 No questions? In general? All right. Here's a problem for you and the vein in which we've been working. Yes. Imagine a rock climber. Most of you know that as a rock climber goes up the face, they'll take with them a rope that they use for protection in case they fall. So we're going to figure out a little bit about what could go into that. As a climber goes up, trailing the rope, they'll occasionally put in a little bit of protection, what's called protection. It's usually in the form of just a block of aluminum with a sling on it. They'll put the aluminum into a crack, jam it down tight, then attach a carabiner to that and then clip the rope through it and then keep on climbing. That way, as the rope trails behind them and they often climb with pink ropes, as they climb higher and higher, that decreases the amount of distance that they'll actually fall. Because they'll only technically fall the distance they are above the last protection they put in, plus then that much more. Because they'll fall down to the protection and then down however much rope they've got behind them. And then, typically, the climbing ropes have quite a bit of stretch built into them. Something on the order of even 10% stretch to act as a bit of a cushion. So what I want to figure is, given some of the possible parameters in this, let's figure out if the climber falls from 10 meters above his protection, what will be the force he feels when he finally comes to a stop. So he'll fall 10 meters down to the protection, 10 more meters before the rope then goes taut again. So he'll fall down to about there and then the rope will start to tighten up and then the rope will stretch a little bit and he'll fall a little bit farther. So we want to figure out how much force is he going to feel that the rope exerts to bring him to a stop right about there. Not knowing quite where that is. Alright, so that will be 10 meters down to the protection, 10 meters farther before the rope even gets tight yet. And by the way, the other end down here is supposedly held by his partner. Hopefully his partner won't panic, throw up his hands and go, oh my god, he's falling! As the rope then goes woo! And he'll fall some little bit of distance farther that we need to figure out. Figure out how much force then is exerted on. Got the picture? Anybody in here a rock climber? Yeah, because all the other rock climbers are dead. So that was a test. They're not really a rock climber. You will assume all rock climbers are ex rock climbers. They either quit or they get killed. So let's say that these 80 kilograms and that the rope has a spring constant of about 4.9 kilonewtons per meter. You want to find the force exerted on that climber by the rope to bring him to a stop. How much time does he have to actually run this calculation? See if you can get it done in time. See if you can do it as fast as he could from the time he peels off a rock. Any idea how to do that? Let's think up a plan. Let's think us through a little bit before we just jump in and start calculating things. There's stuff to calculate. Doesn't mean it's going to be anything useful or going to take you anywhere. So let's think it through a little bit. Well, for kinetics problems, like this is, because the problem involves force on something. You've got something accelerating and some measure or other is going to take some kind of force to do that. We only have two ways to solve those type of problems so far. We have one other coming, but we just haven't gotten through it yet. We essentially have two ways to solve kinetics problems. So you figure, I hope, it's got to be one of those. What's the first way we have to solve kinetics problems? That's the second way. What was the first way? Now, kinematics might help us actually execute the solution because they'll give us extra equations when we don't have enough equations for the number of unknowns. But the kinematics is a separate part. In fact, if you remember, we did kinematics before we even mentioned kinetics. Before I even mentioned force, we'd gone all the way through kinematics. So what's our first way to solve kinetics problems? And good thing we're going over this since there's a test on this next Tuesday, right? So you'll have to look at problems and think this very thing. I've got two ways to solve force problems. Which of the two ways will be the easiest? Will be the most efficient for me now. Lead to the most points. Work energy is one of them. What's the one that came for that? That's not the solution. Well, that's part of it. So I'll give you credit there for that, Joe. F equals MA. Now each of those methods, well, they're really very much the same. Several of the terms in the work energy equation came directly from that equals MA. So it's not like they're mutually exclusive of each other. But they are cast in such a way that they, in the form we have them, they solve different problems a lot easier than the other method might. What type of problems do these two solve particularly well? Pardon me? I think I heard that. I think you're right. Well, the work energy equation, if