entirely different place. Hopefully most of you remember the one rule about where this origin must be. When we pick an origin we put it where. It doesn't matter where, that's the one rule. It's gonna be arbitrary located. All it needs to be is publicly known. If you pick an origin it needs to be understood where that origin is by the other people. Other than that it's entirely arbitrary where it is because we're not so much concerned with the position of the object as we are with the change in the object's position because that's what results from velocity and that's what also then leads to acceleration. So we have some origin, some kind of unit system that goes with it whether that's meters or miles or nanometers doesn't matter. Generally we have some understood positive and negative direction. If unspecified we'll take positive to be to the right, positive to be up just because we're so used to that. Certainly in western cultures we're used to that. It's a little bit different in some other cultures but that's what we'll take it to be unless we specifically say otherwise. All right so at any one time we can have an object at some position we might call S1 at some time T1. Since we know where the origin is, since we know what our unit measuring system is, we can all agree on where the object is at that time. Vitally important in science and engineering and physics that we all agree on what we're talking about before we even talk about how those things are changing. Some bit of time later of course it might have moved to some other spot and is now here at some other time. That's as simple as rectilinear straight line motion can possibly be. Of course there's the possibility that we could move backwards with our next step. Doesn't matter, we can always handle that type of thing. We can go anywhere we want along that line. All we're concerned about is where are we and when are we there. Because from that then we get our first important idea that of average velocity. That's change in position that occurs during a certain amount of time. Exactly like we had it in physics one. We're going to be very specific from the start. We'll take, we'll pay attention to the fact that this is a vector equation. In rectilinear motion that's not a big deal. All that comes into the direction of business is whether it's a plus or a minus. If we move to the right our delta S is plus, if we move to the left our delta S is negative. And that will handle the full vector nature of both change in position and velocity that goes with it. We will always take delta T to be positive. We're not going to talk about any idea of traveling back in time. This is in the physics class. If you want to travel back in time you have to major in physics, get a PhD. Or, well Mr. Peabody didn't have a PhD. Everybody know who Mr. Peabody is? Mr. Peabody and Sherman and the Wayback Machine? Do you know who Brock J. Squirrel is? And Bullwinkle Moose? I don't know. Who I'm going to be teaching in the years to come. You guys don't know who Mr. Peabody is. Well go Google Mr. Peabody. That sounds nasty in its own right, right there. All right. So this is very simple. You do this exact kind of thing in your head on the fly is to travel places during the day. So it's not a tremendously big deal. What's more of concern to us is this idea that if we pay attention to what our position is at certain times and so we have to know it's there and then we have to know it's there and then we have to know it's there. Kind of like what we did up here. We had three different positions that as time went by at T1, 2, and 3 we were a little bit to the right of the origin, a little bit farther to the right then we came back to the left. What's more of concern is not what the average philosophy might be for any one of these. That's a pretty gross measure at times. If T1 and T2 are pretty far apart in time it doesn't tell us a whole lot of what went on in between. So we might have more interest to know what the object did at some of the intermediate times that are missed by the fact that we have this gross delta T here that uses a pretty big time step and doesn't always give us the type of detail we need. Maybe we need to know not what the average velocity that is between two times but what the instantaneous velocity is at a specific time. And if you're looking at this you may recognize this as we're right on the cusp of what's known as the fundamental theorem of calculus. If you recognize that, says that the average velocity over a certain time step, there's guaranteed T to be at least one spot in between that has that same velocity. Somewhere in between the instantaneous velocity is gonna have the same slope as the average velocity does at the end points of that time period. It was exactly this type of problem. How do we define the instant velocity? How do we come to understand at some particular time? How can we have any velocity when at a particular time and in instant time we can't actually move anywhere? If we're, if we have, if no time goes by how can any motion go by? So how can there be such a thing as instantaneous velocity? This was exactly the type of problem that Galileo and Leibniz wrestled with. This idea that as the time step approaches zero, what happens to the velocity? Because the velocity was defined based upon a definite delta T, a finite delta T, not an infinitesimally small delta T. That's