 I have to say, I don't even know what to do with these. Oh yeah, I forgot to fix it. I'm on my duct tape. Oh, hey, yeah, here's the exam, too. It's a long one. No, no, no. I'll read for everybody. We're still good. Fix it? What could I fix it with? I think maybe some epoxy maybe, and a couple of splints. Alright, here's the exams. Before I pass them out and you stop paying attention to me, let me tell you just a little bit about how I grade them just so you can make sure nothing else of the points went right. I don't want to total up things and have it come up short, so I want you to be able to check. When I grade, the way I do it is if I find something wrong, I take off a few points. That way I figure if I miss anything, it's to your benefit. Rather than giving you points when I find something right, because then if I miss something, it's not to your benefit. I do one page at a time all the way through, so I'm more consistent that way. Then I come back to the front, to the first one, and I go through the second page all the way through that. After the first page, I don't know whose paper I'm grading by then, and I'm much more consistent because I keep doing the same problem over and over and over. So down at the bottom corner of the page is how many points you lost that page. Then when I'm all done with the test, I just go and look at the bottom scores on, at the bottom corner of each page, and take that off a 60 points. Just make sure that all those points work out. Not all the points were of the same weight and not even all the answers were of the same weight, because on the multiple choice one, I thought some of these there were wrong answers that were wronger than other wrong answers. So if you were a little closer than being completely wrong, I tried to give you a couple extra points that way. So at data 60, where's Bill? Also make sure that the score I enter on Angel agrees with this. It's pretty easy after entering a couple of the types of score. However, pay attention to Angel's calculation of the percentage total. I have no idea how it does that. Couple I, 50s, very effective. I don't think anybody was under 40, so that's good. Also, I will post a full solution as soon as I figure out what the computer problems are I'm having, and so you can check to see what was the deal there. By full solution, I don't mean just the answer, but an explanation of why that's the answer. Once you've taken a look at that, and still have any questions, then just come see me. Usually though, when you take a second to look at the solution, all the answers are cleared up, and you're no longer mad at me. So much for Samantha's optional day. I don't think she's... Well, the next time, you think you are. Maybe I'll take an option. I'm not really sure either. Your case, I was probably just angry. Man, that's it. I don't have to justify this. I get out my big red pen. It's like a cannon. Check the solution. When I get it put up, I'll put it up right after this class. Check that if you still don't get it, then we can bring it up in class, put it on the discussion board, or come see me. Was it in general okay? Nobody went over time, so that was good. That's always a big word. They're longer than I think they should be. Not out of line in terms of too easy, too hard, not fun enough. They're talking to me. No eye contact. It was fun. It was fun. It was fun? You need to get out more. Talk to Andrew about fun. He'll tell you fun, because he's going to Daytona for a spring break, and it's going to take him at least two months to straighten his head out when he gets back. Back, not only tattoos, but Mary did. If, not in jail. Then he wouldn't come back. Of course, he could be a flight. Alright, then everybody's okay. If you remember, we're looking at what's in general called curvilinear motion. We're going to stick to mostly planar motion or 2D motion. Everything we're looking at happens in a single plane. It may not be a vertical or horizontal plane, but whatever it is, it remains in a single plane. Actually, projectile motion does remain in a vertical plane. At least as we look at it as a free fall problem. Oh, that kind of came across on the test yesterday. What's free fall motion? It's a constant acceleration problem. We are ignoring air resistance. We'll see specifically how that comes into play, but some of the generalities on it were addressed in the book and were addressed on the test. But we're looking at strictly 2D motion as we look at it here. We're going to bring up a new piece of it with some stuff we're going to do today. By way of introduction, let's imagine here we've got a car going at good 60 kilometers per hour. Is that fast? Is that what you guys do? That might as well get out and walk. It's not even worth turning on the car to go 60 kilometers per hour except for reasonable adult human beings. Into a 90 degree corner. Comes out of the corner very simple. Simpleer than something. Don't leave that there. No, I saw the trash can duck. Very simple problem. And all I have is a simple question. And shouldn't have any trouble doing this. Calculate the average acceleration. Kind of a warm up. We haven't been on this kind