 Ok, this is dynamics. Dynamics, the word in general, especially as used in engineering and physics means systems undergoing change. Now that's very different than the way things were in statics, the class that led into this. In statics we had two things that were always true. Well it's not so much at the start, but by the time we got to the end of the term, two things were always true in statics and that's that the forces would sum to zero, which of course means, three dots there means therefore, therefore the acceleration was zero. And also the moments would sum to zero. Well, do jump forward a little bit or actually jump back maybe a little bit if you remember from physics one. If the moments sum to zero, then that meant that the angular acceleration was zero. And that was always true in statics. We didn't want things accelerating in any way, whether it was in translational or rotational acceleration. In dynamics now, and we're looking at mechanical dynamics as opposed to thermodynamics, fluid dynamics or chemical dynamics which are other forces you may come across in the next couple of years, it's possible that either one of those sums will be zero. And I say possible because there are situations where the sum of the forces actually will be zero in some problems that we're looking at. A good example of that is if we're looking at the motion of a car tire as it rolls along. If the car is moving at steady speed, then the tire itself is not accelerating. The forces on it are summing to zero. But the tire is rotating. So there could be angular acceleration of some kind, especially if there's slippage. If the tire is slipping with the road, then we can have those situations. But in general, we're now looking at situations where the objects will accelerate. And that's the main difference between what we did in statics and what we did here. So this statics had a lot to do with structures because it's so important that they don't accelerate. Whereas in dynamics, we're going to look at all kinds of things, lots of different things over the term as we go through this. This is going to follow very closely the type of thing we did in physics one. Most of you I had in physics one, so you'll remember that. This will be sort of like advanced physics one. We're going to go over some very, very same stuff. It's just we're going to go into it more deeply. And I'm going to entrust that you will be developing some greater intuition in some of the things we're trying to be doing. If you remember back to then, we did then, and we will now start with kinematics. We'll do that for objects treated as points. So that's the very same kind of thing we did in physics one. If we're worried about the position, velocity, acceleration of some kind of object that has some finite size like the space shuttle or a car, we're not going to be worried about the fact that the bumper is going to be at one place and the steering wheel is going to be someplace else and the driver is going to be someplace else. We're going to treat the object as a single point to begin with as we do, as we go through the first part of the kinematics that we're going to be doing. Then we'll look at the kinetics of the situation. Remember what that business was? We've been only nine. Bill? Yeah? Once we've looked at the position velocity, acceleration, then we have to concern ourselves with how do we get a particular acceleration. And that's when we got in to play Newton's Laws. In this class, especially as we go through the first part of the class where we treat objects as single points, we will never have a force balance of zero. We'll never have an acceleration of zero. Then we kind of go back and go through the whole thing again for rigid body motion. And that's just like we did it in Physics 1. We did things in that very same order. The difference is when we get to rigid body motion, we'll do like we did in Physics 1 where we look first at rotational motion. That'll start with the kinematics of rotational motion. But if you remember in Physics 1, we stayed there. We didn't do anything other than pure rotation or pure translation. We will finish the course with general motion studies where we will allow things to both translate and rotate at the same time. The very motion a car tire makes as it moves along the road. And any parts of that can have some particular acceleration involved with it. And those are the type of things we're going to look at as we go through this. So, we certainly have this good portion of the first bit that we're doing here is going to be reviewed from Physics 1. So, bear with it. If you feel you've got that more than adequately put away, good for you. But we will be moving on from there fairly rapidly as we get through things. So, we're going to start with the rectilinear motion. The rectilinear motion of a single point representing any object we want to have in the problem. Rectilinear motion is a fancy way for us to say 1D motion. Once we get to 2D motions, things get a little bit more complicated but not terribly. If you remember from Physics 1, once we started looking at 2D motion, it was possible things could accelerate in one direction not in another as an object in uniform circular motion does. It doesn't accelerate along its own path but it does accelerate normal to that path at all. So, things become a little bit