 Okay. Ready? And then to bring up, to consider, to discuss. I spent a little bit of time in the other couple, last couple days looking at the, a couple different ways to look at coordinate systems. So you can apply those as neat, when appropriate for certain types of problems. Sometimes it's just experience. But it's also true that you don't necessarily have to use one and others won't work. The coordinate systems aren't going to change the physics. That's physics. Whatever physics is in the coordinate systems are just different ways to look at the problem. So we'll see how that goes. What we're going to look at today is what's called relative motion. How objects move in relation to each other. We're going to look at two different flavors of it. One is an unconstrained relative motion. We'll start with that. And the other is constrained. The unconstrained is the one you're most familiar with. You've all, you've all seen the type of thing where you'll be in a car looking at someone going a little bit faster than you. And as you sit there and don't necessarily perceive that you're moving, you see that they're moving and it only looks like you're moving a few miles an hour when actually they're moving just a few more miles an hour faster than you are. But you're only concerned with the perception of what you have to see. Plus you've probably all been sometimes sitting in the parking lot and the car next to you starts to move and you panic because you think your car is moving. That's a good example of relative, unconstrained relative motion. So we'll take a look at it in fairly simple terms. So imagine we have two cars. My car, of course, in the lead. Your car, yeah, that's about the car you drive, I think. We need some origin, some place from where we measure things. As always has been the case before, it's arbitrary where it is. My recommendation to you as we do this is simply two things. One is fix it in place. We could do all these problems by attaching the origin to one of the moving objects or some other completely arbitrary moving object. Or it doesn't even have to be to an object, we could just have a moving origin. We could do all the same problems. All this physics would still be intact, but it greatly complicates the problem as you could imagine. My other suggestion, also it's not required but very helpful, is don't put it between any of the objects, if possible. We'll see in a second how that can complicate things. So just for this little bit of problem here, I have decided to put the origin here and we'll call your car A. And so it's at place X, A, and my car, actually my car should be A because A could stand for awesome. I got an awesome car. We'll call it B, X, B. So everything we're going to do, we'll start from that simple fact that now we want to look at the relationship between the two. Where is B relative to A or vice versa? Who's the notation like that? B relative to A in that order. And so for this example, no matter where car A is or what it's doing or how fast it's moving or how it's accelerating, what direction it's going or facing or anything, it sees B a certain distance away in that direction. Just how much we can figure out from the very values we've got there. This is a little bit different than they lay it out in the book. What the book writes, it's the same thing. What the book writes is that XB is XB relative to A plus XA. Very same thing. I just find it a little harder to remember because we have to get these subscripts in the right order because it's not the same thing to say that these would be equal if we just simply reverse the subscripts. They're equal and opposite to each other. We have to get the subscripts right. I like to write it this way because then I always get them right. B A, B A, the two subscripts are always in the same order and you won't mess it up that way. This way I just find it easier to forget which one goes where. You might be a little sharper than me. It's not unlikely. So that's what I need to do to get that straight for myself. But it's exactly the same thing in the book, just slightly different form. All right. Of course that's not all we want to look at. It's certainly possible that these two are moving with some velocity. I will show them moving in the same direction, but it doesn't matter if one of them was moving to the left instead of the right then the value that we find in there would simply have a negative as it's moving closer and closer to the origin. So none of that is going to be of great effect. To figure out what the deal with that is, we simply take the time derivative of this equation which will give us then x dot. Relative velocity between the two is simply the difference of the two velocities. If you prefer B instead of x dot, it was what? Let's see what this can mean. Your car is A and since you're all teenage punks for the most part, you're driving, let's say, 80 miles an hour down the north lane. I'm in B obeying the speed limit, maybe a little bit less to get higher gas mileage. So I'm going 60. You're going 80. I'm