of stuff for two days now. In fact, when we were doing projectile motion the acceleration was constant in a single direction and wasn't even a big deal. We didn't even have to calculate it. I just gave it to you. As I said, A equals minus G. I was going to say, there's a lot of people don't seem to be moving much. So either you already got it or you're completely stumped. So do this for me. When you've got it, think you've got it. Quietly raise your hand. You don't have to raise it real high because it could be up there while we're waiting for the rest. So when you think you've got it you know, little secret signal something to me, let me know you've got it. You think you've got it. If not, keep going. We'll be with you in a minute. Anybody see Jeopardy with the IBM computer last night? I just read an article about that. Is that cool? Oh my God, is he kicking butt? I mean it kicking butt. Didn't it have like 45 grand or something like that? And I want to know what the betting scheme is. How he's figuring out what to do on Double Jeopardy. Well no, he's not very consistent. The first time he got what is it called? Is it Double Jeopardy? Yeah, Daily Double thingy. The first time he bet almost half of what he had, which was a good 13 grand or something. Second time he bet like $600. I'm sure I know those guys look into it. They call them gaming experts I'm sure, and strategizers. There was no limit to who they could call in to help with that. And then on the final Jeopardy it was US cities, which you think would have been a pretty easy one, and he bet 900 bucks. And he already had 36,000 or something and then didn't get it. Oh, US cities. He put Toronto. Because for each thing you can calculate this percentage chance that he's going to get it right. Yeah, but US cities, you thought boy, if he doesn't have data based on US cities. Anyway, so they're all Yeah, I've got to take that and gotten to it yet. It's really cool. It's really cool. And Monday, didn't I tell you Monday my elementary school, the person to whom I lost the fifth grade president election was on cash gap. Actually, I think it was vice president. She thinks it was president. So there's a little bit of controversy. Nobody's hands are up. That's why I was talking. Oh, there's a couple hands up. Ah, there's one. You can just wink at me or something. Okay. Well, can't lift it too heavy because it's broken still. Not playing. Boycott. I don't want to embarrass myself. There's no answer. The wrong answer. Oh, no, I didn't say raise your hand when you have the right answer. I said raise your hand when you have an answer or think you have an answer or under the impression you have the answer or relate to personal self-delusion to the point where you think, yeah, now Phil's hand goes up. That finally got to it. Okay, I'm ready. Hand up. No, working. Joyce is up. Enders up. Mike's up. Land fill. Tyler. Thinking. Thinking or thinking. We don't know what. Alva, Samantha, you were up. You gave me a wink. Mark is up. I think you're the only. No, John and Tyler haven't done anything yet. They're either still working. You're done? Your hand up? I'm done. Wow. Did you hear it slam shut? Who's willing to venture the answer? Phil? Phil is. Phil, thanks for volunteering. What is this? Should I have an acceleration? This is a simple problem. Give me the average acceleration. You can't ask questions now. Your hand was up. Who's willing? You winked. You ready? You can't get it or no one can get it. Mark, where were we with you? Zero. Zero. Oh, a vector. There's a little vector sign there. So a vector of magnitude zero but pointing up. Now that's I've got to say I don't know that I've come across that before. I've come across zero vectors with no units, because when it's zero it doesn't matter what the units are, but zero and up. I like that. I draw it if I could. I guess I could put a dot. Just draw the arrow. It's a long time. I don't know what to do with that. Phil, you said can't tell with what's given. Somebody else said that too? Mike said that. Wrong. Sorry, but wrong. You can't tell. We can't figure this out. You said zero, zero, zero. You know it's not zero, zero, zero, and zero. And you, still nothing. Nothing. One. Tyler Drake. Not zero. He didn't know what it was. He knew it wasn't zero. It's not zero. Let's see if we can figure out why or why not. What's the definition of average acceleration? Louder. That was right. Change in v over change in t. Remember, average means we just take the bulk. The bulk change. We don't care what happened in between. We don't know necessarily what happened in between. Two time groups. We've got a time here. Let's go ahead. We'll call that v1. We'll call this one v2. What happened in between? We don't know. But that's not what I asked about. So we don't even care either. So I've got the straight average acceleration, just like we defined it in that fact. In fact, I think we did that first week. Right? Something's missing though. These aren't equal as written. No, no. This is average acceleration because it's the change in the velocity with time. Change in velocity. The rate at which velocity is changing is acceleration. But