more involved once we get to that but not terribly so. So, the first thing we need to worry about is the position of an object and how we represent that. To relate the position of any object, there's one thing we need first and foremost before we can do anything with it. Remember what that is? You do well and I know you remember because you were always one of Physics 1 and say it. What's the first thing we need when we're talking about the position of any object? If we have some object represented by a single point that happens to be right here, we can't talk anything about the position until we have an origin. We have to have some place from which we're measuring everything. Where must that origin be? Sort of anywhere. It doesn't matter where it is as long as we all agree on where it is because otherwise it's useless for us to talk about the origin if we don't agree where that is and all positions relative to that. None of it makes any sense. But it can be arbitrarily chosen. So, we'll just simply pick an origin somewhere and we'll measure the position of objects relative to that. So, we might call that our first position vector. Remember all vectors have three things. What three things? The units I don't like putting up first. All units have direction as this one does. It's represented as somewhere right of the origin. They have magnitude not only in my right of the origin but on a certain distance right of the origin. And then once we have that magnitude it's meaningless if we don't have units on the number that represents the magnitude. Now, for a rectangular motion we don't need to do anything more than what we did in physics one which is call one direction positive and one direction negative relative to that origin. So, we'll do nothing more than that. Buster minus will be enough to give us the direction of a rectilinear, an object in some kind of rectilinear motion. So, here's our first point there. I arbitrarily chose it as to the right of my arbitrarily chosen origin and we'll see it, say it's there at a time t1. Sometime later of course it might move to here. So, it's time t2 it's going to be there and now there's a new position vector to represent that. Then whatever numbers might be associated with that. Whatever distances might be associated with that. Then of course we're interested in things that are changing. So, we have this changing position, the change in that position vector and hopefully you remember at all times we use this delta symbol it's always the second one minus the first one or the later one minus the earlier one. It doesn't have to be just one and two if we want to skip point two and go to point three we could do s3 minus s1. Just as long as whichever one comes later minus whichever one comes earlier and that would give us just that change in position which is a vector itself because it does have a direction. In this case happens to be further to the right. It does have a magnitude. It's changed by that far and then once we have some magnitude we always have some units. This is specifically known as the displacement between point one and point two whenever and wherever those were it underwent a displacement in this case as shown. That's different than distance. The distance between point one and point two. How so? If we take the magnitude of the displacement vector and I'm not going to write the vector sign, vector sign and the magnitude symbols I'm just going to simply take the vector sign off to represent. The magnitude in the units no sense of direction. That's how we use it anyway. If I ask you in the distance to New York City you tell me it's 180 miles. You don't say well it's 180 miles of the minus j direction or whatever anal thing engineering students would try to get in there and use. It's just sufficient to say what the distance is between the two. And after that perhaps it goes to some third spot. Again it's position measured from the origin and now we have a second displacement so maybe we need to call that delta s one and delta s two perhaps whatever symbolism you need to use to keep it all straight. It is certainly possible and legitimate that we're concerned with the displacement between one and three and we don't care what happened to two for whatever reason. If that's the case then we have a different displacement vector. Now maybe we'll call that one three. Maybe one of them 80 and C. I don't care what you want. But we can do this displacement between any points we want. They don't have to be the two points in order. Once we've done that of course we're interested in how fast this displacement is occurring. And so we divide it by the change in time and that we call velocity. Not just velocity. This is the average velocity. Average velocity because we're only concerned with the position at two discrete time periods. We don't care what happened in between if anything. It's just not a part of the concern when talking about average velocity. I have to be a little bit careful. The book uses a bar over the V to represent average velocity. Then they use a bold face for a vector and I think even a bold face italics and that's just more than I can manage at the board. So of course this will mean vector. If I'm using average I'll write average here down below rather than put a little bar above. And then if it's a scalar or the magnitude only I just take off