 All right, any questions? Today's homework day, right? So if you've got it, you can just let me have it on the way out. Don't bring it up right now. We're busy now. If you don't got it, get it in. I will accept homework even up to the very, you can come into the final exam and hand me homework and I'll accept it for a partial credit. So that's better enough. It shouldn't be any zeros on the homework. Even by then I'll have put up the solutions. You can just copy the solution. I mean, not photocopy. That's way too easy. But you can just copy right down the solutions real quick, turn them in to me, at least get half credit. That's better than nothing. So any questions on those? Yeah, here is the question. 35. 21, number 35. This is one, this is all about, mostly about unit conversions. Dealing with the units. It's sort of an illustration of the way you can solve a problem that you don't know how to do based on just the units alone. So in this problem, you were asked to find how many molecules of water are there in one cup of water? A cup is about 250 cubic centimeters. Now there is something, there's a couple things you'll need to pull out of your chemistry class, such as the molecular weight of water. And once you've got the molecular weight of water, how many molecules does that mean? And those type of things you gotta pull out of your chemistry class. And just high school chemistry would be sufficient. You don't have to take the college chemistry for those numbers. Do you remember things like Avogadro's number? Those type of things. Those are the type of numbers you need. But you can start with just what you're given. I think it said 250 cubic centimeters for a cup. Now that doesn't mean water or anything else. That's just a volume ratio. In fact, if you think about it, that's one of those A over A things we need when we multiply by one a lot. That's one of them there. This is saying 250 cubic centimeters equals one cup. So you're just told to start with that. You're starting with that because you're told that you wanna find out how much is in a cup of water. So that's why we're starting with that. But then there's some other things you need to do. For example, you're trying to find the number of molecules. The only way we have in knowing how many molecules if we bring in the atomic weight. If we bring in the atomic weight, then we're probably thinking of something like, something like the density. And they'll like to start getting all those pieces brought together. And so you can start from here and say, well, let's see. Centimeters cubed on the top. I know I'm gonna wanna get rid of that because somewhere down here, I need what, molecules per cup. I've already got the per cup. So I don't have to mess with that at all. I get somehow from volume to molecules. So one thing you might wanna think about is, well, there's the density of water. Whatever that is. And you can sketch these things out this way without actually having any of the numbers. Just to see, it's sort of like a, you're trying to pick your path through an unknown region. You wanna see if the path is leading under it anywhere or whether you need to take another one. Then there's the, we need to get rid of the kilogram somehow. Remember what we're doing is going along here, canceling this stuff as we go. Kilograms, we're working towards molecules. Well, if you remember something about a kilogram mole, that kind of stuff stands. Kilograms per kilogram moles. See, that's the, you gotta clean out some of the dust in there and see if there's some chemistry in there where it had that sort of thing. And then we do have Avogadro's number with the number of molecule per mole, if I remember the way it goes. And they're without having to look anything up without having to actually do any of the calculation, you've laid out for yourself a calculation that would work and then you just need to go fill in some of the detail. Now you gotta figure out, well, where am I gonna get the density of water? Well, one great place to check is the book just to see if that kind of stuff is in there. That's a standard, that's a standard value. Plus I think I told you the relationship between a kilogram and a certain amount of water, that's one of the ways in which the kilogram was defined. It's a cube of water, 10 centimeters on a side. So you can remember those things. The molecular weight of water, you might have to look that up. Maybe some of you remember it, maybe some of you don't. And then Avogadro's number, another thing you may have to look up. But you've laid yourself out a path that you know is gonna work as long as you can figure out each of the pieces that goes into it. But all of this based upon a judicious eye kept on the units of the problem. If you ignore the units in the problem, you're gonna know, I don't know how to solve it. If you ignore the units, then it's all over the place. That helped a little bit, Len? Yeah. Also with homework, especially if there's a question asked, I prefer if they're asked before they're due, but I understand we haven't been together since last Wednesday. That's a long stretch of time to have done all the problems to see if you have any questions. So if you have any questions and you have to ask them in class like this, I won't grade them between now and the time I leave around 5.30, so if you wanna finish that one problem, then get it to me. That's what, the only time I'm gonna count a problem, a assignment late, is if I've already started grading them all or already finished grading the ones I had, you turn yours in late, now I gotta dig everything out again and start going, it's just a pain for me. So that's any time it's in my hands before I start grading them, I won't count it late. So I'll probably do these if I get up at three in the morning to do these, because they mean a lot to me. You did that work and I think I should, nope, all right. Okay, we've been talking