you remember, the work term has to do with how far an object's moved. The potential gravitational energy term had to do with how far an object's moved in a gravitational field. And then the spring energy part had to do with how far a spring is spread. So it's very much a position dependent problem solver for us. Very good for position dependent problems. Which this one is. This whole problem depends upon where the climber is at any one instant. And in fact, even has something that we're going to treat just like a spring. That's essentially what the climbing rope is. It's a big spring attached to them. It just happens to be a spring in a much more useful form than a spring would be. But it's a very stretchy rope. They don't go climbing with steel cables attached to them because those will jerk them to a stop too quickly in some of those falls and you don't have to fall very far for that force exerted to be enough to kill you. Hopefully, hopefully you don't fall. And if you do not forbid, you survive it. That's the hope. What type of problems are best solved by F equals MA problems? Or by the F equals MA equation that started this whole kinetics thing for us? Works best on, well, general problems that are just what's the acceleration? Here's some forces, what's the acceleration? Or here's some acceleration. What forces do we need? Works pretty well for that kind of thing. Acceleration doesn't appear directly in the work energy equation. It doesn't mean we couldn't get it from it, but it doesn't appear there directly. But it also works best on constant force problems, which then would mean if it's constant mass, it would also mean a constant acceleration problem. So that's where it works best. You can apply either one in many other types of situations, but that is where they work the best. So you figure with a highly positioned dependent problem, including a spring in it anyway, that the work energy method would do quite well for us. And we've been working on that for a couple of days now. So to find the force that takes to stop them for a fall, let's see where that's going to work for us in the work energy equation. The only place we have a force term in that equation is in the work term. That's the only place to form that force appears directly. All the other things have other stuff in them, but force isn't one of them. So is that how we're going to find the force in this problem? Well, let's ask ourself, what forces count in this work term? Remember I told you, not all of them do. Only some forces count in that term. Others we handle elsewhere. Non-conservative forces in this term. Like NPR radio, that's a non-conservative force, isn't it? Did anybody listen to NPR and would even get that? Come on, Don, you get that, don't you? Yeah. Russell and Bob are always complaining that NPR is too non-conservative, too liberal. See, so that was the deal. Yeah. Hey, thank you. Laugh politely. Non-conservative forces. All right, non-conservative forces. Is that what we're looking for here? Are there any non-conservative forces in this problem, even? If there aren't, and this goes to zero, that's sure not going to help us find that, is that force a non-conservative force? And we need to account for it in the work term. It's just about to hit the rock here. You're halfway through your fall, off the wall. What film? Was there air resistance? No, we'll neglect air resistance. He's climbing on the moon. No, no, on that gravity. He was climbing at very high altitude. And you had a question or a statement of it? No, I thought you did. Is that force we're looking for? What's exerting this force that stops it? The rope. And we're treating the rope as a spring. Is this force exerted by the rope on the climber causing him to stop his fall? Is that a non-conservative force? No. No, not for our purposes. That's being exerted by the rope, treating it as a spring. So that's going to be over here. Except there's no force term in that equation. In that part of the equation. So what goods are going to do us? What? Remember? The force exerted by a spring is, it's actually minus k depth. Minus being if you stretch the rope one way, it pulls back the other way in the opposite direction. So that's why there's a minus sign. And if you push it and squish it up, which you can't do with climbing rope, they only stretch one way. So it's not like a real spring that could go two directions. We've got k. If we could find del, then, yeah, we could find the force. Del, of course, is one of the terms in there. So we could solve for, maybe solve the work energy term for del. Put it in there and we know the force on the climb. Any non-conservative forces. Remember? It's nice to go by term by term and get rid of stuff. Does he just make the problem smaller? What? Does he hang the clamp? No. In fact, his wall is much more exciting than that. It's an overhang. So he'll fall free in the space. A lot more exciting. If you're going to fall, make it a good one. I always say. No? There's no limit how far the rope can stretch. No. Well, I mean, yeah, of course there is. It can't stretch forever. You