the vector sign. Hopefully as you remember if we look at these two, some t, one, some time where our first point isn't a certain position, happens to go a little bit farther in the plus direction. Of course this average velocity is the slope between those two points. Doesn't mean it didn't do a lot of other stuff in between those two points. It's just those are the only two points where we perhaps happen to have to add up. Or the only two points where we perhaps happen to care anything about what's going on. However there usually is a lot more going on. Perhaps the path of the object is actually described by something that does more like that and there's a lot more information in there that we need. The question then became and it became exactly this for Newton and Leibniz back in I believe 1690 or something at some particular instant in time when it happens to be right there, wherever that might be, what is its velocity at that point in time? And this was a philosophical point that up until the time of Newton and Leibniz they thought could not be answered because there's no way that we can go down to a single point in time and have any displacement in that time. If no time has gone by, how can any motion have gone by? That was the philosophical stance of the scientists of the time and they believed there was no way that if you went down to an incident in time there could be any such idea of a velocity at that time. Kind of like what's called Zeno's paradox comes in a couple of different forms. It's the idea that if you have to go from one point to another the first thing you have to do is cover half the distance. Then you have a little bit of distance left again so you're going to cover half of that distance. Now you have a little bit of remaining distance and so you'll cover half of that distance and so on and so on forever you'll never actually reach the point you're trying to get to because you always have to do half the distance. What? That's why my life's worked. Yeah it's kind of like school. We make you always go half the distance to your degree and you'll never ever get there. But of course we know that that's not true. Nowadays we have immediate proof of that. At any instant you look down at your speedometer on your car and it is reading something. So even though delta t goes to zero we know that the ratio itself does not necessarily go to zero. And that's what Newton and Leibniz gave us when they developed calculus. This idea of the instantaneous velocity. Now throughout the velocity v without any designation of average on it. Same kind of thing the book does. Again they have really put a bar over it. I'm not going to bother with that. And that's the business of the as you remember from calculus one the slope of the tangent to the curve at that point is found by this and that indeed is the instantaneous velocity. We don't want to write all that. So Leibniz I believe was the one who suggested we use that notation. I think he's also the one that called it the derivative or the differential. I believe Newton called it fluctuations. And that's the instantaneous velocity. The time instantaneous time rate of change of the position vector. Even that's too much to write. So it's very common in science and engineering to use this designation of putting a dot over it. Any term with a dot over it means time rate of change of that quantity. So this is time rate of change of the position vector. Anybody remember what the fundamental theorem of calculus says? Has very much to do with this. It also has to do with one of the ways that speed traps work out on the freeway. It's not so common anymore now that we have radar guns but when I was a hyperactive teenage driver like most of you are we'd see signs on the highways that said speed monitored by aircraft. They'd have planes flying over the freeways and they would have time us at one point, wait until we went to some other point. They'd be between exits. They knew the distance between the exits and they could figure out then the average velocity between those two points. But a couple people tried to say oh but no I know when I got to that exit. I got off at that exit. I was only going 20 miles an hour. I couldn't possibly have been 80 miles an hour if I was only going 20 of the exit. But what's the fundamental theorem of calculus say? You remember what it addresses is? Fundamental theorem of calculus says that between any two points if you know the slope and there's a continuous function between those two points as your position time graph must be, it must be continuous because you can't change positions without a certain amount of time going by. Somewhere between those two points there must be a slope of the curve that's the very same as the slope between the two end points themselves. Somewhere between here and there there must be a point and it looks like maybe it's right about here is one of those points where the curve has the same slope as the slope of the end points. Does that kind of sound like the fundamental theorem of calculus? No I would only not. Chris doesn't it? Intermediate value theorem? No. Well it was fundamental and I took it. Anyway so it didn't matter what you said your speed was at that time that if your speed was the average between those two points they mean that somewhere in between you were indeed speeding as well. So even