about, so let's see, we've been talking about kinematics. That word should sound familiar, though maybe you can't immediately recall its meaning, but I think I told you there'd be four parts to this. Didn't I say that last week, that we're gonna look at kinematics, we look at four parts, we've got three of them. So there's gonna be four little pieces to this. We can make several weeks out of that, but there's really only four little pieces to this, and we've done three of them. What were the things we talked about already in terms of the kinematics? I'm in position. We started this whole discussion wondering, all we're doing is looking to see what objects are doing, and that's all engineers and scientists do. We either study objects or we study ways to make objects do what we want them to do, whether they're molecules down at the nano college in Albany or whether it's the space shuttle. It's something we need to make do what we want it to do, so we have to study it in some detail. One of the simplest things we can start with is wondering when is something at a particular spot? I think I used an S there for that, for position, just a generic position. We might be a little bit more specific, very shortly we might talk about position in terms of X, Y coordinates or something like that if that's useful, but for right now, we're just speaking sort of generically, so we talked about position. So we want to know when things are and where they are as we go through these problems. Then once we had the position in hand, we did something else. What came next? Velocity. Velocity. We worried not just about them being in some point in time, but the fact that we don't want them to just sit there forever, that's pretty boring. It may be that we need to make them sit there, and we don't want them to move. That's certainly the concern a structural engineer would have, a structural engineer wants to put up a structure and wants it to stay put. Wants not only the building to stay put, but wants the top floor to stay put. It's catastrophic when top floors go somewhere else in a building, especially if there's people there. So we started looking at that and that was all based upon the position. And we haven't gotten to our fourth one yet. That's coming up here. So this velocity, let's see, we talked about two flavors of velocity. We had average velocity. Everybody know what those three bars mean? Is defined as. Maybe the math teachers will come running in and say, no, no, that's not the symbol. But that's good enough. Average of velocity is defined as, and we did this on Wednesday. What was it defined as? Yeah, check your notes. If you don't remember, it's in there. I know it's in there. I put it on the board. Mike, help us out. The initial position minus, I mean, I'm sorry, the final position minus the initial over the final. Okay, you're right. The more detail we'll get there in a second, but it was the change in position divided by the change in time. And this, you don't even have to, this is an equation that you don't have to memorize because you know that when you get in your car and you drive somewhere and you're driving miles per hour, the units tell you what you could do with that to figure it out anyway. So this is not like it's any great news to you. In fact, you're very familiar with this idea of velocity. But then Mike went in a little more detail, just mostly letting this, reminding us what this delta symbolism means. It's always the later minus the earlier, the later minus the earlier. It's gonna be very common that our time, our initial time t1 will be zero. And so we don't always have that in there, but in general, if it's in there, well, we can handle it. It's not that big a deal, we can handle it. So we looked at some things like that. This position measured from where? Because it doesn't make sense that we all should agree on where we measure things from if we wanna study something about the position. So let's agree on where we measure all positions from because you don't know where I live. And don't try to find out where I live because I will get the restraining order against me. Again, since I've had Joe before, this would be the third or the fourth, I think. You know something like that? Al, I just got an open one with the sheriff's office for you. Where should we measure the position from so that we can all talk about this and agree what we're talking about? You know, oh yeah, I'm at five miles from where? The origin, five miles from the origin. Let's agree on where that should be. Phil, right? Phil, I mean a point of convenience. What if something's convenient for him that's not convenient for you? Huh? Position? Yeah, well, that's definitely true. We don't wanna place an origin. Let's start working on a problem and have the origin move. That's dastardly description to any problem is trying to understand it. Because then the numbers don't mean anything. They're changing as we go. So, Phil, you said convenient? No, who said convenient? That was Phil. Don't be shy. What do you mean by convenient? Two subs, so we're gonna measure it. Well, that certainly makes it kind of nonsensical if we all agree on a single origin and then everything's measured from there for that specific problem. And it may be that when you draw it on your page you've got an origin that's different than mine but the problem lays itself out on the paper in the same way. But what's most important to remember about the origin is wherever it is, it's arbitrary. We can put it anywhere we want. If we do wanna put it somewhere that it's inconvenient, you have the right to, if you do wanna measure everything from where you were born, the hospital room where you were born or the barn you were born in or wherever that might be, because then I'm talking to some Adirondackers, a couple of you know, this might be the case for. You can do that, it's gonna be inconvenient, but students are known for