 Take it, take it, take it, we're on tape. Here's something I have for you that I would have given you last week if we hadn't had the snow day. It's a summary of the constant acceleration equations. Anytime we have a constant acceleration problem, it falls out in exactly this way. Every one of the constant acceleration problems do. Now, I know a lot of them I think were due on the homework. That's coming in today, right? So a lot of them are already in there anyway. So this would have been a nice help. What my students in the past have found to be very helpful is to get these equations tattooed somewhere. So you always have them available. Some of them on their forearm right there, that's convenient. Others like to tattoo someone or somebody else to share. I have mine on the inside of my eyelids so I can blink and I read them and it's right there. And I've got all my, in fact, I've got all my graduate school and everything, it's all right there. There's her phone number. And I wanted to call. Oh, well, a little late now, it's been four years, she's not waiting for my phone call. Also though, and I think I tried to hit this in class, all these constant acceleration problems are done in exactly the same way. And I think I talked on how to pick the equation to use. You've got these four equations, but which one do you use for the problems? It's self-explanatory right there and how you do it. Each constant acceleration problem has four variables in it. There's five possible. They're all listed here. Figure out what four are involved in your problem. That tells you which equation you use and you're done. I don't know how you gotta watch units a little bit, some other things, but basically, they're all pretty straightforward. We'll look at actually another constant acceleration problem then tomorrow, and we'll take it up in a little more detail. I don't know where she's sitting, so I can't give her a paper or so. We'll take it up in a little bit more detail than starting Wednesday and the like. Start touching on a 2D constant acceleration problem then on the last model. There Joe, Samantha came in. Somebody pissed her off and she left. She's sitting there. That was her stuff, and then she's sitting here. I don't know if she's sitting there. What is, not only is it an optional day, it's evidently a musical cheer day. There's Badgers. Oh, that's where you went. Mike, you went and got him? Thank you. Is that okay? Is that gonna work for you? You want to switch him? You want that one? I'm okay with this. The regular desk was a small one, and we don't know what she's doing. Angel, I have put up a slightly revised schedule. What I'm gonna do is cut back a little bit on the vectors that we would have started on Wednesday. We would only use part of the time for that anyway. We'll finish up vectors today. I've got back a couple on the couple of the homework problems for the vectors for that one assignment. And then we'll be pretty much back on schedule. So if there's a few changes, may not even be worth printing out a whole new schedule, but at least double check it because some of the problems will change a little bit as you drop a few. Also, everybody who gave me the first lab report, your marked up version should have been available for you back on Angel. Joe, you had yours, right? Pretty, it's just there. You just, I don't know where you click exactly because I don't see what it looks like from the student side. So I only have to kind of guess what you need to do, but evidently it's, yeah, well, Joe can figure it out. We pretty much figure it out anyway. Remember, as I put comments on those, I can't always explain in absolutely clear and full detail what I'm thinking. Some of the marks, some of the notes are a little bit cryptic probably. No sweat, do what Joe did, bring it to me and say I'm not sure what you meant right here and I'll explain it in a little more clarity as we go through these, as we start to learn how to do these things. One of the things I was looking for in that first report is whether or not you've read the things I told you to read about report writing. So if you didn't, then you got dinged for that. So if I give you some resources to look at, please do so. All right, so you can bring those tomorrow and we can talk about them because that's kind of the lamb thing anyway. All right, any other questions before we get going? All right, our step into two-dimensional motion. We've so far just been looking at one-dimensional motion. One-dimensional motion is some kind of time called rectilinear motion. Do a little bit to prepare ourselves for what's called curvilinear motion. Very little with three-dimensional motion. One, because it's not all that much difference as you're gonna see from what we do with rectilinear and our step to curvilinear, it doesn't change all that much. It changes even less when we go to three-dimensional motion. Plus it's very difficult to draw, both for me at the board and for you in your notes and on your paper. Some students find it very difficult to conceptualize three-dimensional