going 60. So just to put in these numbers, it's going to look as if I am moving relative to you at minus 20 miles per hour. As you sit in your car, you're going to see me coming closer to you at 20 miles per hour. It's a little difficult to perceive that perfectly because any time you're in a car, you know the trees are going by and you're feeling a little bit of bump in anyway. But if you can put it out of your mind, that kind of start look only at the car in front of you going 20 miles an hour less than you are, it appears that that car is coming back. That's the kind of thing we're trying to get out of this. That type of relative velocity. Then of course the next step we can take is the relative acceleration, simply the time derivative equation right above. Surprise, I wouldn't think that all of this got in the same way with the symbol. When we get to the acceleration in some detail, there's a little bit extra that comes into it but we're not too worried about it yet. Seeing fairly straightforward, if you like it. I didn't draw the whole car, that's just the front end. So I don't get all worried that something just doesn't make sense there. All right. What we're concerned with is the windshield which is at an angle of 50 degrees from the vertical, tilted back 50 degrees, just as it passes underneath a bridge or some kind of overhead that's dripping water. I want you to determine the velocity, the relative velocity of our windshield. So I'll call that D and C. A few other details. Let's see the car is going 100 kilometers per hour and the overhead is 6 meters of the drop as it hits the windshield. Of course it'll be falling some but it'll also appear to be coming towards the car. You'll notice really the car that's moving, at least our usual frame of reference. It'll look to the driver as if it's falling and coming towards the windshield. So I want you to find what? Pardon me? That doesn't look like you. I don't have great chalk, is that what you're plugging in? We have everything you need to find the velocity of the drop relative to the car. Supposed to find, write down the definition of it, I just gave you. Got the same velocity as the car, of course. Two together will give us some relative velocity between the two that will make a particular angle with the windshield. Stupid boxy cars, that sign and that Nissan cube thing, windshield straight up and down. We've got these two vectors. If you do that, then it's just this vector subtraction. Figure out the rest and what it might have to do with the windshield. Don't forget if you put down some angle, let me know what it is. It's lazy with the vectors and the vector notation. It makes traps for you. Hopefully everybody's got the velocity vector for the car, simply 100 kphi, or what is that in 27.7 meters per second. That's the easy one, velocity of the car. What's the velocity of the drop? It's the car. Well, let's take that to be simple free fall from Physics 1. How fast is something going when it falls from rest a distance of 6 meters? Delta Y for the drop. What do you want? Final stretch. Yeah, velocity of the drop. Now that'll be just the magnitude of it. There's no such thing as the square of a vector. This is not a vector equation. Don't do it in kilometers per hour. Well, you can if you want, but you've got to use the same units on the two vectors. Good bless you. What's the velocity vector then on the drop at the moment of collision? It's fine. We hadn't established a downward direction, but that's usual, so we'll go ahead with that. It's arbitrary of which way we call positive and negative, so I think it makes sense. 10.85. Nope. Use per second, J. Oh, 2.8 meters per second. Check. Punch B minus BC. And BD we've got now. As 28 meters per second in that direction minus BC. That's minus. We need to do that to subtract vectors a little bit better relative to C. That's what it would look like to use the driver that the drop is, what the drop is doing. This is what I got. And we agree with that. We're assuming it must fall as it is anyway, from nothing more than that straight definition of our relative velocity. Any questions? There's a couple of homework problems, a couple of sample problems that you might suspect, a plane going one direction and another plane going another direction. You just need to figure out the relative velocities and the salarations using this. It's usually fairly easy to set those up from the problems and then you just do a straight subtraction of the vectors to get the relative result. Any questions? Oh, we have an exam in here in a week. Same thing as a strength exam, open book, open notes. Okay. Not much more we can do with this right now in this unconstrained motion. It's just exactly that. So we'll go into constrained motion because usually there's a bit more to do there and a little bit more confusion at the initial start of things. That's an unconstrained motion. That, remember, is the two objects, or more objects it could be, are able to move independent of each other. It's not completely so on the highway