doing things inconveniently. So wherever we put the origin, it's arbitrary. So I'll put it kind of in the middle of the board and you'll put it kind of in the middle of your page and that's good enough. Once we put it there, it's not gonna move. Then we also need some kind of scale of measurement what we're doing with things. For most of our problems, it'll be as simple as a straight line. Doesn't have to be horizontal, doesn't have to be vertical. Depends on the problem, but this is as simple as things can get, I think pretty much everybody understands this, I hope. And then we look at things, maybe it was there at time one and that's the position then S1. We understand what that means, especially if we have numbers up there to represent certain distances, whether we're talking about meters or centimeters or whatever, and we went through this in some detail. We looked at other spots as to the position and time to wherever that might be and whenever that might be. Well, we measure that again from the origin, that's certainly an illustration why we don't want that origin to move anywhere. We don't want these things to pretty much stay put. All right, so we looked at a couple things. We looked at the average velocity. If this is where it was at time one and time two, whatever those were, what was it doing in between those two times? This is the kind of thing where maybe we're taking frames of a movie. So we took a picture of it when it was right here, then we took a picture of what was right here, but what was it doing in between? Where was it in between time one and time two? Was it right here at midway? Was it over here? Maybe, we don't know, we have no idea. We don't have that information. We might feel confident about making a guess. We don't have that information. If at a whole bunch of other times, it was all evenly spaced at every one of the next times, then we might think, well, I have a pretty good idea what was going on, but I'm not absolutely sure. We just don't have that information. A lot of different things could have been doing in there. So another way that sometimes is helpful to look at this is if we make a position time plot out of it, where we graph those as our two coordinates. So at an early time, it wasn't very far away, so maybe it's something like that. Then at a later time, as implied by the number two, rather than the number one, it's a little bit farther away, so maybe it's something like that. We don't know anything else. I don't know where it was in between. I don't know where it was before or after, at least not in terms of the discussion we've had so far. And this average velocity we're talking about here is simply the slope between those two points. The fact that this slope is positive tells us what, or does it tell us anything that that slope is positive? Certainly is, we don't have any dispute whether that's a positive slope, do we? Does that tell us something? It tells us direction. On average, this is moving to the right because my distances were increasing. I didn't actually say it, but it's implied here, and you seem to all have picked it up, that I was calling this the positive direction, this the negative direction, so that this position one was less than this position two, and in fact the numbers got greater as we placed that. If I drew the next point, point three, such that it has negative slope between those two points, is that even possible? Is that possible? Yeah, of course it is. It just means that the object, whatever it was, moved right or left? Left, just mean, in fact, you can see that, it's moved back to some spot, a little bit somewhere between one and two, a little bit closer to, or some reason, whatever was going on, it got back to there at some later time, that's all we know. Nothing more, nothing less. We started talking then on Wednesday about the desire, the very reasonable desire, to know a little bit more, that these times that we see these things might be just kind of far apart, and where you don't know what's going on. This, you wouldn't want to be driving on the freeway with your eyes closed most of the time and just blink them open every once in a while to see where you are, I don't think you would. Well, some of you actually do that, you just look down at text for a couple of minutes, you know, the average time of looking at the text screen when driving is five seconds. You imagine going down the road and just closing your eyes for five seconds? Most of you can drive straight even when your eyes are open. Oh yeah, yeah, you're thinking, oh man, you talked to my parents. So we want to know a little bit more information than this. The average velocity may just not be informative enough for us. So what do we talk about next? What is it that Newton and Leibniz did to answer this very problem? They said, well, this delta T may be too big. I want to know what the velocity is at not some big macroscopic period of time. I want to know what the velocity is in an instant in time. Even though an incident time, no time actually passes by, so we couldn't actually cover any distance in an instant, we know that we've got some velocity at every instant. You know that, driving in your car, you look down at your speedometer, it's reading some number, and you only look there for an instant. It doesn't go down to zero all of a sudden because it's an instant and you can't possibly move in an instant. And so they came up with this concept of let's drive this time period down to nothing. Down to indeed, on incident time, because that's how long an incident time lasts for zero time, and that's what we came to know as the instantaneous velocity, and that's where it was that exact problem that the Newton-Lydnitz wanted to address that led them to the development of calculus. And that's the first part of why I need you to have already had calculus when you get here. So that gives us the idea of an incident time. So let's