what we're talking about here. So this was our idea of the instantaneous velocity. Hopefully all of you recognize that as the derivative. And it was exactly this idea of the instantaneous velocity that led both Galileo and Leibniz separately to their definition of the instantaneous velocity and the definition of the derivative. Which to us, very simply, is the slope of the position curve at any one particular point, the slope of that tangent line. We will in this class also make use of an alternative notation. We're concerned with changes in the position vector. We'll designate that at times with just a dot up above it. That is universally known as indicating a time rate of change in whatever that quantity is. But that's exactly what we have. The time rate of change of the position vector, also known as s dot, and will be said that way if you talk to other engineers, you say s dot, they'll know then exactly what you mean with just that. So we'll use that and makes not exclusive use of it, but we will make particular use of it when it is useful. In two and three D motion, it can be very useful to use the dot notation because then we can write x dot is the velocity in the x direction and y dot is the velocity in the y direction. And we can keep them simple and straight that way. All right, hopefully that looks familiar to most everybody. All right, then we look at changing velocity itself. The fact that the problem is a little bit more interesting when not just the position's changing, but the rate at which the position's changing itself also occurs over some time period and we understand and remember this to be the average acceleration. The rate at which the velocity is changing as some particular time period goes by. What our points one and two we can define for any problem as we wish. It is a vector quantity like all the others we've talked about so far. Well, time, I guess it could be a vector. It does have direction, but it only has one direction. It only can go forward, only go to the right down the time axis. Just like we did with the velocity, we might also concern ourselves with the instantaneous acceleration to find an exactly the same way that as this time period approaches zero, which is the perfect definition of an instant in time, we then get the instantaneous acceleration and again this is just the time of rate of change of the velocity vector. Using our notation, we can also write v dot. Since v itself is a derivative, then we can take out the velocity and replace it with the time rate change of the position vector. Then this is also known as the second derivative of the position with respect to time or we can also write it as s double dot. Standard notation, any one of those, if you choose so to use one and the exclusion of the others, that's fine. I recognize any one of these and you're welcome to do which one you want, just have to do it correctly, that's the only thing. If I put up a notation that you don't quite understand, just ask for clarification. Don't mean any of this to be particularly confusing. All right, so that's all review, I hope. Now we're gonna get a little bit more specific stuff. We didn't quite hit in any great detail in physics one because we didn't do more stuff than we did in physics one. There'd be no point in taking this course. So let's go into all of this in a little bit more detail. We're gonna be interested in three cases, three possible ways these rectilinear, kinetic, kinematic problems can shake out. Actually, it doesn't necessarily have to be rectilinear, it could be curvilinear and spacelinear, I guess. But we're interested in three possible cases, not in any particular order, but those are the acceleration is known as a function of time in some problem. That's one possibility. As time goes by, we know what the acceleration is. From that, we want to get the rest of the details. What's the velocity? What's the position at any of the points in time? Second possibility we'll look at, and then remember, and it's not in any particular order, just the order in which I happen to do them, acceleration as a function of position. Remember, these can all be and properly should be, I guess, as vectors, full vectors, but we'll just use the plus and the minus. Don't have to fuss with the vector signs right now. We will, of course, when we look into two and 3D motion, but for right now, hopefully we can understand that even though these are vectors, all we need is a plus and a minus in the rectilinear motion. The other possibility is that velocity is a function, sorry, acceleration is a function of velocity. Any one of those three could be a possible way a problem could shake out, and from those, we're trying to figure out more about the problem, maybe what the position is, maybe what the velocity are, maybe the time that happens in any of these problems that goes in between them. So those are the three cases we'll look at. Case one, that of acceleration as a function of time, how do we take a problem like that and bleed out of it, something about the velocity, something about the position? Easiest way is to start with our definition of acceleration, collect variables, since acceleration is a function of time, we want it to be with the time variable itself, so we'll move things around a little bit, just some simple differential algebra, if you will, and then of course, this is just trying out to be