something's missing. Pardon me? Yeah. They're not equal because this is a vector. Has a little vector symbol. This doesn't. So I've got to put that in there. Now they're equal. Now they're equal. What does, oh, I forgot to give you, hey, Phil, I guess you were right because I didn't write down everything you needed. I didn't. I forgot. Five seconds. Is that what you were looking for? Why don't you just ask for it? Now you're ready to say zero? Yeah. No? When you take this again next year, I won't make that mistake. Okay? So you'll be ready to jump all over me again if I do. What's delta V mean? Yeah, but what do I write next so we can do some work here? V2 minus V1. Changed in the velocity vector. So V2 minus V1 over delta T. What is V2? The velocity vector 2. This is actually the speed because I don't have a little vector symbol over it. But if we put it with the direction, then I have magnitude, units, and direction. I've got a full vector thing. So what do I write down for V2? I want to give it some direction. Start with its magnitude and its units. But I want to give it some direction. So we'll use the unit vectors and just to be exciting we'll pick them in the usual direction. Sort of our X, usual XY direction. So this is 60 kilometers per hour J hat. Positive? Yeah, positive. It's in the positive J direction that I drew. Minus what is the vector V1 that I can write in there? Positive what? No, hold. Yeah, magnitude, units, direction. But positive. Minus sign? He said positive, but I wrote minus. Yeah, that's from the definition of changing. We can't do anything about that. That's just pure beautiful algebra. Alright, delta T let's do this. Change the 60 kilometers per hour to meters per second because I have time in seconds, so I don't want hours and seconds mixed together. So change that kilometer to 60 kilometers per hour to meters per second. Should get the idea that I need to hear the units play plus some just double check and see what other people think. Oh, what they had. You guys said 67 or something, didn't you? 16.7. He says 217. So let's see if we can come to some kind of agreement. You say 17. Anybody know how many kilometers per hour in a meter per second or vice versa right off the top of their heads? Not always off the top of their heads. So what can we manage? Let's see. How many kilometers on the top, so I want kilometers on the bottom, meters on the top. How many meters in a kilometer? Thousands. So if you look at the units, you don't have to worry about whether you divide or whether you multiply. They'll tell you what to do. We gotta multiply thousands on the top. I got hours on the bottom I want hours on the top, seconds on the bottom. How many hours in one second? Who knows that one. How many seconds in one hour? 3600. 60 seconds, 60 minutes, 60 times 60 is 36. Oops. Divide. I mean I don't multiply 3600, I divide. So 60 times 1000 divided by 3600 is 16.2 thirds. We're engineers. We've got real work to do. Give it to me in a decimal. 16.2 thirds sir. 16.7 meters per second. Why don't you say that in the first place? 16.7 known in uneducated points in the world as 16.2 thirds. Oh wait. Divide it by the 5 seconds. Let's go ahead and put that in. So we'll have the acceleration then. In meters per second squared. 16.2 thirds divided by 5 is, bring in a calculator then actually use it. See Joey did it but won't talk to me. 3.34 meters per second squared. And that's the same on both so I'll just factor that out. Leaving behind j minus i, is that right? 60 and a 5 and all that pulled out to the front here. Trust and mark. Anybody check him? He's right. Left behind the j minus i. Just the way it was if we want to turn it around oh my goodness, we can't have a commercial thing on screen, on tape. They're not sponsoring this class. They might have to see it. There, take that. They're not giving me free coffee today. Simple algebra if we wish. No difference. Just sometimes people don't like to lead with the minus sign. Sometimes they do. Sometimes people want to lead with the i and then the j and then the k that they got them. No trouble. Either way was right. All the information was there. Let's draw that. Let's draw that acceleration vector. 3.34 meters per second in the minus i direction. So where do I put it? I can put it about anywhere, I guess. I'll put it here. There's 3.34 meters per second squared in the minus i direction. Is that right? 3.34 meters per second squared minus i direction. That's the first component. Second component 3.34, same length plus j direction. So I'll put that on here. Yeah, that's pretty good. Pretty close to the same length. 