things going on on a two-dimensional surface. So it's just not that crucial that we push into three-dimensions since we do this well enough. When you do need to hit three-dimensions, if you do, it's not that big a step for you. You'll be able to make it for the most part on your own. One of the biggest tools we're gonna need in two-dimensional motion is the concept of describing various physical quantities through the use of vectors. The easiest of all vectors to conceptualize, which means it's a good place for us to start, is the position vector. That's where we started our rectilinear motion anyway. So it's a good place to continue with curvilinear motion. It's a pretty simple idea. A lot of you have already done some vector stuff in high school anyway. Again, we need an arbitrary fixed, agreed upon origin, just somewhere from which we measure things. Then anytime we have an object and we need to know where it is, we can simply draw an arrow from the origin to that object and use that arrow to locate the object for us. That's as simple as a vector is. It's got all vectors, all vectors that we're gonna deal with. Well, there's one minor exception as we'll get in a second. All vectors, though, at least all vectors in my classes, all vectors have three things. If, for a vector or a quantity you should understood to be represented by a vector and you don't give me all three of these things, you don't get it right. So these three things are important. For the most part, at least two of them should be pretty obvious. If this is our position vector, maybe we'll call it S. We could have used R or P or anything else, it doesn't matter. That's our position vector. It locates that object for us. It should be pretty obvious what at least two of the things are that all vectors have. Go ahead, Mike. Magnitude. Magnitude. In other words, if this object is a certain distance away from the origin, we need to know how far. So all vectors will have a magnitude. The length of the vector represents how much of the quantity being displayed. In this case, position is pretty obvious. That might be 6.2 miles. So now I know that that object is 6.2 miles from the origin, and we got a pretty good start. If you didn't tell me that 6.2, I wouldn't know where to find the object. We'd have a lot of trouble getting started and talking about what we're talking about. So magnitude. A vector twice as long as another one represents twice as much of whatever that quantity is. Pretty simple, straightforward idea, I hope. Another thing all vectors must have. Direction. If you tell me 6.2 miles, but you don't tell me in that direction, I'm gonna waste a whole lot of time looking for that thing. So we need to represent direction in some way. Perhaps it makes sense to just tell me that angle from some reference line, maybe from the x-axis, or maybe from Bay Road, or maybe from just the horizontal, and then some elevation. However, it's done. And there's lots of ways that we'll see to do this. If you don't tell me the direction, you haven't told me enough information. Just tell me how far away something is, but not in what direction. And your dad probably did that to you during that Easter egg hunt last year. Well, I see an egg sounding feet away, and then you started crying because he didn't tell you where and what direction on her feet. Len, is that coming back as a bitter memory? Yeah. See, I know just what's going through your mind, you guys. I know exactly. All right, three things. There's two of them. Doesn't matter what order they're in. It's just the order we might give us magnitude first, then Andrew gave me a direction. What's the third thing? All vectors have to have these three things, or you're not getting full credit from me. Position. Nope. Position, that's already, we got the origin, we already agreed to that, and we got the magnitude and the direction. That's a position right there. Units. If you told me 6.2 at 40 degrees, what am I gonna do? 6.2 steps, 6.2 miles, 6.2 light years? I need to know. That shouldn't be any great surprise because I've already told you, I need units on numbers if you want full credit. So all vectors have all those three things, or you haven't told me the whole thing, the whole answer. Full credit. So there's position as easy as that can be, but remember what we did with rectilinear motion, we started looking at the possibility of that position changing. So we wanna do that kind of thing too. So imagine an object is there, and then at some other time, a little bit later, we find it there. What is it in between? I may not know. I may not care. But all I know is it was at 0.1, a little bit of time later, is at 0.2. I'm gonna need to call that theta one, because this is now S1, this becomes S2, and maybe this becomes theta two, and I get a good idea now of where it is. Not terribly in light of what you do as you put these kind of things together yourselves. All right, so one of the things we're gonna want to do most is exactly what we did back here at rectilinear motion when we made this very same step in one