integrated, integrated between V one and V two, since that's the variable of integration, and from T one to T two, I'll trust you to integrate this side, who's done? Has anybody integrated this side yet? Remember when you first learned differentials, what the name of the integral was, you weren't told the term integral. Your, it was called anti-derivative. Integrals, undue derivatives, so then this is then simply integrated to V, between the limits of V one and V two. Is that the symbolism you use for the limits on the integration, evaluating the limits? Well, we can't do this integral if we don't know what the acceleration function is with time, so we're gonna have to leave this one alone, but this one we can condense a little bit, that becomes in V two minus V one, or delta V. Remember that acceleration is a function of time, that doesn't mean time's time, it means it's a function of time, and so there's our solution for case one. If we know acceleration as a function of time, then we can fairly easily, depending on just what that function is, determine what the change in velocity is. So as an illustrative example, we've got some function of acceleration with time, whatever it happens to be, whatever is causing that, who knows, there's lots of things that we can insert here. Between any two points in time we're concerned with, T one and T two, we can figure out what the change in velocity is between those two times if we know this function. Whether we know it as a mathematical function or whether we know it as a graphical function doesn't matter, the integral in terms of the graphic gives us, when you integrate a curve, you get the area under the curve. So the area under this curve is the change in velocity between those same two times. If we know what the velocity is at one of the times, we then of course can figure out the velocity at the other time. And we can change those times, we can make them as small or as large as we want and figure out as much detail about the velocity as we could possibly want. So this will give us the area under the AT curve. It may be easier to actually just calculate that area than it will be to actually do the integration. All right, simple as it was, that's case one. Case two, that's where acceleration is a function of position. All right, let's see. We're gonna have to bring two things together here. Remember that A itself equals dV dt. That was just simply our definition of the instantaneous acceleration. So that's certainly true. And the other thing we remember as V is equal to dS dt. Those two things are always true. No matter what the functional form of acceleration, these two things of course are always true. No matter what the motion is, we know those things to be true. So let's do this, let's see. Let's move this around a little bit. Make this dT equals dV over A. We'll do the same type of thing with this one. This becomes dT equals dS over V. Since those two things are always true and they're both happening at the same time, we can combine them because they're both equal to dt and it's the same dt for each. So we'll just bring these together. They both equal dt. They both must equal each other. So dV over A must equal dS over V. Or dS, remember this was a problem where A was a function of position. We now have it in that form. We could integrate this if we wanted to. Equals V dV. Or we could write it as S double dot dS equals S dot dS dot. If we know acceleration is a function of position, sorry, yeah, acceleration is a function of position, then this is the solution, this is the method with which we'll be able to figure out what's happening with the velocity. All right, we can do a little bit more with that. We'll leave that there just so we have it for reference. We can do just a little bit more with that. A B S equals V dV. We can integrate this from S one to S two. Integrate this one from V one to V two. The left hand side, well, if we don't exactly know what this, if at this instant, this moment, we don't have to know what the acceleration function is on position, then we can't actually do this integral. We could do it graphically, but we'll leave it as just an integral we have to do once we have the functional form between those two. With that right hand side, we can integrate. That right hand side, you can integrate. The integral of V dV. Who's my integrating genius here? Well, almost. Remember that, increase the power by one, bring that power down. So we'll have one half V squared between the limits V one and V two. So that comes, the one half comes out. We've got V two squared minus V one squared. Kind of looks like we're right on the edges of kinetic energy. If you remember, one half V squared was a big part of that. In fact, we can finish that link in a little bit. It's always necessary for me to do this, so I'll do it. This is change in V squared. We have V squared at point two. We have V squared at point one. We're subtracting them, this is the change in V squared. This is not equal to the change in V squared. Mix that up. Usually about a third of the students do. I've got eight people here, so two of you are gonna screw that up. If you'd like to volunteer right now, so we know who that is. No, we'll leave it as a surprise for later. So this then tells us that the area under the AS curve, whatever that might look like, or whether we actually know that function, have it written out, we'll