think he's going to go down, just roll home into bed and then the rope will stop him in bed on his pillow. Go climbing with Alan. You want to fall. Alan will climb anything, get to the top and then just jump off. Get him going home. Yes, of course there's a limit to how much the rope will stretch. However, we're not going to let him hit anything else. That would be a non-conservative force. Once you hit the rock, you're not going to be able to get that force back when you turn it back the way it was. That's a different type of problem. That's an impact problem. We're going to get to those shortly. You said at the end of the second 10 meters, the rope is top? Yeah, let me redraw the picture here. Put in some protection. It's now 10 meters above that protection. He's got a rope going from the protection up to him. I won't draw the rope that comes down to his partner, the belayer, the person who's supposed to tighten the rope in case he falls so it doesn't just play out. He's going to fall down the 10 meters, down another 10 meters before the rope gets top. Then the rope gets top and we'll stretch and bring him to a stop. Except Alan's rope will just keep going. Well, there's a delt, right? I'm trying to call it that. Wouldn't this right here be delt? Because that's right when the rope gets top. Anything after that is rope stretch. So if we could find that, we just put it in there and we'll know the force exerted on the climber by the rope. So let's look and check if we can get rid of any of these terms. Any non-conservative forces. No? What about the rope itself? Exerting the force on him. Yeah, that's a spring force. So it's a force, indeed, but it's a conservative force. Presumably once he's able to grab the wall, climb up a little bit, take the stretch out of the rope. Presumably everything's back to normal in terms of the rope. He's pooped his pants, but climbers do that. That's part of climbing. Right? See, he's wearing white pants. He's a very great climber. Somebody had a hand up for him? No, poop is pooping in general. Finally he found something he wants to write down in his nose. Any non-conservative forces. Kinetic energy is not a force. Remember, forces are caused by things I can touch, feel, look at, draw a picture of, I can't draw a picture of, I can't touch kinetic energy. Plus kinetic energy is a term over here anyway. Any non-conservative forces. No, as long as we keep them from hitting the wall, there are no non-conservative forces. Remember what I told you? The situation we have when we have no external work being done. Remember what I called that on Monday? That situation, that particular situation? Conservation of energy. If this term is zero, we have conservation of energy. However much energy we start the problem with, we finish with, we have it all time, it just transfers from one term to the other. In fact, that's what a fall is. He's got lots of gravitational potential energy. He loses it all, and it all turns into kinetic energy, and he's moving with great speed. That's what falling is. So any change in kinetic energy from start to finish, we'll call this the start, and this the finish, because that's where we're trying to find out how much force was exerted on it. It's not what I asked of. We're trying to figure out what del is so we can figure out what that force is. Any change in kinetic energy. In fact, there's no kinetic energy at all, here or here. So knowledge does not change. It's zero at both places. It's pretty high in between, because he's moving at pretty great speed when he starts to tighten the rope, but until then, there's no change in kinetic energy. But of course there's a change in gravitational potential energy. He falls, and of course there's a change in spring energy, because the rope is taught just to the point where there's no slack in it, but in here it's being stretched. So there's actually a lot of potential energy being stored into that rope from this potential energy. So I want you to figure out then how much force is exerted on him by finding this del using the work energy equation. Just to help us out a little bit, especially with what comes, do this. Well, I guess we don't need to yet. So just go ahead and work on it, and then we'll set some other boundaries in a little bit. So now we have a situation where nothing's left of the work energy equation but that. So use that to find what del is, how much stretch is in the rope, how much farther beyond that bottom point there will he fall. Figure out how much force is exerted by the rope to do that. I think I've got to get in all the terms you need. You've got his weight, you've got the 10 meters, and you've got the spring constant, and the fact that the rope is pink. What a dumb climber. Of course that's redundant, isn't it? Anybody stuck need a little help to get going here? Len's looking through his back issues of his climbing magazines. Good hint right there. Don't get involved in any hobby where the back of the magazine has an obituary section. I'm your advisor, there's my advice right there. You got it already on? I think I got