dimension, now we're doing in two, is we wanna have some kind of understanding, not just of where it is and when it's there, but of what the change was between the two. It should be fairly obvious that if we had the positions one and two, we moved from one to the other, that it's that change in the position that's gonna be as important to us as anything we look at. So one thing we're gonna wanna do is learn how to figure out what this change is, as well as some other combinations of vectors. That delta S is intimately a factor of whatever the original vectors themselves were, because if we're at any other position, then that delta S is gonna be different for any other two positions that we could have come up with. So we need a way to look at vectors to be able to do these type of things. All right, one thing we're gonna need to do is to add vectors. This will be vitally important when we start talking about force, because we're gonna be adding forces together. And it should be pretty common sense to figure that forces are the type of things that are gonna have magnitude, how big is the force, and of course what units it is. But also direction, a force in one direction is very different than a force in some other direction. So force is exactly the kind of thing that's gonna be very helpful for us to represent as a vector. As much as important as position is. So we're gonna need to add vectors. Easiest way to do it, imagine we have some vector A, we want to add to it some other vector B. Easiest way to do this is to imagine exactly how you'd add two trips you took together. If you took a trip A, and you want to add to it a trip B, you'd pull it up there, put them tip to tail, the result of where we started and where we finished is the addition of the two vectors. That's the easiest way to do it. Simply put them tip to tail, tip to tail, however many you've got, whether there's two or seven, doesn't matter. Notice that as I moved vector B, I tried to preserve both its length, its magnitude and its direction. We can add any vectors this way, so it doesn't matter if they're position or force or any other vectors, we're gonna look at all vectors can add this way. Some of you may have learned of what I guess is called the parallelogram method for adding vectors, where we take vector A, put vector B to its tail, and then draw parallelogram. That line is parallel to this one, this line's parallel to the other side, and then cross the parallel. Notice that within my ability to sketch this at the board, the result is the same either way, same direction, same magnitude. So I don't care which one you do, whatever works out best for you is fine with me. For how many of you is vectors a review? You've had them before. For some of you, they're new though. Okay. A lot of this one chapter are doing some problems like these. They're giving a couple trips that a boat took based on distance and direction it traveled, and you have to add all those together to figure out what the total trip is. We'll actually hopefully take a look at some of those. The one thing we do have to worry about when adding vectors together this way, by the way, this is considered a graphical solution of adding vectors, because we're actually drawing these things, is you have to have the idea of some kind of scale. Most likely you're either going to be drawing vectors that can't be drawn to life size. For example, one of the first problems, I think it's problem three in the homework assignment, is about a ship traveling distances over the ocean, where it travels distances of 18, 19, 20 kilometers. I don't think you wanna draw that to life size. Okay, a couple of smiles so you realize why you don't wanna draw to life size. I don't want you turning in. Alan, I don't want you turning in paper where these are drawn to life size. I don't care how many times you think you can fold it up. I'm gonna have to take a piece of paper as you can't fold it more than eight times anyway. Everybody's heard that? Except I believe a girl in Australia proves her wrong. Oh, that's nice. Now, I'll tell you. So you want some kind of scale, and generally if you have a graphical solution of the vectors, you have to have your scale shown there too, where it might be as simple as saying that much distance represents maybe one kilometer, and then I'll know I can take the scale and say, okay, that represents about four kilometers. It's not real, it's not a full scale, but it is drawn to scale so that the solution then can be put back into the scale and figure out, okay, now my answer is eight kilometers or whatever it is. So it's important that you have a scale. You might also wanna do it where you write one centimeter on the paper equals one kilometer in the problem, something like that as well. That's a pretty easy conversion. Use simple conversions for yourself, simple scales, so you don't make mistakes there. If I have an 18 kilometer trip I need to represent, I easily know that it's 18 centimeter vector, and it's very easy to draw it that way. If your conversions have fractions