if the speed limit was much less by the time you got to the exit the fact that you got to that exit so fast indicated you were speeding elsewhere. Alright the full part that velocity vector for statics that was a constant for dynamics this class is going to be quite variable. So as we look at how it varies with time we then come up with this idea of what we call the acceleration. In this case the average acceleration between two distinct time periods we look at what the velocity is. How that velocity changes is the acceleration of the object. This is a lot less intuitive than is velocity. Velocity we're very familiar with. If nothing else we always see it registered on the cars. Acceleration is a much less intuitive quantity and you have to be a lot more careful with it. There's a lot of times when it's negative when your intuition would say it might not be. As we take the limit of that as well as the time period goes to zero and we get the instantaneous acceleration. If you need an idea of what it looks like when you see the needle on your car moving, the speedometer needle moving depending on which direction it's moving and how fast it's moving can give you a bit of an idea of the acceleration. Of course we don't want to write all that so we write the much simpler form dv dt. Of course we don't want to write all that much simpler form v dot. But remember v itself is a time derivative of the position but we don't want to write all that. We're very lazy people. You guys should be thrilled to be going into this field because of how lazy we are. And now often we come up with simpler ways to write things. Of course we don't even want to write all that since it's the second time derivative we can if we want, right, double dot. And that's a very commonly understood form of notation that you'll see throughout your career. Especially common in heat and mass transfer subjects where you have mass flow rates and heat flow rates. It's very, very common to use the dot notation in that. All right. Sound kind of familiar as we go through this? Okay. Let's see. We've got to do a couple things now that we didn't do in Physics 1 because things are getting better. Now if you have this this reviewed knowledge of velocity acceleration. Since this class depends upon that acceleration not being zero, that's the one we'll look at the most. We're going to look at three particular cases where we have some acceleration. One thing we didn't do really at all in Physics 1 is look at changing acceleration. We'll look at changing acceleration a lot more. The first one we'll look at is acceleration as a function of time. It doesn't matter what order these three things are in. It's just the order I am having. We'll also look at acceleration as a function of position. Both of those very often the acceleration of function of position has a lot to do with the way you drive. You only accelerate at certain places. You always accelerate away from a stop sign. You always accelerate into it. Well, the reasonable drivers of us will actually stop there. We'll accelerate into it and accelerate out of it. Time isn't necessarily as common. It's more like a rocket launch might be that at certain places through the launch, certain times through the launch, the acceleration is a certain value. More likely it's a combination of the two because as position changes for a rocket launch, so does the gravitational field change, so does the mass of the object. It gets a little bit more complicated in the real world of doing these engineering things, but all this stuff is. Then the third one we'll look at is acceleration as a function of velocity. This is what happens for the most part with drag. Drag forces is one object moves through the fluid. The drag is a function of the velocity and the acceleration is a function of the drag and so we get this intertwined idea of these two things going between each other which leads to the course we call differential equations where you'll start to address that third case. But we can take at least a preliminary cut at it. Okay, thanks for writing my software. We'll see. The thing we got a pro here and those have a push and button on. I don't think they work. Alright, so the first thing we'll look at is acceleration as a function of time. Probably the simplest of them in that if we need to see what happens with it, well if acceleration is a function of time, let's put those two together. I'll leave off the vector signs for now because the integration of a vector doesn't make quite as much sense, though it's just sort of a technicality. And then we can integrate between V1 and V2, integrate between T1 and T2 and we have the ability to, is this nothing more than what we might have thought. Anyway, the change in velocity is the area under the acceleration time curve. So if we have the acceleration as a function of time, we can now figure out what the velocity changes are. We can't necessarily figure out a specific velocity, but we can figure out what the velocity changes are. If we're given one of the specific velocities, then we indeed can figure out then what the changes are. But between T1 and T2, the area under the acceleration curve will be the change in the velocity. One thing you need to be careful with, the possible situation, oh sorry, that's, yeah, that's acceleration. The possibility that we have an