3.34 meters per second squared in the plus j direction. Is that right? That's what we got here and everybody was happy up to here. So all I'm doing now is drawing this. There it is analytically. Here it is graphically. So if I add those two vector components together, I get something like that. Tyler's right. Wasn't zero. Phil and Mike were right because at the time they answered, I didn't have all the stuff on the board. I'd forgotten to put five seconds there. But everybody else is evidently thinking, no matter what the time it's zero. But that's definitely a non-zero acceleration. That's not zero. That's definitely non-zero. The speed did not change. How come there's acceleration? If you were going around, say the 60 kilometer per hour was on the speedometer all the way around, that needle didn't budge the whole way around the corner. You ask any dope out on the street and there's no shortage of them. Were you accelerating around that corner? They're going to say, no. The needle didn't budge. I wasn't accelerating. I wasn't decelerating. How come we don't agree with the dopes out on the street? One because we rarely do just on principle. Why isn't this acceleration zero? The needle did not move. Don't throw that at me. You can answer this without skipping ahead in the book a couple of chapters. No? This is a change in the velocity vector. The dope out on the street is thinking in speed. That doesn't work for us. This is change in the velocity vector. If anything about the velocity vector changes, we have acceleration. If the magnitude changes, we have acceleration. Well, it didn't here. If the direction changes, which is as much a part of any vector as the magnitude is, if the direction changes, we also have acceleration. Sometimes both change. Maybe when in at 60 came out at 45. Then both direction and magnitude is changing. And that's most certainly acceleration. In this case, we just happen to have a change in direction only. Any time we have a change in direction, we have a change in vector. We have non-zero acceleration. So to us, those of us who are infinitely smarter than the dopes out on the street, and by the way, they all have access to this video. So when we're done here, I'd appreciate if everybody filed past, got their face on camera when they left the room. Today, just help me a little bit. Any time there's a change in the velocity vector, we have accelerated motion. We need to be more precise. We need to be more complete. Because, well, we're figuring out how the world works. They're out there in philosophy classes where anything works. You could say God is the great spaghetti monster, and that'll fly. Does everybody know about the great spaghetti monster? Oh, yeah, Google the spaghetti monster. And you will find a religion that you can get your hands around. That one's there. That's a great spaghetti monster. Can you Google that on here first? Yeah? Show off. Alright, so let's take this a little bit farther. Make sure we know what we're working with here. Let's be a little bit more. Alright, so we just had average acceleration delta v to change in the velocity vector. And we had that over here in this case. What do we do when we first talked about average acceleration? What do we do next? About three weeks ago I introduced average acceleration and we almost immediately went to what next? Instantaneous acceleration. Just like we did with velocity, too. From average velocity to instantaneous velocity. And then we did that at the same thing with acceleration. How did we define instantaneous acceleration? Yeah, the exact reason I needed you to have taken calculus before you came here. Because we had to do this limit as delta t goes to zero of that average acceleration. Which is the derivative with respect to time. It's exactly the type of problem that Leibniz and Newton were trying to answer. If you'd drother, don't forget we can use this dot notation if you wish. Notice that every single little piece on either side of the equal sign is a vector. Otherwise they wouldn't be equal. I can't have a vector equal to something that isn't a vector. That's not a rule I made up. Just to be anal. Yeah, I know I am anal. I'm an engineering professor. It goes with the territory. But that's just the deal. At least it can't be equal if any part of these aren't vectors if one part is. These can't be equal. So watch that kind of stuff as we go through here. We really got some stuff coming up where a lot of you are going to forget that you're working with vectors at every single little point. And you're going to screw up. I'm trying to help you. Trying to avoid screwing up. Now, the next step on this, on what is the instantaneous acceleration, the acceleration at any little point we could have taken along here, even if that 60 kilometers per hour didn't hold at every single little spot. All we know is it went in and 60 kph came out at 60 kph. Maybe it did something else in between. We just don't even know. Doesn't even matter. Because we can take an instantaneous acceleration at any possible point that was coming up and we would be able to figure out an instantaneous acceleration if we knew some more about the motion that was going on there, which we will in a second when we get to it in greater detail. This is worked out in the book, what this deal is. It turns out to be V squared over R where V is the instantaneous velocity. If we did go around here, you could look down at your speedometer needle at any instant. You know what the instantaneous velocity was. That's what your speedometer reads. Instantaneous velocity. Look down for an instant. There it is. So that's the instantaneous velocity. Doesn't say V average. V divided by R, which is the radius