and all kinds of other things in it, you're bound to make mistakes that anybody does when you make things more complicated. So try to keep your scale something simple. All right, we're gonna add vectors a lot in this class, especially when we get to the next part of the class after this kinematics, when we get to the kinetics. Add vectors a lot. We also, though, need to subtract vectors. It could be that I have a problem where I don't want A plus B, I want A minus B. I'll use those same two vectors there. So I'll scientifically transfer their length and their direction by doing this. Does that look like A? Like the same A? See how accurate that method is? Now, subtract from it B. Here's how I do it, because I'm a simpleton. So I need to keep things simple. Here's how I do it. I remember that subtraction is the same as addition. Because adding vectors, I already know how to do. I don't have to learn anything new, which is good for a simpleton. Keep the number of new things down. So I wanna add to A, just like I did here. I move from here to here. I wanna resume my trip from there, but I wanna add to it, not B, but minus B. So what's minus B look like? Here's B. That's what sort of a B look like, but if I put the arrowhead on the other side, I have minus B. Now that, I can add to A. So I take minus B, move it up to here. I go from where I started to where I finished, and there's A minus B. Isn't that easier than what some of you were probably taught, where you're taught to put, oops, let's see. A looks about like that. When you subtract vectors, they tell you to put them tail to tail. Let's see, so I wanna subtract B from it. The trouble is, once I've done that, I always forget, do I go up or do I go down? I always forget. This way, I never forget. It's impossible to forget. A minus B from where I started to where I finished, I'm all done. Now, since I already have this answer, I know to do that. Same answer as it should be. It's just, from my small mind, this is so much easier than this is, because I just forget. Where's the arrowhead go on the A minus B? I can never remember. Maybe you can. How many of you were taught to subtract vectors this way? Nobody? This way you were? Yeah? So I'm not the only small one. So we're all your other old teachers. All right, so we're gonna do that a lot. We're gonna do both of those. The reason we need to subtract, you may have already seen it. Coming up here, if we're talking about, remember we had two position vectors here because the object moved from one place to another, you may have already noticed that this is the subtraction of those two position vectors. So that's probably the number one reason we need to subtract vectors. The reason we need to add them is whether we have multiple trips going on or we're adding multiple forces together or all kinds of different things as possibilities. So we've already got that knot. The very useful part of this is we know the average velocity would be the change in position which we now have divided by the amount of time it took to move that distance. So now we can subtract vectors. We can already make our step to velocity. We can make our step to instantaneous velocity. None of it matters now that it's vectors. Doesn't matter that it's a vector, all of this stuff still applies. A little bit of trouble here though that we'd like to avoid. Doing these kinds of solutions, these kind of addition and subtractions of vectors is dependent upon your ability to draw a straight line, that represents the proper distance, proper magnitude of the vector. Also, can you properly get the right angle on it? All of those things depend upon the ability of you, the user, to hand draw each of those things that need to be done. I notice not very many of you have rulers on hand so that makes these vectors even a little bit tougher to do, get an accurate solution. It all depends on your ability to draw all this stuff. So what we'd like is something more accurate, something more direct, more precise, more repeatable in its ability to get us to the correct answer. To do that, we're going to solve these vector problems and so many of what we have left to do are vector problems. We want to solve these vector problems analytically. So here's some vector s representing position. Should make sense, I guess, that even velocity could be a vector. How fast are you moving and in what direction? We were handling that before with just a plus and a minus sign direction left to right, up and down. Now we can handle directions in all kinds of different places. But what we're going to do to solve vectors analytically is to use a mathematical skill I hope you've all got. Some of you may need to review it a little bit, but everybody who's here should have already had trigonometry. Is that a foreign word to anybody? Trigonometry? Nice guy. Phil, are you friends with trigonometry? Maybe not good friends. Maybe you're acquaintances. You met. All right, that's what we're going to work on a more. The idea being that a vector, any vector we've got can be broken down into what we call