see what that would look like. Let's take the same points there. Let's make the very reasonable assumption that as it went between these three points, it did so maybe like that. That's not unreasonable. If we had more information, if we were able actually to track where it was, we could come up with that. Then the instantaneous velocity at any instant in time, so I'll let's see, I'll pick an incident time. There's an incident time. What's the velocity the object is moving at that instant? Do you remember what the derivative tells us is going on with the function itself? In this case, it's the position function. What's the derivative tell you whenever you take the derivative of a function? The slope. At that instant, that curve has a slope that looks something like that, I guess. At that instant, that is the instantaneous velocity. The slope of that line. Now in high school physics, how many of you took high school physics? Before you took calculus, you may have had some kind of position curve like this and then they told you, now draw the tangent and then get the slope of that tangent. Did you do that type of problem with it? That was graphically getting the slope at one particular point. That's what the tangent is. And then graphically figuring out what that slope is and that was your instantaneous velocity. We now know, since you know calculus, how to do that in a more exact way. Because everybody's tangent was a little bit different than everybody else's so everybody's answers differed a little bit. And if we go to any other instant in time, we get a slightly different slope. We get a slightly different velocity. Just as you remember from calculus, the slope changes different places with different functions. So if we happen to have this function, if we happen to know actually the functional form of position as a function of time, we can deliberate it and we'll know the velocity at any instant in time. In fact, a couple of the homework problems in chapter two are exactly that. They give you a function of s as a function of t. You deliberate it and you've got the instantaneous velocity just like that. Nothing too complex. Most of these things are polynomials so if you can take the derivative of a polynomial, you'll do great with it. Everybody remember how to do that? Okay. All right, so that's where we got to on Wednesday. So now things will change a little bit. So we're able to figure out some of the velocities of these different things. Let's see. Let's say right here, right at time one by whatever means, maybe I look down at the speedometer and I know what the velocity was or maybe we had the function and we are going to take the derivative of it or by some means we know what the velocity is right at point one. And right at point two, I can also know what the velocity is by whatever means as simple as me saying, hey quick, look down at the speedometer. You got it. Took some data. Notice I made the V2 arrow a little bit longer. It'll look like that to you. A little bit longer. Not a lot, but a little bit longer. Was that maybe in sloppy you think or what should we let that represent? Yeah, maybe the velocity is a little bit higher. Nothing worse than here you're going 30 miles per hour. Here you're going 45 miles per hour. So we can represent that if we need to. So let's put some numbers to this and see what we can make of this business. Let's say this is, we'll just keep it simple. Three meters per second. We'll call this one four meters per second. See, well we'll just let those numbers there represent seconds. So at one second, it was moving three meters per second. Does this make sense? I have it pointing to the right and this three is a positive number. Does that make sense? Do those two things go together? If you remember on Wednesday, we looked at the possibility of things coming back this way. When they do that, they get a negative velocity. So this tells us it's moving with a certain velocity to the right because it's positive. That's our positive direction. These positive signs are going to become immensely important to us in this class. They're as important as the units are. They're as important as are the numbers themselves. Maybe more important, because we're just making up the numbers as we go. So we've got to watch those kind of things very, very carefully. So we'll let these numbers actually represent times, seconds, when they were there, just to keep it easy for ourselves. Because now what we're talking about is the possibility, not just the position is changing, but the velocity is changing as well. So now we have some idea like this. Certainly what's going on is certainly the type of thing you've done in your car before and we can even put these numbers in here. We've got V two, it's four. V one is three meters per second. T two is two seconds. T one was one second, two minus one meters per, sorry, not meters per second. That's just seconds. We can put numbers in there. What do you call it when you're driving in your car and the velocity changes? Your speed changes, your speedometer needle's going up or it's going down. What do you call it? Acceleration. Acceleration, specifically when we talk to the schmows out on the street, we say, you're in your car and the needle's going up. What's that? And they'll say, oh, that's acceleration. And what if you're in your car and the needle's going down? They'll say, oh, that's deceleration. That's the way most people out on the street would say it. So it's a concept you've already got for it. Now be careful, we're going to realize that it doesn't have that full intuition that it might think it does like we do with velocity. But that's the average acceleration. The average rate at which the velocity is changing. That's very much an indicator to us of how quickly that needle on your car's pedometer is moving. The faster it moves, the greater the acceleration. The faster it comes down, the greater