of the corner, was a nice 90 degrees circular radius turn. Any trouble with this equal sign? Anybody about to howl and protest? That sounds like Alan is. That's not a vector there. There's no nothing here to make it a vector. What should I do? Is there a vector sign over the V? I can't. There's no such thing as the square of a vector. It just doesn't exist. It's not that it's beyond your mathematical capabilities. It isn't. It's not there. It doesn't exist in this world. There's no such thing as the square of a vector. So I don't want a vector sign over the V. So I've got to do something else because it turns out the direction of this is always changing. And some of you have already figured out where we're going with this. In fact it almost came up a second ago. Was it Phil? I think you said it so you know where we're going with this. So what I'll do is say it's towards the center of the circle. That's the best I can do because every place, every different place along here has a different direction towards the center because it's going around this corner. So I can't say I or J because it doesn't always work. But I can't say towards the center because you can find the center of the circle, the center of the corner and so we've got a directional component to it. The derivation of that is worked out in the book. So I'm not going to repeat it here. It's not terribly complicated. It's just not that productive to go through it for the sake of showing it. So let me introduce then our next major topic. We're going to look at circular motion. Things going in circles or portions of circles. Like we just looked at was a one quarter portion of a circle. We're going to look at circular motion. Careful now when you write this down. Stop, Mike. See I left some space in here. We're going to look at circular motion. A lot of what we've been doing last, especially the last couple weeks, we looked at projectile motion and just general motion from curvilinear motion from rectilinear motion. And it was always helpful if at any time we could identify what things were constant, what things weren't changing as we were talking about them. What then, in that mindset, defines a circle. Can I say something's constant, something's not changing or something's changing always in the same way or something like that that I can say that helps us define a circle. Of course I can. What is it? Somebody. What's a circle? No, constant in a circle. The radius. The radius is constant. That's a circle. R equals constant. At the first time I've written down something like this. It's helpful. It helps us. What it does for us is it helps us get rid of some of the problem because the less stuff is changing the more stuff that isn't. And then it's easier, a little bit easier to grasp. So, circle and motion. R equals constant. Yeah, well, of course. That makes perfect sense. Alright, you've got to draw a circle on your paper at least as good as that. That's not too bad. That's a good one. Joey, not too bad. It's easy to do it when you're tiny. Oh. You're enough. You could have used your... Oh, man. Not bad. Samantha, not too bad. Not willing to give a try. Yours is invisible. It's a snow circle. So, there's our circle. An object with constant radius. So, we're going to need to know where the center is, of course, because then that's where our radius is. So, we're going to look at things going around in circles or in portions thereof. We don't have to go through the full circle to talk about what we're talking about here. But, I want to put a little bit more in it here to be a little bit more specific about what we're talking about. Let me add uniform. Uniform circular motion. Point of focus for our two-dimensional motion. What's uniform mean? No, no. I already got radius constant here. That's circular. So, it's got to be something else. Huh? Oh, so it's r equals constant is constant? Oh, no, no, no. Come on. If this was... it would be a boring one, but this is a raised track. Yeah. That doesn't get up and walk away. No. She said velocity is constant. I said no. Alan said speed is constant. Why aren't those the same? Velocity has direction. Velocity has direction. So, this is speed is constant. Is that what I just wrote down? Don't agree again. You got to get married. You disagree so consistently. Yes? No? Did I just write down the speed is constant? Well, not in those words, but in words we understand did I write down the speed is constant? Yes, I did. Because there's no vector sign here. So, I'm not concerned with direction. I'm only concerned with magnitude and units if we had the actual numbers in there. So, the speed is constant. Now what do I write down? Uh, equals two greater than, less than, not equal to. Uh, what else could I write in there? No, I'm not going to write down the chain. I don't know. I got a little tiny space. What am I going to write down in there? Not equal to. There's one vote. Not equal to. There's two. Alan, did you check with Samantha first? So that you two wouldn't disagree. You're on your honeymoon after all. What did you say? What did I put in there? Joey, M-A-W. I said Google that. Don, velocity isn't constant. The speed