its components. We typically use components that are perpendicular to each other. We do that because our XY system is perpendicular to it. The X and Y directions are perpendicular to each other, so it just makes it a very easy way to break vectors into orthogonal or perpendicular components. The simple idea being that say I want to go, I want to know where whatever this position vector represents out to some point, I could get to that same point by also traveling in this direction for a little ways, turning at a right angle, and traveling in that direction for a little ways. And I would still get to the same place. The idea being that the part of that position vector S that goes in the X direction added to the part of the position vector that goes in the Y direction will get me to the same place. S is made up of a little vector in the X direction plus a little vector in the Y direction. Very simple idea. You would do exactly this. If you lived here and you're asked for a play date over here with your best friend, actually the moms would call and make a play date, you wouldn't cut across everybody's yard to get there. You would go part way down Elm Street, turn and go part way up Main Street till you got to your friend's house. Isn't that exactly what you do? Especially in a town where the streets are laid out as a rectangular grid. Some of you would probably jump on your dirt bike and actually take this trip straight across. But the rest of us civilized beings would. All right. So that's the idea of what we're gonna do here with all these vectors. However, what we need to do next is figure out just how big is SX if we have S? Because we need to be able to get from S to its components and from its components back to S at any time we need. And how big is that? For this we use trigonometry because it's so easy to do once you get the hang of it. It's very repeatable, it's very accurate, it's on every one of your calculators. Everybody's got a calculator, not everybody here has a ruler. Given that this angle might be theta, then the magnitude of SX, and notice I've taken the little vector sign off here, so I'm only talking about the magnitude when I do that. In the book they do boldface for vectors and non-boldface for just the magnitudes. I have to, I can't do boldface here at the board so I put on a little vector sign. Another way to do that would be to put that kind of thing and I think most of you have probably seen that notation as well. They mean the same thing. This is just an awful lot easier. Just take off the vector sign and I'm done. I don't have to add a bunch of stuff. Remember, ink is expensive, especially chalk ink. Hopefully most of you remember from your trigonometry that it's the hypotenuse side adjacent to the angle S cosine theta and if anybody could do that on their calculator, I hope. The magnitude of the little vector in the y direction that helps us do the very same thing is then S magnitude of that hypotenuse sine theta. If you can get those two things, the cosine and the sine, every time we need to, you've got this part of it not. This'll be a breeze for you. Yep. Are we gonna be working with radians later? We'll be working with radians later. I don't think we bring it in yet. I think most everything we're doing in degrees to start with, we'll deal with radians later. It's just another way to measure the angles for those of you that haven't heard that term before. Just another way to measure angles tends to work better in some of the problems we're doing as we'll see. All right, the way we're gonna use this the most is when we're adding two vectors together, we need to add vector a to vector b. Sometimes I put the vector sign here. This letters right by a vector should imply that it is the label for the vector. Sometimes I forget. Sometimes I try to put it in just to be a little bit clearer. The way we're going to add vectors is by adding the x parts to each other and the y parts to each other. Let's see how that would work. All right, I have two vectors I need to add. Drop a line straight down, tell me how long. There's a x. Here then go up that line is a y. And of course, those are perpendicular. Be careful with this. If you draw it sloppy, then you're gonna end up with something that may not really help. We can put into two parts, bx there, the by there. Then the vector a plus b, and now get into ways. The old fashioned way, here's a, a plus b is gonna look something like that. Or I can do it using the components, do it using this. Let's see how that would work. A x is just that. Add to it bx. Then the same direction. They add together something like that. I add on again now, a y. A y, that's right here. So I put it there. Maybe it takes me up to something like that. There's a y. I add on to that b y. Offset them just a little bit, and notice I get the very same answer within my ability to draw the sketches at the board. Little cartoonish looking, but either way, we get the same solution. Why might this be favorable? Being, because you can trigonometry, we can calculate all of these, and