the deceleration. All those kind of things go with this. You might suspect that we want to do with the average acceleration, the exact same kind of thing we did with the average velocity to get the instantaneous velocity. Can we get an instantaneous acceleration? Is the velocity changing at any instant in time? Sure could be. Sure could be because when we slow down, and we don't do it in little chunks every once in a while, it happens very smoothly. That needle drops with a nice smooth, well, the guys like that needle going up as fast as possible because that makes the tires squeal and that makes the girls go, ooh. Right, happens all the time in the parking lot, the guys squealing the tires and oh man. And we'd like to come up to the stop light, slam on the brakes. Get that needle to slam down there. That's the man, that's the action we're into. We have the very same concept here that we can look at all of these things as they happen in an instant in time. And it's just the time derivative of the velocity function if we happen to have that. Some problems we do, some problems we don't. Ooh, hey, wait a second. Velocity itself is already a derivative. So this is the time derivative of a, that's too much to write, I don't want to write that much. So we'll call it d squared s over dt squared. Are you familiar with that? With the second derivative? It should be if you came out of calculus one. Do you remember the second derivative test? Do you test for concavity and all that business? Well in terms of what we're working on, that's all still very useful stuff. If we have the time position curve and we'll pick something real simple, maybe it looks like that. Any instant we can look and see what the slope is to know what the velocity is in any instant. Well just like you did in calculus one, you can graph that because it'll be a new function. Remember though that the velocity is the slope of the position curve. Position curve has a slope of, kind of looks like, well you know what, let's do this, let's go ahead and put some numbers on here. Let's just call this something like that function is two t squared. Does that look like a parabola? Do you ever take a little bit? Well if the position curve is two t squared then the velocity curve, which is the derivative of the position curve, bless you, who can do that derivative in their head? Two t squared, derivative of two t squared. Remember the power comes down and you reduce the power by one so this becomes four t. What's a line look like that has a functional form of four t? Isn't it a straight line? Positive slope got to match what was in the position curve. Notice the slope down here was pretty close to zero so the velocity's zero. Notice that the slope increases steadily so the velocity increased steadily. These two things go together. They mean the very same thing, frontwards and backwards. Whatever information we've got up here in terms of the slope is what's graphed down here precisely. And we can throw in this next little piece to it. dv dt, v happens before t. What's the derivative of a function four t? Just the constant four. So in this case, for those numbers, I have to pick two t squared. And graphically, this is a picture of the slope of the velocity graph. Slope of the velocity graph, well it never changes. So the slope of the acceleration is a constant function all the way across. These three graphs all tell the same story in a way. And so there's the first part of why I needed you to take calculus before you got here. Also, the four things we needed for kinematics, we've now got our fourth. It was time, position, velocity, acceleration. Those are the four things that's all we're gonna look at in kinematics for the first couple weeks here. All right, let's see, let's clear a little board space and look at this in some more detail. Because this is not nearly as intuitive as is velocity. Because everybody, not only lives velocity every day, you have to have some velocity to get here. You know that the velocity had the change in certain spots, if you're going through traffic or in a stop sign or you're pulling in somewhere, you gotta go around a corner, you know that your velocities changes. Not only that, but you have a pretty good idea what it changes from and what it changes to. Most of you have been driving off long enough that you have a pretty good idea what your speed is at any time. Not that you can always control that speed but you pretty much know what it is. That's not so with acceleration. Acceleration's not nearly as intuitive as is velocity. Let's be careful here because I've got two terms I've used and as we speak colloquially, we use them pretty much interchangeably. In here though, they mean very different things. You ask anybody out in the street, what's the difference between those two things and there's not gonna be much difference between them. Both of them have to do with distance traveled in a certain time. Both of them actually are right from delta S over delta T. So what is the difference between the two? Velocity, as we're gonna use it, as I had it up here with this number line, depends upon what direction you're going. An object with a velocity in that direction has a very, very different velocity than one. I just wanna make sure I made the arrows the same length than one with a velocity in that direction. Velocity depends upon direction. So in what we've been talking about so far, not only did we need the number, whatever it is, four or 6.8 or 12.9 or whatever the number might be for a problem, and of course we need the units because we need to know are we talking about meters per second or miles per hour or millimeters per decade? Well those are all very, very different. But we also need to talk about the direction. What we've been using so far is just simply plus if it's moving to the right, minus if it's moving to the left. That was sufficient. That's velocity. Velocity has