it. There's only one way it's going to work, so the units work. Del is going to be your unknown. You're going to have one equation, one unknown, and Del will be the unknown. You want to find Del? I guess we have two equations, two unknowns, because then we want to end up with the force in the end. But with work energy equation, you'll have one equation, one unknown, and you'll have Del. Remember my suggestion is work out each term one at a time so you get each piece of them, you can check the signs on them, you can check the units on them, and a much more manageable efficient way to solve it. Go on to the next term then. You've got to do it sometime anyway. Go on and do it and see if that helps you. Mike, you doing okay? Got it? In here I'm screaming. That's the best part. It's very cool. First you hear him screaming, then you hear him whizzing, whizzing by, and you hear him hit the bottom, and it falls down. Yeah, better stuck. Crumpy. Crumpier. I don't know which one that went wrong. Not both of them. Where am I looking? Nope. Where? Nope. Watch your units. I didn't say that, because I'm not sure you were done what you were writing down. Maybe you were. Or you can come and sit over here and work with Alan because of your units at all, and just circles numbers on his paper. No, you're doing okay. Careful though, it's minus and minus. Because the whole distance is 20 plus del, but down it would be minus 20 minus del. What's that? Do you have a question to ask? No? Do you have just better than the one half width to make a whole width? Is that the math? Climbing? Climbing. Climbing. Climbing. Climbing. Climbing. Climbing. I can't say the song. I'm not afraid of flying, but I'm afraid of riding. It's a little tired. Is one of the terms of the other giving you more trouble? Is UG giving you more trouble than UE? Which one? UG is the easier one? Yeah, usually it's for students. You don't need to give the other one. You don't need to give the other one? Of course. But that's not any of them. Only Del is in there. Del is in both terms. Yeah. Alright, a couple more minutes. We'll start working. What you got, Tom? How are you doing? I thought you were on your own. Well, you're in the right order of magnitude, yeah. That's a lot of different programming. Phil, how are you doing? You're quiet. Are you working with Lynn? Without actually disclosing anything in two days? Competition? I think if I were you, I'd just leave it as Del, find Del, and then plug Del into the little equation. No. No, why not? Here, let's do the first term together. See if we can just make sure that one goes okay, right? Md delta H, don't leave off the delta H. A lot of students do that. This is like a vector equation. If we have vector here and vector here, then they can possibly be equal. If we have delta here, we're going to have delta over there for them to be equal. Of course is the mass of the climber, the kilograms. G, what do you put in for G? What's so funny, Mike? You put in for G. For the millionth time, he's falling down. Gravity is down. So what do you put in for G? I'm going to pay the money. My students would fall for that every single time. Well, not all the students, but there'd always be one student at least. Still remaining. We want that minus sign. A couple of you did yesterday. A couple of you tried pretty hard to get that minus sign in there on G. It's not there. Minus hasn't even come up yet. Delta H. What's the distance he falls? Del minus 20 meters? 20 meters. He's going to go from here to here. How far is that? You held up the tape measure. That's 20 plus Del. Put the minus sign out in front. That'll do. Worked just as well as anything. If you'd rather, you could put the minus 20. Sorry, that's 20 plus Del. You could put the minus here to be minus minus 20 minus Del. But that's the same thing. The 20 plus Del is the distance. The minus shows us again that he fell in the downward direction through this gravitational field. Do the units work out? What should the units be? On every one of these terms. Jewels are Newton meters. Newton meters is usually good enough. Kilogram meters per second squared. That's a Newton meter. As long as Del is in meters, it's going to work out just fine. So whatever number we come up with in the end, for Del, we already know at a time it's got to be meters because that's how it's going to work out. That's what we've got here. So you can multiply this thing through. You get what? Minus 15, 696. Is that right? Minus 785 Del. And we already know that all is going to be Newton meters. Because we worked that out. Anybody can knock at those numbers? Del remembers the unknown we're looking for. You know, there it sits. We've got to figure out some way to solve it there. Delta Ue. How do we calculate that? A region is one-half minus Del one squared. What are we looking for here? Del one or Del two? We have a Del here, but it doesn't have any subscript on it. Is it one of these? Remember, when he gets down here to 0.2, we want that stretch in the rope. That's Del two. So what we're looking for is Del two here. We want the amount of stretch when he's at 0.2. What is the stretch in the rope when he's at 0.1? Isn't there 20 meters of rope? In fact, isn't there at least 40 meters of rope down to his partner? So what's Del one? It's zero. It's the amount of stretch in the rope. As he's climbing up to that point, remember, the rope is just hanging there behind him and doesn't really have any stretch in it. So Del one is zero. We've got one-half. I gave you K. That's in kilonewtons, which will make this equation kilonewton meters unless we make a change to it. 4.9 kilonewtons is how many newtons? 