we don't have to make these sketches that are so prone to mistakes, and fudges, and little goof-ups. We can figure out all of these using trigonometry, and then just simply add them together. And we'll make very, very little mistake doing so. So let's, let's, let's see how that might work. Let's look at problem three, three in the book. It's a sign, if you don't have your book here, no trouble, I'll just tell you what it is. But it's one of the ones that's signed, so we're not doing any extra work, it's actually gonna save you some work. Pred, Pred, Pred, Pred. Okay, page 88. Page 88 is a three-legged trip by some aircraft. Plies from initial position P, final position P prime, by three separate legs. And those, it's actually sketched out here for you, so there's a lot less chance of going wrong. And you're asked to find the answer graphically. That means you have to actually sketch this out to scale. So you have to take the first trip, and get a protractor. And at 60 degrees, you've got to draw the first trip by the aircraft, representing 18 kilometers. Again, not to life's size, that's a lot of paper. So maybe 18 centimeters would be just about right on a piece of paper. So you draw a line, maybe 18 centimeters long, but it represents 18 kilometers. So it'll have direction, magnitude, and units. 18 kilometers, 60 degrees to the right of vertical. I guess that would be, might be north or something. Doesn't actually stay in this problem, if I remember. From there, the plane flies a second leg down 60 degrees from that, about nine and a half kilometers. So you draw that vector at 60 degrees and make it about nine and a half kilometers long. And it looks like, yeah, that's about where it shows up on that picture. And then at another 60 degrees, and this one's from the vertical, it then says you go 12 kilometers. So all that is, is a resketch of just what was in the book for this problem. What it asked you to do, though, is reproduce it on your paper to scale. Maybe this one in the book is to scale, maybe it's not, I don't know. The trouble is, it's really small. The smaller these drawings are, the more likely you are to make a mistake. You have a vector that's two centimeters long and you make a half a centimeter error. That's terrible, compared to a 20 centimeter long vector with half a centimeter error. That's not so bad. So make these drawings as big as you can on your paper. This one, you'll already be able to figure out about how much paper you need. If that's 18 kilometers and maybe 18 centimeters as you draw on the paper, you can tell if that's gonna fit on your paper or not. And you know, it'll start way over at the left when you start to draw this. But you're, as a first part of this assignment to draw this to scale. That means you need a ruler and you need a protractor. This drawing in your homework should look very nice. Nice straight lines, not the sketchy wobbly things. Like if I was flying the plane, that's the flight attendant to bring me another couple of free drinks. You're to find what, where is the plane now in relation to where it started. Maybe we'll call that R. That's pretty typical for resultants. The vector that results from all this other work, quite often then we'll call that R. You draw it to scale with a protractor. You then lay the protractor down, figure out this angle, take your ruler, measure the length of that vector. If it comes out to be 10 centimeters, then you use your scale, you know that represents 10 kilometers. That's how you solve this problem graphically and that's the first part of this one problem you have to do. And if you do it carefully, or if you're really good at this stuff anyway, some of you may have taken drafting in high school. Possibly, some of you may even know how to use AutoCAD. Know what I mean? Some may know how to use SOLIDWORKS to do it on SOLIDWORKS. You can draw these angles at 30 degrees, make them however long you want to make them. You can do this right in a drafting program if you want. Just to be back, there's people that have a standard, I would think, which way the angle is. Like in this particular question, you've got 60 degrees from vertical, which is actually a 30 degree angle. No, no, no, no, no, no. It's 60 degrees from vertical, leave it at that. There's no question. This is 60 degrees. There's no question what direction this is. In the book, they show the y-axis. This is 60 degrees from the y-axis. No question. Don't do anything more to it. Anytime you make another step where you say, well, I want to look at that angle, I want to look at this angle. Every time you do that, you run the risk of making another mistake. Every time you change the numbers from what they are to something new, you run the risk of making a mistake. I'm more referring to the answer with R. Would you have to write from the x-axis down or would you just say, do whatever you want. You're going to lay your protractor down here and measure that angle. You don't want to measure this one, measure this one. Just tell me on the drawing which one it is and I'll have