acceleration that happens to turn negative for a while, that is positive area. As integrated, we'll give you a positive change in velocity and this is a negative area. We'll give you a negative change in velocity. So just as they integrate in positive and negative senses, it still carries the physical meaning that we need in terms of what happens to the velocity. So as simple as it is, there's the first case and not much more earlier that we can do now. Well, there is. There's a little bit more we'll get to in a second here. Case two, possibility that acceleration is a function of position isn't too terribly much different. So the acceleration as a function of position and the acceleration is, again, still just dv dt. We can't really regroup this because we're not talking about a velocity. The velocity would change with position, but it's the acceleration as a function of position that we're working with. So we can't really change this around to recombine variables and just simply integrate like we did over here. We could do this because acceleration was a function of time. But we can move things around a little bit to get to there. So it doesn't seem like it's much help. But remember that velocity itself is a time rate of change of position. And since these things are all happening at the same time, then we can solve both of those for the time component, set them equal to each other, eliminate the time component, and we get dv over a equals ds over v. Doesn't seem like that's a huge help. But we have some more rearranging to do. We can collect variables so the velocity parts are together. We can also put the acceleration over the position component since that's the function we have anyway. And then we can call this vdv equals a ds. Or if you prefer s dot ds dot equals sw dot ds. Now that we can integrate. Let's bring it back up here. Just re-arriving it there. And if you're doing something besides rectilinear motion where we truly need a two-dimensional vector, it's not a big deal. Both of these are done in either the component directions as needed. Alright. So we can integrate this between v1 and v2 to integrate this between the two end points. Presumably we know something about how the acceleration is changing the position in there. What's this integrate to? Because we can do this integral without knowing what the velocities are. Since it's a function of velocity, we can integrate that. Joey, you don't like to integrate anymore. This will integrate to one half v squared between the limits v1 and v2. Is that right? No, you see it. Are you familiar with this notation of the integration limits? Okay, good. This one, we can't do anything with that other than say that's the area under the AS diagram. So if we have our acceleration as a function of time, whatever that might be, the area under that graph is one half change in the quantity v squared. I don't know. That might not look like it helped a lot. It's not as obvious as the area under the AT curve was. Anybody recognize this one half v squared quantity? Or sort of recognize it. Kinetic energy. Actually it's what we call the specific kinetic energy. It's the kinetic energy per unit mass. So that could be a little bit of help then. This is the change in the specific kinetic energy. Change in the kinetic energy per unit mass of the object. Now a couple little bits of warning here for you from someone who's pretty experienced in teaching this. This is an easy one to forget that it's available to us. It's not one you've seen before. There's some times when if you remember this it'll solve a problem you're working on in a couple short steps where if you don't remember it, you're not going to be able to solve the problem. So that's the first warning is don't forget this beast. Draw an arrow by it and a box around it. David, I won't check yours. The second warning is this business here has to do with delta v squared. That's what we have here. We put v2 in minus squared minus v1 squared. That is not equal to delta v squared. Delta v squared is v2 minus v1. There's delta v and then we square it. And those two sides are not equal to each other. In a class of 11, at least two of you will make that mistake called these two things equal. Who are those two? Does anybody want to volunteer for that right off the start here? Alan, didn't you do that in Physics 1? You never. I think you did. I never did and I never will. All right, just don't forget that these two things, these two different sides are not equal to each other. It's a pretty common thing to forget for some reason. I'm not sure why. Just because you're yawning and pulsing, I guess. It is a little easier, I guess, to look at that side and want to do it that way just because there's one less exposure. But it's not necessarily correct. If v is zero, it's correct. But that's a pretty boring problem. All right, we'll do a sample problem here. Between two plates, let's say we have an electric field and we'll just keep it simple now because this is not an electrostatics or electrodynamics class. We just want to look at what happens to an object whose velocity pursues acceleration is a function of position. So let's imagine we have these two electrically charged plates, about 200 millimeters apart. And right in the middle, we release an electron from rest. Electrons, in case you didn't know, are blue. Or