because they are supposed to stay on the highway themselves, but the cars can be, the objects can be going in any direction. What one does does not affect what the other does. In constrained motion, the objects themselves are directly linked, whether by actual mechanical connection, like a rope or a chain between the two or something else like electrostatic or magnetic interaction between the two. However one moves, it affects how the other is going to move and respond to changes in motion by either one. Simplest way to set this up is with a simple pulley system. So we'll start with the simplest of all pulley systems. A single pulley, single rope with two weights on it. We'll set up everything from here and then we'll take it to more complicated systems. What we want to do with these types of things is, given the motion of one, or one part of the system, figure out what the other part of the system is going to do. It's pretty easy with this one because obviously if A goes up at a certain speed, B is going to go down at a certain speed. But we'll set up what we need to do here with this easy system and then make the next step to more complicated systems using the very same idea. And in this one we need to establish an origin from where we measure everything. And again I make the very same recommendations that I did with that before. Make sure the origin is fixed. You could attach it to one of the objects but things get a lot more complicated. Also I recommend it's outside of the two objects. We could put it between the two objects or then motion of one in one direction might have a minus sign while the other one is moving in the other direction it would also have a minus sign. That just gets a little confusing, a little hard to understand. So if we pick our origins in particular places then everything is going to be an awful lot simpler. Alright, so easiest thing to do I guess is we can either set it to the ceiling or set it to right here. It's not going to make any great difference. But what we're trying to find out is the motion of one, motion of the other by looking at the time, rate of change of those positions. And if we want to find the acceleration of one once we know the acceleration of the other then we want the second time, rate of change of either of those positions. Alright, so let's add one more thing. Just to get a start and we'll say that that pulley has a length off. Because here's the deal with these ropes and pulley problems. It's all based on determining the length of the rope. You'll see why that's so very useful in a second. I just want to determine the length of the rope. How much rope did you have to go to, to Earl at Ace Hardware, make a purchase, come back and set the system up. Alright, we've got one piece of rope that's x a long bit over the pulley so that, let's see, circumference is 2 pi r so that would be just pi r. That's how much rope just loops over the top of the pulley. Then I have enough rope to get down to xb. Yeah, look right. Simple as that. Here's why that's useful. Remember what we're trying to find is if we know the velocity of one, what's the velocity of the other going to be? Here's what we're going to do. We're going to take then the rate at which the length of that rope is changing. Hell dot, if you will. Anybody have an idea of what that will be? At which the length of the rope is changing. Huh? Doesn't change. Remember, I buy only the best ropes from Earl. I buy stretchless, massless ropes. You guys don't know that but I've never bought anything else. You know this is going to be zero. Let's take the time derivative of the right-hand side of the equation. Well, the time derivative of xa is xa dot or va, if you prefer. The speed at which a is moving. Time derivative of pi r. You have to do what's it called? You have to change the product rule on that. And it's zero, of course. Those are both constants. And the other piece of that. From then, we get dot a equals minus x dot v for this problem. Any other problem we're going to have a different answer for this? Well, that's exactly what we expected. If a is moving up at a certain speed, b is going to move down at a certain speed. The minus sign means that, well, they're moving in opposite directions. But if we actually had numbers in here, it could mean that xb was decreasing. So if I have a positive value in here, xa is increasing. Giving a negative value over here, xb would be decreasing. Again, it's just what we knew anyway to be moving in opposite directions. And if we're more worried about their acceleration, time derivative of that equation. So simple as that. Calculate the length of the rope. Take the time derivative of the length of the rope itself, and you'll get right out of the velocity of the objects and the problem. So let's do such a problem. Maybe this will happen this afternoon. We don't know. Let's see. Here's your car. Sad-looking little thing that it says in what you drive. And you're stuck. So we got to get tomato. You know that is, don't you? Tomato? Tomato, is it? You don't know? You