is, the velocity's not. Take a particular instant. We'll take, we'll take this one we've got right here just because the radius sign happens to be pointed there. So, there's something, whatever it is, going in uniform circular motion. At that instant its velocity would look something like this. At that instant, does that mean it's departing from the circle and the radius is no longer constant? No, because an instant later, by the way that's how big an instant is because you didn't know, an instant later it's now going like that. So it's still on the circle. So r is still a constant. Is the velocity constant? No, no. The velocity is not constant. Is the speed constant? Yeah, that's, I haven't erased that. It's up there in ink. What do I have to draw to make the speed constant? Or can I, can I even draw the speed being constant? Here's a little sketch. There's the velocity vector in perspective. We'll go ahead and call that v1, then v2. Some amount of time went by but we'll let that get real, real small. So we're talking about instantaneous. What do I have to draw so the speed is constant? Or can I even do that? I can do it. In fact, I don't think I did too bad a job showing the speed is constant. No? No, Andrew, what? Yeah, the magnitude of these vectors are pretty darn close to the speed, the same length. Not bad for a sketch cartoon at the board. Remember, the magnitude represents speed. The magnitude and direction represents velocity. So the speed being constant, same magnitude is the velocity constant. Now the direction is changing. Even though the magnitude isn't what the direction is and that's a change in the velocity vector, that's accelerated motion. Anything that happens at velocity vector is acceleration. I don't care if it gets longer. I don't care if it points in different direction. I don't care if it does both. Any change whatsoever in the velocity vector is accelerated motion. In fact, an even tighter definition of circular motion is any motion where the velocity is always perpendicular to the radius which it would be for circular motion, any time the velocity is always perpendicular to the radius, you have circular motion. Because we go anywhere we get some other velocity vector, but same magnitude. And even though it's way over here it's still perpendicular to the radius. So uniform circular motion is this business going on all the time with the magnitude of those velocity vectors never changes. Not anywhere not know how. Magnitude is always the same. We don't consider a change in units necessarily to be a change in that, because that's just a simple conversion. We're going to leave these in the same units. That'd be all kinds of trouble to draw. We're drawing different units from the same drawing. You've got to be a knucklehead and do that. We're not. So we know that we've already worked out instantaneous acceleration for uniform circular motion is v squared over r towards the center. That's a vector this isn't until I give it directions. So I'll say towards the center. Actually there is. And if and when you take a course we call dynamics we'll have a unit vector that always points towards the center. See how that's different than our i, j, k unit vectors? They never change direction. But a unit vector that always points to the center, the unit vector itself is always changing direction. So that's a different calculation to make and we'll do that if you take the course we call dynamics that I'm actually doing this spring right now. So we can draw in the acceleration vector. It's a different vector so I'll draw it in a different color, but it's always towards the center. I'll label it a... See Samantha's got color in there. She's working the magic. Now Joey's going for some colors. What do you got in there Joe? Shooters. He looked over and saw the neat pink highlight and thought I need something to eat. Here's the next question we're talking about uniform circular motion. We've laid out a lot of detail already. We're ready to take another vote. Do I need you to put your head down on your desk for this vote so there's no cheating? No? We can trust you to vote honestly. Is this... Well let me ask a preliminary question first. What's the magnitude here of the acceleration vector? No it's not equal to the radius. The radius is a distance the acceleration is an acceleration it doesn't have the same magnitude. I mean they can travel the same length but they're completely different vectors. V squared over r. Magnitude is V squared over r towards center. V squared over r towards center. V squared over r. Got it there. Alright here's what I want you to vote on. Is this a constant acceleration problem? Because if so that sheet of constant acceleration equations comes into play. That tattoo you got is still good. Is this a constant acceleration problem? Think about it for ten seconds. Get ready to vote no cheating. No peeking. Mark is sitting in front but we'll look at that me. I'm ready to pull the guys out. So that nobody's peeking you can do this right in front here. Say one finger and no you're not allowed to choose whichever one finger you wish. One finger if you think it is constant acceleration two