4,900. To stretch a climbing rope one meter, you need to apply 4,900 newtons to it. All a lot of apples out there. Del two squared. Will that give us units of newton meters? Yeah, of course it will. We have newtons per meter times meter squared. This will be newton meters as long as Del is in meters, and we already said that. So Del two has got to be in meters. That equals what? 20, 25, 50? All right. I do that number right, 24, 50. Well, now we put these two back together like that and see what we've got. We've got minus 15, 6, 96, minus 785 Del two plus newtons squared. That looks like a pretty simple quadratic equation. One equation, one unknown. Anybody here downloaded the quadratic equation solver? So if I were you, I'd look around the rest of the room and see who's willing to borrow it for five bucks. You can't go below what I'm going to charge you for the answer. Now, remember most quadratic equations have two roots. One of them is going to make sense here, and one of them is not. John and Bill, do you have the same roots? Since you're the only two, two points are enough to go get the quadratic equation solver. 2.7 negative 2.4. Bill, you concur? Or you forgot how to run your solver? Maybe give John five bucks. You pay him three and then charge Allen five. He's being very polite about it. What do you got, Bill? We'll go with John. Oh, let's see. I already know that's meters, because that's the only way everything else worked up above. So there's two roots there. Which one makes sense for us? John said he thinks 2.7 makes sense because it's positive, and that means stretch. Do you hear it? Sounds pretty bad. I was just rephrasing. What? It's interesting. What's the definition of del? Let's remind ourselves and see which one makes more sense. What's the definition of del? I don't think so. It has to do with that. It has to do with that. You've got to know what del means here, but this equation is not going to help you. Check your notes. If it's in there, I know I put it on the board yesterday. I remember doing it. We've taken an hour or two to run back through yesterday's. Take, or you can check your notes. It's in there somewhere. It's in there somewhere. Check your foot back a couple pages. Ow, stop. I see it. You see it? I do. Right there under your nose. I'm not. Let me help you there. I circled it. What's l? The British should say what the l? I'm just kidding. It never would. What's l? That's the length of the rope at any time in the problem. If we're looking for del 2, we're going to put in l2 there. His rope, let's see, is 10 meters. Then he falls down. The rope is still 10 meters long. At least the effective length we're worried about. And then, he'll stretch a little bit farther. So, it'd be 10 meters plus del. Essentially, the length of the rope. The length of the spring at any time. So we're looking for del 2. So it's the length of the spring at l2. That's the length of the spring. In this case, the rope at rest, out of the box. Which for our purposes is the 10 meters. Because that's when it comes tight. So, this is 10 meters. Trying to decide whether down here should be positive or negative. If it's positive, if we use the plus 2.7, that means l2 is longer than the rope at rest. If we use the negative 2.4, that means l2 is shorter than the rope at rest. Which is the case. It's longer. Got down to the 10 meters. There was still essentially slack in the rope. That's when the rope started to tighten up. And it stretched a little bit as it brought him to a stop. So we want the rope here. It's a plus 2.7 meters. Minus signs are very important in the work energy equation. The force was exerted on that climate. Now we know he's going to fall 10 meters, plus another 10 meters, and then another 2.7 meters as the rope brings him to a stop. Do you want it to be longer? But why would it be shorter? Now, remember it's the dell we have here. If this is 2.7, then we know the length of the rope when he finally comes to a stop must be 12.7. Which means he's he fell the 10 meters down to here, another 10 meters, then another 2.7, so the effective length of the rope from the last protection piece is 2.7 meters. Is Matthew okay with that one? Is Joe okay? Yeah? Whoever you're texting is okay with it? What? He's okay with it. I don't want him to worry. Hey, wait a second. If you're texting him what we're doing, he's not paying tuition. That's the set of services. You watch the videos, too. Videos come down tonight. People all across the world are stealing our videos. Actually, you can't get our U-Tunes. I don't think without logging in, I don't think. As a student, you've gotten onto them. Do you have to log in as a student to do it? You remember? Yeah, my iTunes. Just clicked on the iTunes loop, and iTunes opened. We are getting all this for free. Found my greatest test. What's the force? Well, we've got the 2.7, multiplied by the 4.9, came out to be who? Negative. Negative. What's negative mean? It's a force opposite of the direction he was moving. He was moving down. The force is going to be opposite to that. Come out to be 13 kilonewtons or 1.3 kilonewtons. 