watch the movie Cars with the tow truck in it? Going to come pull you out because you're stuck with that silly little car ears and the ball tires and you're in the snowman? I'm going to tow you out. I'm going to do it with a pulley system looks something like this. We'll attach a pulley to the front of your car. Attach a pulley, yeah, a guardrail there in the middle just because it's available. And we'll run the cable back two pulleys to the front of your car. So we got a wind like that. All right, there's the setup. Got to set an origin, a place from where we measure things. Any suggestions? Good suggestion though? Well, you've been listening. It's to a fixed object which is a great idea. But it's also between the two objects because my tow truck is going to pull you out. We want to know depending on how fast this truck moves, how fast is the car going to move in response. Don't want it to be too fast. Well, sure, why don't we put the origin right here on the tree that's there. How'd you miss it? It's a Christmas tree. It's beautiful. So we'll put the origin right there and measure everything from that. Make it simple. We'll call that XA. See how we do with that. Got to figure out the length of the tow cable. We're still in particle motion. So we're not going to worry about this little bit of distance between the car and the pulley. All that stuff is constants anyway. And when we take the time derivative, all those constants drop out. So we don't have to worry about any of the details. We don't have to worry about how big the pulleys are or how big the links are or just where those things are attached. All of those things are constants and we don't have to worry about them. Matter where it is? Well, let's see. We've got this piece of the tow cable there which we could call XB minus XA. I'll put parentheses around it just so we remember just where that came from. That was the top piece there. Then we have a piece here. Does the length of that piece depend upon where the guard rail is? Yeah, of course it does. If the guard rail is up here, that'd be very short. Guard rail is way over here. It'd be real long. So we better pay attention to just where the guard rail is. We'll call that XC. So how long is this middle piece of tow cable? XC minus X. So we'll add that on. Again, just putting parentheses around it to aid me in understanding what it is we're doing here. That's the second piece. So it's the second set of parentheses. And how long is this piece? Same thing. So we'll just put a two there in front. Plus, there's a little bit around there and a little bit connected there and a little bit around there. That's just constants. We don't need to worry about it. It's just stuff that's in there. We're going to take the time derivative and that's going to all disappear. All the stuff that doesn't change disappears. And the only thing left is the stuff that does change, which is the amount of rope between each of the objects of concern. So you take L dot XB minus three XA plus two XC plus a bunch of constants for rope around employees. Is that right? That's an algebra there, and the derivative of this we get XB dot the velocity of the tow truck minus three XA dot plus greater change of this, which would be X dot C, the velocity of the gargoyle. We hope that is zero and the constants of course are zero. In fact, that whole thing then, since the length of the rope does not change, it's a whole thing that needs to be zero and we know that the car will have one-third the velocity of the tow truck. The tow truck takes off at nine miles per hour. We know the car and we know in the same direction because there's no minus sign in here to say that anything other than both of these are going to get longer together or they're going to get shorter together since there's no minus sign between them. Jake? I mean, that doesn't have to do with like, there's not three pulleys, but I was thinking of it like if there's three pulleys and how like the mechanical advantage has come down by like three, but that's different. No, that's exactly what's going on here. If we did a free body diagram on this, you cut through three cables, you then have this great looking auto of yours with three times the force on it. That's the mechanical advantage that remember comes at a price, the price being that it'll move one-third the speed. That's not much of a disadvantage when you've got a car, you're trying to get out of the ditch. What about the acceleration? Who's got that done? Everybody, I hope, just falls straight out of this, just take the time derivative of a linear equation. And they're all done. Now we know this is the acceleration, this one-third acceleration of the car is one-third of the tow truck. Question, just before I give you one for your own, now we'll see what you guys didn't even know who tow meter the tow truck was. Why did I know? So I went and watched the movie Cars. I have kids. You are kids. You should have been going, your parents should have been going, your parents won't take you to movies. You go home