what? We said they're green. They're blue. So we release it from rest right there at the midpoint and want to know then what the velocity is when it strikes the plate. And I'll give you the acceleration as a function of position. Let's just say it's 4s meters per second squared, where s is the position measured from the first plate. So s1 is 100 millimeters midway between the two plates and t1 is zero there. That's where we'll start the problem and we release it from rest. And we want to find out what the second velocity is. So we can use this because right there's the second velocity we need. We can go ahead and sketch that out. Then we've already done that integral, so let's not belabor it. You know that the left hand side integrates to that and then that on the right hand side will be the integral of that area with, or sorry, acceleration with respect to position. We know the acceleration to be 4s and between the two positions. Actually, no, I think that this wouldn't work with the plates charged that way would it? Because if it was here it would stick to that plate, acceleration would be zero. If it was down here its acceleration would be the greatest but it would be in the opposite direction. So whatever the plates are, there's some magical electric field between the two plates. Maybe a magnetic field would work better. It doesn't matter. We're looking at the upper dynamics problem a little bit. With that we can find out what v2 is. So take a quick second and do that. Figure out what v2 is. We'll put some magical charge on these plates now. You can integrate this side with 4s and then solve for v2. Remember this 4s is in meters per second squared. The limits are in millimeters as written, so you've got to straighten that out. Your acceleration is 4s. Yeah, second squared is a little confusing. It's four times, it linearly increases with position. Up here is the acceleration zero, down here the acceleration is four times two. So wherever you are it's four times the current position. And that puts some odd units on this four because s will be in meters. So this is... 4s is in millimeters, right? Well, for it to work here it'd have to be in meters. Always cover it with a... Be careful, there's two s's in there, one for seconds and one for position s. But it's just a straight linear function. It is linear, so mind it. That's the part you slept here with. Yeah, Bill got it? Meters per second. Meters per second. 0.346. Yeah, that looks like what I've got once you change the limits. Anybody got anything radically different than that? Okay, slightly different question. And this one isn't as straightforward as that one was. How long does it take to go there? How long does it take to cross that half of the plate starting with a zero velocity? And reaching the bottom plate 200 millimeters later. Okay, that one's a little bit less intuitive to get the delta t. We need dt, but remember the dt we don't have there. Now we eliminated that, but we can put it back in if we remember that that was ds, a change in position with time with the velocity. And then we could do that now. If we had the velocity as a function of position, because that's what we're going to need to do this integral, to do this integral we need velocity as a function of position. We have acceleration as a function of position, but since the acceleration as a function of position is known, we can also find the velocity as a function of position. To do that, we go back to this side and generically integrate it not between the two positions, but between the start position and any other position. Then we have the velocity as a function of position. Where now we generically integrate up to some position s. That will give us the velocity as a function of position s. So we can do that. Let's see what happens. Raise the power by one, divide by that. We get 4 over s, 4 over 2, s squared. Is that right? Yeah, 1 half s squared. Evaluate between 0.1 meters the initial position and just some generic n position s plus the factor of 2. So then integrating that, we get v equals 2, 2s squared minus 0.1 squared. Oh, and then that's all square root. No, just the parentheses first. If you do that there, it will do it between those limits. You should get that. And then now you know that velocity has a function of position and you can do this in a row. All the way down to the end plate and get the delta t. Oops, sorry, it's 1 over then. 1 over that velocity. And you can do that in a row. We're not going to do it here. It's not the point of this class. You might remember it involves a log function. Double check that you get 0.658 seconds. So a quick example of a problem where the acceleration is a function of position and we need to pull that into it. But it can also be done for the time even though the time is not an explicit part of that relation. What's that? Case three, right? This was an example with case two. This came from case two. The acceleration is a function of position. That's how we got to there. So case three, acceleration is a function of velocity. That's more likely if we're talking about fluid drag. As the velocity goes up, the drag goes up, as the drag goes up, the acceleration goes down. As the acceleration goes down, the velocity changes in different ways. So we have to keep all that in the problem. Now, this splits into two possibilities. One