it. I'll understand. Well, that's kind of what I'm asking. Yeah. If I put... It's not a negative 30. No, no, no. It's from the x-axis. Come on. If I drew, if I wrote that, is there any question what this angle is? Does anybody have any question what this angle is if I write that? I don't need a minus sign. I don't need some other angle. It's done. This is 32 degrees. Technically that would be negative 30. Drop the technically crap. You're not happy unless things are more complicated. Alan, stop it. You're going to screw yourself up. You're going to screw me up. If I write that's 32 degrees, you're done. It's right there. It's 32 degrees. Don't say it's minus or plus or anything. That just comes. Then what's a minus thing? Did I do it wrong? Should I have done it behind my head for a minus or what? It's just trouble. All done. 32 degrees. Onto the next problem. Don't dig yourself a hole because you're not happy the one you're in is big enough. All right. In fact, I'll give you the answer if I have it on my paper here. 27 and a half degrees. There's the answer. We'll get something close to that. When you do this graphically, do it by hand, you might come up with 30 or 25 or something. That's close enough. Remember, the graphical solution of actually drawing these is approximate. We're going to do it, though, analytically much more often using trigonometry. Here's how we do it. I'm just going to reschedule this. I'm going to make it nice and big. I suggest you do the same thing because you've got a lot of stuff to fit in here. So nice and big. Sketch it as best you can. All right. 60 degrees, 18 kilometers. Same drawing, just rescheduling it a little bit. And then we come back 60. That makes those two parallel, doesn't it? I believe it does. Third tripper are actually, they left the stove on it, come back. All right, here's how we're going to do it analytically. We're going to break each one of these vectors into its component pieces in the X and the Y direction. So this first vector here could be made up of one vector that way. Let's call this one A just for reference. So this is AX and there's AY and they're perpendicular. Let's figure out how long those pieces actually are. Because remember what we're going to want to do is add all the X pieces together and then all the Y pieces together to get the same final vector. So just to emphasize that they're to be kept separate, we'll do the X pieces there and the Y pieces here. We know this angle to be 60 degrees. What's that angle then? Also 60 degrees. Remember the business, you have two parallel lines with a diagonal line across then the two opposite angles are the same. I forget what they call it. So there's our right triangle with the known angle. So AY is the side adjacent to the angle. How do you find the length of the side adjacent to the angle? You use the cosine. The cosine's always the side that's adjacent to the known angle. So this is 18 kilometers times the cosine of 60 degrees. Notice if you really wanted to, that's 30 degrees. You could have done 18 kilometers side opposite the 30 degrees. You'll get the same thing. Who's got a calculator? What's, what is that? One half. One half, one half a kilometer? Oh, it's nine kilometers. Nine kilometers. So we already know that piece is nine kilometers. Well, it kind of closer to what that one was there. So that's okay. What's AX going to be? We're now at the side opposite the 60 degrees. So it's the length of the hypotenuse times the sine of 60. All right, Mike, what's that one? No, no, no, no, radical threes. I can't think in radical threes. Give me a ruler that has radical three on it. Give me a distance, then I know what I'm talking about. All right, Mike's going to do it. Oh, he's got to go to the calculator. Phil's laughing at him because Phil's already on the calculator, so I'm going to do it the easy way. Come on, Phil. Your voice hasn't changed yet, has it, Phil? That was, I thought I'd heard you talk. Oh, what, Samantha? 15-six. Are there more numbers than that on your little screen? 15-six is fine. Take all these homework and test problems and there'll be three, maybe four significant figures and just don't even worry about the rest of it than that. All right, that's the AX and the AY piece. Now we do it for, let's call this vector B. Vector B is made up of two pieces as well. A little bit in the X direction, a little bit in the Y direction. Oops, not A, B. So that's BX, that's BY. We have now a right triangle and the known angle is the 60 degrees right there. So how big is BX, how big is BY? You do it. Let's see if we all get the same thing. Did somebody take a note? I don't know, but weren't you guys having no fingers doing this? We're all doing the same problem so we all come up with the same answer and the solution and they might have it wrong by the time you get to the end. Got it already? Did you check with anybody? Or are you too embarrassed by your really high voice? Come on, Phil, if you're towing almost anybody, if they start laughing at you, just beat them up. Get an answer, check with