13 kilonewtons. 13,000 newtons. Just about the survivable limit of a human being. Well, less so if he actually did fix the rope around his neck. That was a dumb thing to do. He should have done it around his harness, around his waist. They have rather elaborate web harness things that they wear to spread out that force even more. But that's about the limit of what a human being can withstand, at least from some of the mountain climbing that I was able to pull up where I got some of these members. All right. What if I'll ask it anyway. What if he fell at only 5 meters past his protection? Do you see more force or less force? Less. I'm not going to You know, if you panic here on the wall you're going to freeze. And the Colorado Mountain Club has to come get you. Of course. In fact, it's pretty careless if a climber gets too far beyond their last protection before they put some more protection in. All right. So we're going to develop this whole idea then a little bit more. So we'll keep the same numbers. We're just going to look at it all in a slightly different way. Keep the same numbers. Look at it in a slightly different way. Let's see. We've got our work energy equation in general of work on this one. We know that the work term went to zero for the two points of interest that we were talking about. The delta K went to zero as well. We're only left working with these two terms. Those two terms are very much a function of a function of position with each other. As one goes up, the other goes down by the same amount. Because for the problem we were doing, these two terms were both zero, which means if one of those increases the other's got to decrease and vice versa. That was a situation of conservation of energy in fact at any time in this problem the total energy was constant where the total energy is the kinetic energy because he started with no kinetic energy and he finished with none but he certainly had some kinetic energy in between. Those three quantities all added up together are always a constant. And in fact for this problem we can calculate what that constant is. And then as any one of them goes down for instance his potential gravitational potential energy drops as soon as the problem starts he's immediately losing gravitational potential energy. This term starts to go down what's this term do? Starts to go up because he's picking up speed. What's this term do? Let's say we're only talking about 10 meters or so this one's going down he's losing 10 meters worth of gravitational potential energy this one's going up because he's picking up speed that's what a fall is what's this one doing? Nothing. The rope is still essentially at rest as he's falling. The rope doesn't come into it until it starts to tighten up down here. So as he reaches this point what's UG still doing as he reaches that point just as he's reaching this point as he's going past this point here in the middle where just where the rope starts to go tight what's happening to these three constants that always add up these three terms that always add up to a constant this one's still dropping because he's still dropping what's this one doing now just as he gets to this point right here where the rope starts to become tight that one starts to go up this one's still dropping because he's still dropping what's this one start to do this one now starts to go down so you can imagine his kinetic energy is probably a maximum right here because he lost a whole bunch of gravitational potential energy and he still turned into kinetic energy but then some of the energy starts to flow into the rope and he starts to slow down there because now finally the rope starting to exert a little bit of force on him that force grows as the rope stretches more and more and more until the rope stretches a maximum the force is a maximum he comes to a stop at the bottom of his fall and then what these are still all constant when added together what's the kinetic energy at the very bottom of the fall zero what's this well we're not sure if it's zero it depends on where we measured h from but it's certainly at its minimum it's not going to get any smaller because that's the bottom of his fall what's this maximum zero that's at its maximum we're at the maximum stretch then what happens he actually bounces they take a pretty good you've seen fungy jumpers do that they jump off the bridge go down they hit the bottom and then they bounce they take a pretty good bounce back up just in case the front of their pants are still clean so they