from here and what, they put you back in the closet, or under the stairs like Harry Potter. Oh, do you know that reference? Alright, here's a problem for you. Hanging from the ceiling, one pulley on a length of length D, lined up with it is another pulley hanging from it on a length of length C is our weight, the cable runs like that is our other weight, which of course could be you, maybe it's you pulling on that line. All that A likes you to find the same thing. We found for the velocity of one when the other one is known. And it doesn't matter which. If we know the velocity of B, we know A. If we know A, we know B. Just find the velocity of the two. Set it up however you want. It doesn't matter. We should all get the same answer. A pulley system like that's not going to do one thing for one person. It's going to be completely different for somebody else. What, the distance D and C? No, from A to the top pulley. A to the top pulley, and that down to here? I mean, yeah, I do know. No, they're just somewhere. C is not from the ceiling. C is just this strap here. It's like you're taking a course in a circle drawing. So the velocity of either one and what's the velocity of each other one? Acceleration two. Acceleration is directly related to the mechanical advantage because the force is directly related to acceleration. Well, let's see if I can't get you started. What's one of the first things you got to do? Get an origin. Because then we can, once we establish that, then we can say things we need to, like if one's moving in a particular direction, we know that then should be positive or negative. We can relate to what we find for the other one. So what do you suggest? The ceiling. Sure, that'll do. Just as long as it's fixed and not moving, that's as good as any. Certainly the speeds of these things isn't going to depend upon where we happen to lay our measuring tape. So we've got then distance down to A and a distance down to B. It's not that we're really concerned with what those are. What we're concerned with is how those are changing. That will then tell us the velocity, not only the speed but the direction in which the things are moving at any one time is in terms of those. Because you come over, that's you and you pull down on the ropes because you need to raise this. I thought that was a second pulley. The second pulley is a dead end. He didn't know how to pull that. He thought it was stuck right there. If you pull on it, it would just stay there. No, the rope. It looks like it has the ceiling. I thought that was a stationary. Alright, here's a little bit. Here's the rope. Here's your hand. You're doing some work. You're going to pull on that for us. You volunteered. I thought the two pulleys were attached to something so they could move. How hard to pull on that rope? It's not that we're having it. That's the symbol for is attached to something. Is attached to something that does move. That's ED. Let's see. This length of rope is xA minus D. xA minus D. A little bit the wraps over, we'll just dump that in a little bucket called constant. Now we have this length of rope, which would be xB minus C minus D. Alright? So the second length of rope is xB minus C minus D. And then we have this length of rope, which is xB minus C. A little bit that wraps over there and little pieces that don't really matter. Because they're all constant, they won't change anything. It's 2xB 2D minus 2C, which are constants. So I'll just dump them in the constant bucket. So we'll set that to zero and we're all done. That's what you got? Frank, that's what you got? You got it or we're getting there? Who's calling? The minus sign means A gets greater, as it would if you were going to pull down. xB is going to get, that's as we pull it down, that's going to cause B to go up. If we had the origin in between those two, the minus sign would have been a lot more confusing then. Because then the minus sign wouldn't tell us, simply tells us here, they're moving in opposite directions. If we had the origin between the two, it would have just been a lot more confusing. Because then the minus sign could mean they're moving in opposite directions, but towards each other, all kinds of stuff gets confusing. So we just don't want to mess with that, simply put the origin on the outside. Redo the problem real quick, origin right here, since that's a fixed distance. Who's done? You're done? What turns it to make? Not all it does is it takes D out of this equation, but that's a constant anyway, so it didn't make any difference. So you can put it wherever you want, just fix it, and put it away outside of the two objects, which in this case were the two pieces of the road. All right. Do you want to get started on one so that we won't finish in time? Or knock off just a few minutes early? Keep going? Do you just ask me to like regular reference? Oh, sir, I would stay forever to listen to you, since your reference wasn't attached to a $50 bill which was customary. All right. Any questions about that? We're more involved one then on Monday.