direction. Speed does not. If I ask you what speed you were going on the north way, you'll answer me. You won't ask, well what do you mean? North, going north or going south? If I ask what speed you're going on the north way, you'll just answer me because it doesn't matter what direction we're moving when we talk about speed. So speed has no directional component. Velocity does. So speed could be, well there wouldn't be any minus signs if we're talking about speed. Even when you go backwards in your car, your speedometer doesn't go below zero. So we have those, that concerns. We'll talk mostly about velocity in this class. We'll be a little bit even more specific with it on Wednesday. So let's see, let's do something. Here's a little picture of some way for us to measure position. Let's see. Let's make a little chart. Let's put position down here, velocity here, acceleration here. Is it possible for me to have a positive position at any one time? Is that possible? Yes it is, just means I'm to the right of the origin. Is it possible for me to have a positive position and a positive velocity? Is that possible? Well we already looked at that. That's maybe to the right and moving farther to the right, farther down the number line. So if I was here and then later on here, that's a positive position and a positive velocity. Is it possible for me to have a negative position? Well yeah, of course. I'm just to the left of the origin. Remember the origin to arbitrary, so that could certainly be a possibility. Is it possible for me to have a negative position? A positive velocity, is that even possible? Got lots of heads nodding. I don't think anybody's shaking their head. What do I need to do to have a negative position yet a positive velocity? I need to be to the left of the origin but moving to the right at that instant. So if I was over here, that's a negative position, negative s, but that's a positive velocity. That's certainly very possible. All right, I'm gonna change things up a little bit with either of those two situations. What if this was the case? What if I was here and I'm going to move to here? So I'm gonna start here and I'm gonna move down to here but I did it this way. So I'm gonna start at my first spot and I'm gonna move that way. Now what happened? Now what was my velocity? Started here, that's represented by this spot here and then I moved down to a farther spot in the positive direction. But I'm going backwards. Now what's my velocity? I had a positive vote here. Would it help to tell him he's full of it if you knew his first name? Anybody know his first name? Yeah, it's easier to tell him he's full of it when you know his first name, isn't it? So you can say, go ahead, Joe. Say John, I think you're full of it. I'm here, I'm moving that way. That's our positive direction. I'm facing backwards. Now what's my velocity? John said positive. Gonna stick with it? Yeah, you bet. What do I have to draw a picture with my nose on it so I'm just going to tell him to face it? I'm here and I'm moving to here. Was that this situation where I had positive position and I moved in the positive direction or was that this? I had positive position and negative velocity. Which was it? Or is it both? Be strict what we're talking about. Nothing I've said before up until this instant had anything to do with who was facing which way. That never came into it. In this part of the course, that doesn't matter. I don't care which way I'm facing. All I care about is I was here, then I was there. I don't care if I was facing this way. Actually, you want to, how awesome is that, huh? Or if I'm facing this way. None of that, neither one of those mattered. See, I don't think my feet's showing them. The tape got there. I have to get up on the table and do it there. I'm gonna crash right off. Look at your legs higher than we want. So this is a case of positive position, positive velocity. Because none of this has to do with which way I'm facing. Which way is that dot facing when it moves? Well, that's me. I'm that dot. Or if we're talking about a car, that's the car. Or the space shuttle. I don't care what. We don't care about which way things are facing. We only care about where they are and when they're there. That's all we care about. So we decided this one was possible. A negative position with positive velocity. Yeah, that just means I need to stand over here more. But still, I need to move in the positive direction. Is this one possible? Positive position, negative direction. What do I have to do to do that? Start to the right of the origin. That gives me positive position. Move to the left. That gives me negative velocity. There you go. Doesn't matter which way I'm facing. In fact, if I want, I can probably get sick and throw up on your joke. I could twirl on it. None of that matters. All I care about is, what are we doing in relationship to what we've established here with this location, this locator system? Pick the origin, pick the positive direction, everything else is set. Another possibility, negative, negative. Can I have negative position and move it in negative with a negative velocity? Could you do it? I mean, I couldn't do it just because I'm amazing. Could you do it? Could you have negative position and negative velocity? Maybe we should all get up here and do it. What would you do to have negative position, negative velocity? Left of the origin gives negative position, moving left gives negative velocity. Everybody's pretty comfortable with that, I bet. Because we're very, very familiar with velocities. Because we do it all the time and we quantify it all the time. Let's do the acceleration column. All four of these are possible. So I don't need to play with any of those. Each one of them has two possibilities for the acceleration because the acceleration could be positive or negative. Let's see if we can figure out which is which. So we'll split this one off. Is