fingers if you think it's not constant acceleration three fingers. If you can't tell four fingers if you don't want to play along. No fingers if you're getting tired of voting I guess. One if it's constant acceleration. One if it's constant acceleration two if it's not. Three if you can't tell. Four if you have stain. Which doesn't make any sense because you just voted which is not a stain. So anybody who votes four gets an explanation to do. Go ahead and vote one or two or three. I'm watching which finger if it's one finger. Is that a vote or is that a piece? I'm cool. What do you do? I got a two. I got a two. Who else did two? Len, what did you do? You did two. I had three twos. No I had four twos. And all the rest ones. Patrick, you said two. What was the two vote? Two was two. You voted two. Two was this is not a constant acceleration problem. Is that right Phil? Is that what two was? This is not a constant acceleration problem. Is that what two was, right Joey? So you're changing to one. You're changing to one. Let him. Maybe he's gone to the dark side. So I only have two people sticking with two. Joey and Phil, since there's only two of you and you think it's not a constant acceleration problem, explain it to us. Why would you vote that? Do you have his cell phone number or why don't you give him a call? Because I want to hear the answer. I just asked you. That's your reason. I figured that was because direction was changing but now acceleration is more involved. So here is a constant. I want to know what to put in between there. You two guys are saying a not equal sign. Everybody else said equals. Even the two that jump ship. You say not equals. And why? Because the direction is changing. He says no it's not. So you're jumping ship. Is this just you, Joey? Joey says... Joey says that. Is that right, Joe? It's always been my philosophy as a teacher to never purposely put anything on the board that's wrong. And I just put that up there, didn't I? So I agree with you, Joe. Sorry, pal. I just want to know who is closed. Joey, what changes if it's not constant? That's because something's changing. What's changing? Direction. It's pointing in that way here. It's pointing in a different way here. It's pointing in a different way here. And in fact over here, over here it's pointing in the opposite direction. That sounds to me like a change in a vector. So this is not a constant acceleration problem. That's right. The center's always in a different direction from where it is at any instant. It's not a technicality. Not a technicality or reality. Let's say you're on this big circular track. You're on Earth. You like northeast, south and west. So there's north and east. And you're at this position here. And your mom says, quick! Look at the center! You would look at about 45 degrees south of west, wouldn't you? Then a little bit later, you're over here and your mom says, quick! Look at the center! Are you going to again look 45 degrees south of west? The center's not there. It's in a different direction. It's now over here. What is that? South of east. You get to here. What's your mom say? No, she's not going to say quick look at the center. She's giving up on you. Joey's the only child she loves anymore because he's the only one on the way to look. Once you're looking out the wrong window. Only Joey knows the center's in a different place now. What a lot of you are thinking is that the car we're in but what did I tell you about the direction in which the objects we're talking about are facing? Huh? No? I said it doesn't matter. This is all particle we've been talking about for a couple weeks here. And it never matters in which direction the object is facing. So you can't say but it's always out the same window. Because we don't know what the car is doing. That's not our concern here. We're way out in outer space. Looking down always see for the car is a little point. A little point is going around the circle. Well points don't have windows. So the center is always off at a different direction. This vector, if we describe this acceleration vector in I and J notation, that I J notation would constantly change. I J notation for this vector would be very different than the I J notation for this vector. Because it's a different vector. It's changed. The magnitude's always the same. The direction's always changing. But that describes it perfectly. From where it is. That's what I always drew. You never questioned it. We're talking about this object accelerating. So what's the acceleration vector from that object? With velocity we have speed to represent velocity without direction. Very easy. Take off the vector something. Yeah it doesn't have a new name like velocity and speed does. Velocity and speed are everyday concepts. Everybody's got at least a feeling for velocity and speed. But remember when we first started talking about acceleration, I told you it's a lot less intuitive than the velocity is. So we don't have some other word that describes acceleration magnitude without direction. Maybe it helped if we did but we don't. So that's it. And some books would do that kind of thing. You're okay to do that if you