is that delta t is in the problem. Either it's given or it's asked for. Now, it's not the situation we had here because here was non-acceleration as a function of velocity like we're talking about now. If delta t is in the problem, then it's simply a matter of using the relationship that we already had, then just integrate for it. We have the velocity-based integral on the other side. So we can find out delta t from that, just a straight integration if we know that acceleration as a function of velocity. That's not the area under the acceleration velocity curve that describes this because this is one over the acceleration, not the acceleration directly. And the other possibility is that delta s is in the problem. It's either asked for or it's given. Didn't I tell you to forget this? Didn't I put a box around it, draw an arrow to it, and say, don't forget this? And then you forgot it. The reason this is useful is if acceleration is a function of velocity, we can recollect the variables. Then we've got the ds on the side that we need to integrate directly just doing that integral if we know the functional form of the acceleration as a function of velocity. That one's hard to draw. Not really anything I can do with that. So we can solve the three possibilities assuming that those three situations are exclusives of each other, that we have the acceleration as a function of time but not as a function of time and position. All right. Let's do a quick do of the special situation that of not functional forms of acceleration other than if the acceleration is constant. And if you remember that led to four constant acceleration equations. And then we'll do lots of problems with constant acceleration. So not in any particular order if acceleration is that as it always is and it happens to be a constant then it becomes nothing more than acceleration change of velocity with time. In other words, the average acceleration and the instantaneous acceleration are the same. And these actually are all vectors. However, when you do it in its component directions you can just use the straight equations. All right. So that's actually a special case one with acceleration as a function of position. We also came down to this equation when we got to it. However, acceleration is constant then it comes out of this integral as a constant. You can just finish the integration in there or then solving this in a form that you're a little bit more familiar with v2 squared equals v1 squared plus 2A delta s. So now we have two constant acceleration equations and we know there to be four. So we've got two more. Actually, we're at the end now so we'll save the next two constant acceleration equations where they come from for Friday. You're right. I had a different schedule every year. I guess I was hoping to get lunch before technical premium sketching. It's not going to happen. All right. So let's see. So there's two of them. The third one, well, we kind of have to develop it ourselves if we have a velocity time graph with constant acceleration, what does that graph look like? If acceleration is constant, what's the velocity time graph look like? It's linear. I'll go ahead and draw it like that. And so between two times T1 and T2, we have two velocities v1 and v2 and the change between those is linear. Then the third equation is that the average, the mean velocity, just the distance midway between those two is equal to that midpoint. In other words, the average velocity, which we know to be delta S over delta T is just simply the arithmetic average between the two velocities. And so that's our third equation, again not in any particular order. And then the last one, and then we'll stop because that's a more natural stopping place. And I've got you all decided for stopping anyway. It is a little bit different formulation than you've probably seen before. We know that the change in position will be the integral of dv over between v1 and v2. That's just that form I had given you before. I told you you'd forget and you did already. AdS equals vdv. But if the acceleration's constant, it can come out of the integral. And I'm going to do a little step here with v2 since the acceleration is constant v2 is going to be v1 plus a delta T. The velocity v1 is going to change by an amount a delta T because the acceleration is constant. Now that we can integrate to get the position 1 over a here. This integrates to 1 half v squared between those two limits. And I've got the 1 over a on the front. Oh, might as well put the 1 half out in front as well. A little 1 over 2a. And then just square the parts in there. Let's see. v1 squared plus 2a v1 delta T plus a squared delta T squared. That's the top limit squared. Took the a out, took the 2 out, took that limit and squared it and we got that. Minus it reevaluated at the lower limit. Well, that's just v1 squared. So far, no goof ups. Well, we have v1 squared minus v1 squared. So those two things cancel. It gets a little tight here. The 1 over 2a we can now take through. We've got a 2a there. So those will cancel. We get v1 delta T. The 2a goes over here. We'll have the 1 half. We have a squared. But one of them will cancel. So we have just a. And then we have delta T squared. And you have the last constant acceleration equation that you probably recognize. Delta s equals 1 half at squared plus vit.