it possible to have positive position, positive velocity, and positive acceleration? What do you need to do if that's possible? What do you need to do to do that? On a positive position, I have positive velocity. So I'm moving towards point two. What's gotta happen at point two for the acceleration to be positive? What, Tyler? I'd have a greater velocity. So I'll draw a longer arrow and I'll make it bigger. Moving in the positive direction, in fact speeding up in the positive direction. That's not hard at all to do. Start in the positive direction, move a little bit of speed, and then move faster, faster, faster, faster, faster in the positive direction. That's easy to do. What about positive position, positive velocity, negative acceleration? Is that possible? I have to do what? Slow down a little bit. Still moving to the right. See, because the velocity's positive. Still moving to the right. So let's, and then doing what? Slow it down. I'm not moving to the right quite as fast. Very possible. There's a stop sign up ahead, so I've gotta slow down for it, but I'm still moving in that direction. I don't care remember which way I'm facing. I could be backing up going down the highway. There's still a stop sign I have to prepare for. I still need to decrease my speed. Doesn't matter which way I'm facing. So both of these are possible. All right, let's go down to this one. Negative position, positive velocity, positive acceleration. Let's see, let's put a little sketch for that. Negative position, positive velocity, positive acceleration. What happens next? For example, if I get in here, what's gonna happen there? Is it possible to do this? By the time I get to here, if the acceleration's positive, then the velocity in that direction has got to be greater. So I have even a greater velocity. Maybe I'll put two pluses on it. If I could do that, I guess. That's certainly possible. You know, your home is in Glen's Falls, but you just came through Albany and then later you're gonna go up to Lake Placent. So your home is in the middle and you're just speeding up on the way home because you can't get home to mom and dad. No? You didn't feel that? So that's possible is negative position, positive velocity, negative acceleration possible. What do you have to do in that case? In that case, this second arrow would simply be a little bit smaller. Negative acceleration, moving from high velocity in the positive direction to a smaller velocity is negative acceleration. Decelerating, maybe there's a stop sign right here. Certainly possible. Haven't found one of these that we couldn't do yet, but we still have a couple of cases to look at. Positive position, negative velocity, positive acceleration, is that possible? Let's see, positive position, so we're over here, negative velocity, so I wanna put it here to start with, I guess, because we're moving in that direction. That's certainly possible. We can certainly be at point two, moving to the left. What does the velocity arrow look like here? Because that's the direction we're moving. We're moving in that direction, so we're gonna be here sooner or later. What must the velocity arrow here be to give us positive acceleration? Should it be longer or should it be shorter? I heard longer, I heard shorter, I heard longer. It looks like of those, do we need to take a full ballot? All right, now, this is not like the presidential election where it's not important and you shouldn't vote. This is important, so you have to vote. Should we put somebody's pacemaker? Oh, they're playing with the sparking things over there. Should we put our heads down on the desk when we vote so there's no cheating? No? Man, you're not, you're not game, all right. Should the next arrow here be longer, it'd be shorter, split about half and half? Who didn't vote? Somebody didn't vote. You're boycotting? Positive position, negative velocity. So we're on this case now, what do we need to do to get positive acceleration? So by the time we get here, we have a positive acceleration. The arrow should be shorter or longer. Let's do this. Let's not guess. Let's just make up some numbers. Let's call this velocity, we'll call it negative four meters per second just to give it some number. And one second later, we're here. So delta always stands for what? But more specifically, the later minus the earlier. In this case, this one happened later than that one. These are just labels, these are like mile post markers. So this is really going to be V one minus V two in this case, isn't it? Because one came after two did since I'm moving to the left. It's always the later minus the earlier. No matter what the numbers are, those are arbitrary, I just picked some numbers. I could have picked letters. And they've got to be in the same order down here. We'll let that be just something simple like one second. We're looking for a positive velocity here. V one is already negative. That's going to be minus because that's part of the equation. What must this number be to make this whole thing positive? What kind of number do I need to put here to make this whole ratio positive? It's got to be something negative itself, is that right? It's bigger than four, isn't it? Because if I had a negative two in here, this would still be negative. So I need something bigger than a negative four like maybe a negative six. Yep? Doesn't the four need to be on the other side? Oh, I'm sorry, that was V two. Thank you, Mike, good catch. The V two is here. So this is the one we don't know minus a minus four. Thank you, Mike. We would have been all screwed up if that had stayed. And we'll call them the time difference just one second. That better? Thank you, good. Don't leave mistakes up here, they happen to see. The trouble is I'm living in three different time zones right now because I just said something. I'm writing something and I'm thinking of what I'm going to say next. So it's very easy to