somebody. We're all doing the same one, might as well see if you're doing it. Len, you got something? Check with Phil, introduce yourself. Wrong, he's wrong, wasn't he? Oh, kilometers, thank you Joey. Do you want it to sign on the left here? Do you agree? All right, and you can look at, if you've done a decent sketch, you can tell that you're getting these right, you're not getting them mixed up. Clearly BX is shorter than BY. So if it comes out the other way around, you probably mix the sign and the cosine. It happens pretty easily as we start looking at these triangles all turned in different directions. You're all used to this kind of right triangle, maybe not that kind. So you gotta be careful with these. So what do we get for BX? 4.75, wrong, it's so hard, wasn't it? It's a little bit. A little bit? It won't at all by the end of the semester. Anybody get something different? So let's check, double check how we got that. Anybody get anything different? How about BY? What do we get for BY? 8.2, nope, 8.2 kilometers, nope. Negative, because this is going in the minus Y direction. So we have to put a negative on it. Negative, what'd you say 8.2? We'll need that when we go to put all these together because clearly AY and BY are in opposite directions and now that's reflected right there. Once we lay down this X, Y, ordnance system, that kind of stuff was for the most part determined. All right, now we gotta get CX, CY. Let's see where that triangle would. So CX is gonna be something like that, CY something like that. Those two separate legs would have given us the very same movement across the 12 kilometers of given angle. Nobody in class you like? Is that there's nobody in class you like yet or nobody like in class still? Or nobody you like in class anymore? What's orthogonal? Are these orthogonal? Both of them, yes. Are what orthogonal? This and that? Well, you can tell that's 27.5 and that's 30. We all agree, anybody disagree with someone you ask? Bill, you got CX, what did you hit him? Did you just hit him? Did you warn him? Negative, why negative? Yeah, it's going to the left. So negative 10.4 kilometers. Bill, we have a side opposite the 60 degrees and my partners are 12, so it'll be 12 sine 60. And you have to put the negative on because we can see that's the way it's going. CY now, Joe, you got that one? We buy that, negative six kilometers. Now, those together, those all together should equal the resultant vector, which will be part in the x direction, part in the y direction. And all we have to do is add up the two separate directions. We've already got the minus signs in there, so we're careful so it'll handle the fact that we had to backtrack in certain respects. So sum whiz kid, 15.6 plus 14.7 minus 10.4, 9.95, we'll call it 10, it's a minus, isn't it? Oh yeah, I'm sorry, I'm looking at the VX. VX is positive, so we'll call it 10. A little bit less than 10. RY, we do expect to be negative and we got about negative 14, so it's going to be about negative five, something like that, what is it, five, one, two? Oh, there's, and the negative takes care of itself. Grease, just with our drawing, just what we expected. Let's double check two things. One, what's that angle there? It's actually the same angle as here, isn't it? I'm going to take that one because that's the one I happened to list already. How do we find this angle when we know the two components? Theta equals the arc tangent or the inverse tangent of side opposite over side adjacent, RY over RX. What's that give you for an angle? 27.5, 27.5, good, just what we thought. Depending on just what you put in for RY and RX, it may have a minus sign on there, so we know it's not telling us it's in the other direction. It's just, it's that angle when we're done. And what's the magnitude of RX? 27.5 degrees, but how far? How do we find that? Yeah, you remember this from fourth grade or something, this Pythagorean standard. So the magnitude of R, which I can denote with just the bare letter itself, R, is Pythagorean's theorem, square root of 10 and 5.2. And notice the units work out. What do we get for that? 11.3. 11.3, all right, so 11.3 kilometers in that direction. And we can go rescue that boat if that's what happened. Sank right there. All right, this takes some practice. For some of you it's kind of new. It's pretty well spelled out in this chapter. There's several homework problems just like this. We're gonna be using this a lot. So if you're struggling with it a little bit, either come see me, practice some stuff, go down to the math lab. Does everybody know where that is? If you get on to the math lab with a trigonometry problem, they'll be able to help you. Go down to a physics problem. They may or may not be, it depends on who's on duty. Any questions? All right, then we'll start with two-dimensional motion on, well actually we'll start with it in lab a little bit tomorrow. We'll look at a two-dimensional experiment, analyze it a little bit, and then as we learn stuff in the next couple days, we'll be able to put it together in some more detail.