do a problem like this in a second, that between those two times, the area under that curve is the change in V squared. So if we know acceleration as a function of position, we can then fairly quickly get to an understanding of what the change in the velocity is. Directly, we get the change in velocity squared, but from that, we can get the change in velocity. Oh, yeah, yeah, thank you. That was inflation, good jack. Oh, I always have at least one student awake, every very last period. Try to have at least one. All right, so any question about case two before it comes down? Case three is a little bit more involved, so I think what it'll do is try to keep it all together. I'm just gonna move that center part over to here so we have it. We've already done that one, that was just where we simply used the definition of the acceleration to that one. This one we just finished, so it doesn't give us information of the velocity directly, but it's not a big step away from it, especially as we have to remember the one half is in there. Now we'll do case three, the acceleration as a function of velocity. This is a very realistic situation because this is exactly the type of thing that goes on when you don't neglect air resistance. Remember in physics one, almost everything we did, we neglected air resistance. When you don't neglect air resistance, acceleration becomes very much a function of the speed. As you know, the faster you drive down the highway, the more the air is rushing over the car, the more your gas mileage drops, all of those things are part of the velocity. It doesn't really matter what your position is in those situations, it matters whether or not you're driving real fast. So that's a very realistic, but a much more complicated situation. So we'll deal with some of it here. The thing is with this one, there's two possibilities. There's two possibilities that can sprout from here. We have to handle them each a little bit differently. This one say what we wanna find out is how much time is going by. As an object is accelerating, with an acceleration that depends upon the velocity itself, maybe what we need to figure out is delta T. Maybe what we need to figure out is the change in position. So those are two possibilities we need to work out and we have to do them separately. Neither one's terrifically complicated, but they don't have the overlap that the others did. All right, so we'll do, we'll do this first part here. We need delta T. For some reason, that's the way the problem was laid out for us. Well, we have a pretty good link of them with just the definition of acceleration. Right there we have acceleration and velocity length so we can rearrange this a little bit. In a way that's collecting the variables, if you remember doing that, when you were learning about integration, we have the integrate variable of integration is our velocity and our functional dependence is velocity so it's good to put those two things together. Then we can integrate both sides. The left hand side just becomes delta T so there's what we're looking for in the first place and then, well, we can't do anything with this unless we know what the functional dependence is there and we could actually do the integration. So that's the general solution to the first possibility of our third case. We knew how A depended upon B, we could do that integration. It's not as easy as just calculating the area under the AB curve because this is DV over A, not ABB, but you can always grab one over A as a function of B. All right, so there's that possibility there as we look through this. The other possibility that we need delta S. All right, let's see, in this case, let's see, A is a function of V but as we know from, well, as we know from case two, we know that this part is just from A, DS equals B, DV. That was our solution to case two. Remember, we're looking for delta S but A is a function of V so we'll recollect variables a little bit here, separate variables, I guess the term is. We'll get the DS on one side. Remember that's what we're looking for so you can already tell that's where our integration's gonna be and this is V over A, DV. Remember that A itself is a function of V so those are not two constants divided by each other that's a variable and a function of that variable dividing each other. So we can integrate both sides. We directly get our delta S there. Integrate the other side. At this point, actually, there's not much we can do with that because we don't know what this, at this point, we don't know what the functional dependencies of A on DV so we'll just have to leave it like that until we do know what that dependence is and then you can just do the integration. If A is some polynomial of V or what's most common in fluid mechanics is A is either a direct proportionality to V squared or V cubed depending upon the situation then that's a fairly easy integral to go ahead and put that functional form in and finish the integral. All right, that brings us to the end. What we'll do on Friday is we'll re-look at some of these for a quick special case and then continue it from there, okay? There's homework assigned, there's quite a few problems but you haven't helped Friday to do them so I'll get started on them. They're not due Friday but keep on top of them so you don't get behind. Well, about the only way to fail one of my courses is to not turn the stuff in.