get little mistakes like that you can't leave on there, you've got to fix them. In fact, 100,000 X credit points to you. Second week of class is already got 100,000 points. How many of you guys have? I don't know, you're open for four on the homework. Wow, good job, Mike. All right, so this is already a positive number. Don't forget though that we're moving with negative velocity. So this is also going to be a negative number. How big can it be though? We're looking for the size on V one. Could I put a negative six in here? That would make this whole thing negative and that's not what we want. Could I put a negative two in here? If something like that would work because that's positive, that's negative but smaller. So I want a smaller velocity vector assuming I'm still moving in the minus direction. That's positive acceleration. I was moving to the left but I was moving to the left less and less fast. That's positive acceleration. It's also not as intuitive. Gotta be careful with it. But you can always just make up some numbers and see what happens and it'll tell you. Thanks for the catch on that one, Mike. We all messed up early, just had it. Oh no, just cleaning up. All right, all right with that one. We have positive position, negative velocity, negative acceleration. Is that possible? So positive position, negative velocity which puts us up to here or some little bit of time later. How big should the arrow be here? I heard longer. Oh, but he, I see, like parliamentary procedure. Yeah, got a couple votes for longer. You're with us on any of these? Longer? Yeah, it's got to be just to make it illustrative. We'll make it longer. So maybe now we're going minus six meters per second. The speed was increasing but it's negative acceleration. This would actually be termed if we were going to speak loosely like that. This would actually be deceleration as we define the term where direction is imperative to the study as much as anything else is. But we don't, that's why we don't use that term because this sort of doesn't look like deceleration, does it? Not to me, not to you. So that word, deceleration causes confusion and that's what you were worrying about in my office this morning, that very thing. Let the numbers, let the equation, let these things do their own talking and you'll be fine. So this is certainly possible. On the positive side, moving left, moving even faster to the left. Is that possible? I'll leave that to you to think about those two possibilities. All right, as we look at accelerated motion problems, we'll look at that a lot. It's just a lot more interesting when things are accelerating than when they aren't. That doesn't mean there's not a lot of things we can do with constant velocity problems but things get a lot more interesting. There's a lot more physics, a lot more engineering involved when we have constant acceleration problems or acceleration problems. 99% of our problems will be constant acceleration problems. We won't look at too many others that aren't exactly that for us. Whenever we have constant acceleration problems, as soon as we can identify a problem as a constant acceleration problem, then immediately four equations are available for your use for solving these problems. Not that they are magical in any way. They're all derived in the book. One is, and I'll just put them in the order in which they appear in the book. I think they appear in that order. Yeah, one is the average velocity itself. Now, let's draw ourselves a picture of just what's going on with that situation. So velocity time, what's a constant acceleration problem look like on the velocity time graph? Nope? It's a straight line. We're already, that's in fact what I had up over here. Remember, the constant velocity came from a straight line. So it might look something like that. So if this is the velocity at time one, and this is the velocity at time two, that's not supposed to be two. Between time one and time two, what's the average velocity? Velocity is changing, because it was down here and now it's up here, but what's the average velocity? How do you find the average of two points on a straight line? No, much easier than that. No, much easier than that. It's the point in between the two. Since they lie on a straight line, the average value is right dead center. If that's a straight line, it couldn't be any place else. So you just take the average of the two velocities, add the two numbers together and divide by two. So that's your first constant acceleration equation. I'll get the other ones up here real quick, and then we'll go on. The other one is just the acceleration itself. V two minus V one over delta T. By the way, in the book, this is equation 2.4. This one is equation 2.14. The third constant acceleration equation looks something like this. Delta S. And if you look at these, in fact, especially this one, this is just the integral of A equals a constant. One half A delta T squared plus V one delta T. Tells us how far something has moved for a certain amount of time it's been accelerating. And that depends upon the initial velocity they had to start with anyway, which makes sense. If something's already cooking along with some speed, then the acceleration's gonna make that speed even greater, it's gonna travel even farther. So that makes some sense too. That's equation 222. The fourth one, the only reason they're in this order is because that's the order they appeared in the book. No other order implied by it. And this one is V two squared equals V one squared plus two A delta S. And that's equation 225 in the book. Now that brings us to the end of class today on Wednesday. No, maybe tomorrow, maybe tomorrow we'll do this. I'll show you how to choose which one of the four equations to use in every single problem. Full proof method. It's just knowing you have four equations to choose from doesn't help you choose which one of them it is you wanna use.