want. I think it's easy enough just to take off the vector something. Our book does bold face for vectors. If you haven't noticed I hope you have. If you haven't you haven't opened your book yet and that explains a lot of things. I can't do bold face at the board so I'll do a little vector sign. So this is a non-constant acceleration motion. But it is constant speed and it is constant radius. Here we go. A little problem. I'm going to go with it. Top Gun Pilot. Top Gun is the Navy fighter jet pilots. Some who go on to become overweight brothers in law. Who we hate. Do you know who my brother in law is? Do you? No? Nobody knows who my brother in law is? An astronaut. Actually an ex-astronaut. He quit about a year ago. Been up on the space channel twice. Been up on the space station for six months. Six happy months of my life I'll tell you. Because do you know what it's like for me to go see my in-laws and go to a dinner party with them and they introduce me. Oh here's our son in law. And everybody goes oh the astronaut. And my father in law says no. He's a professor. And they all go and walk away. He wants a light for me. But I'm not bitter. Is it really bad? In my mind, yeah. He's fat. They kicked him off the space station and they opened the door. All right. So here's what the Top Gun pilot can do. And his fighter jet going let's say 400. No let's do the let's see. Where's the number? We'll say about 640 meters per second. We'll change that in a little bit. Goes into a circular loop. Not too bad. It's not too bad. We'll leave it. Does one 160 degree loop on a circle we'll say is about 5000 meters. We'll change that in a little bit too. But that for now is good enough. Goes in. Does one full loop. Takes 48 seconds to do that full loop. All the way around. So this is V1. Comes out. 48 seconds later. Having just done one full loop. And I don't care if it was a vertical loop or was a horizontal loop run. None of that matters. It's just a circle. And he went into 640. Came out of 640 showing just like that. 48 seconds later. Calculate for me. Ring tone. Is that what that was? I'll run the tape back. I'll find out what that was. Mike, you're redder than anybody else. Find out for me the average acceleration. When you're done just look up. Raise your hand. Somehow indicate to me you're finished working. Put down your pen. Do something. Let me know you're all done. Find out for me the average acceleration. Nobody done yet? Done. Phil, you'd be done if Len would let you copy. Len, be a sport. It could be your all day. Yeah, I know. Should all be done. You should not only be done. You should have been done quite some time ago. In fact, this full calculation you should have been able to do in your head instantly. Am I right, Alan? Did you do it almost instantly? Len? Len just threw it in reverse. Mike, what? What are you thinking? Andrew, what do you think? Anybody got it? The average acceleration, he said zero. What did I say my philosophy as a professor was? I'll never do what? I'll never, huh? Right on the board if it's wrong. Yeah, I'll never right on the board if it's wrong. I'll wait a second. Mike, explain to me how the average acceleration is zero. Is that the definition of average acceleration? Was there a change in the velocity vector? It was 640 meters per second in that direction. It's now 640 meters per second in that direction. Was there a change in the velocity vector? The average acceleration was zero. The average acceleration was zero. I don't know how to make zero vector. There you go. I don't know what that means. Instantaneous acceleration. Who's got the instantaneous acceleration? Which instant? What's that matter? They're at the same place. They're different instances in time, but they're at the same place. So what do you say Mike and or Andrew, now that Andrew's on board? We'll just call it 81.9 meters per second squared. Where? Towards the center. Why not zero, Alan? There's its magnitude. There's its direction. That's not zero. That's its instantaneous velocity. And that's not infinite. That's the only way that could be zero. That's what you're going back with. This will be your ticket of admission on Monday. Same thing. Slightly different. I'll give you a little bit more detail. Let's say now that the velocity is per second. That turns out to be still about the 640 meters per second magnitude. 24 seconds later, that was half of the 48 around the whole turn, until this time. And now with minus 400, minus, oh sorry, minus 400 I, minus 500 J meters per second. I'm not going to tell you the radius this time. I want you to find for me the instantaneous, the magnitude of the instantaneous acceleration anytime. We'll assume it goes with that constant speed, uniform circular motion. So I want the magnitude of the acceleration. We already know it's always towards the center. So we don't need to do anything with direction. Then we just have to sketch that whenever we need it. You don't have that? Don't even bother coming on Monday. That'll be your ticket of admission. And I'll be punching tickets. Alright, I'll post the damn solutions. If there's any questions after that, let me know. Wow, it's a rough day.