go back up from more terror and more excitement taking notes like crazy today alright so what I want to do is we're going to look at these two terms together as the potential energy term because both of those are potential energy terms both of those involve conservative forces we're going to look at that potential energy term and it is a function of y where y is his position in the air measured from some spot and we can arbitrarily pick that like anytime we can pick an origin so we'll say right here is y equals zero I just arbitrarily chose that well I didn't arbitrarily chose it I know where the problem is going and that's a little bit easier place to pick it but we could pick it anywhere that would be the same alright so let me clear some board space here we're going to look at his potential energy terms as a function of y with y equals zero right here it involves two terms in terms of y with that as y equals zero what is the potential energy term potential energy at any height y what is this term I'll give you the mg it's not delta there's no delta here so I don't want a delta here what's christmas well remember the deal with the potential energy term in general it's mgh where h is measured from where wherever you want to set the origin I did that for you so this term should be mg and not y one because we're going to look at him through the entire fall mg y because h is height at any particular moment we're now using as y that will be our variable of height if you want to do h as a variable no sweat a lot of times we see h as a constant for some reason I don't know why but why we always see is it can be a variable what about when he passes this point it's still mg y because his y would just be negative his gravitational potential energy would be negative because we arbitrarily set a zero point here what we're most concerned with is the changes and we're still going to get that alright let's see what's his his elastic potential energy at any one moment well no del 2 is a single value at this point and I want this as a variable function of height so we can see how these things all change and interchange among each other would it be 1 half k y squared that would certainly make sense down here wouldn't it because that's actually the number we're using when it gets to the 2.7 meters that would be the maximum stretch we have the maximum potential energy so that would be okay but if we use 1 half k y our equation would say he has elastic potential energy up here which he doesn't how much elastic potential energy does he have in this region he has none he has no elastic potential energy because remember the rope has not come taught yet it's still slack it's just trailing behind him as he falls the slack is lessening until all the slack is taken up and he starts to stretch the rope after that we need that term to always be zero but I want this as a function of y so I don't want 1 half k del in here del squared because then I'd have two variables I want one single variable because then I can graph it so how could I do how can I handle this term so that it's a function of y below here that's fine this is this is 1 half k y squared work fine below here above that it wouldn't work no that's not going to quite work either you're looking unfortunate, well not unfortunately I don't blame you one little bit you're looking for some elegant single solution and there isn't one we need a piecewise function we need this for y greater than or equal to zero because the rope doesn't even come into it yet and then below that we need well the n g y still applies he's still following he's just gone below our arbitrary chosen reference point then we add on the 1 half k y squared there for y less than or equal to zero does that make sense is that what the deal is between 1 and y equals 0 only n g y is coming into play the only thing that's going on there is he's losing gravitational potential energy they both can't be 0 sure they can this just says these two functions have to be continuous at y equals 0 they're both 0 in the functions match what we don't want is whatever this function looks like to take a huge jump at y equals 0 because that's not going to make any sense all of a sudden a whole bunch of energy came from somewhere instantaneously or a whole bunch disappeared so the fact they're both equal to 0 just means the functions are continuous the slopes might not be but the functions are continuous familiar with that term for pre-calculus or something continuous functions alright so we can plot as a function of height above our arbitrarily chosen y equals 0 we can plot his potential energy y equals let's see what's the biggest point at 20 meters how much energy does he have not including what power bars he just ate and how much data he drank what I'm talking about here how much gravitational potential plus elastic potential energy does he have at the 20 meters at the 29 it's mg y where y is 20 meters figure it out how much 15,690 you're going to buy that sounds pretty fair 15.7 kilojoules what did you say 15.7 kilojoules right kilojoules as you said 15,700 joules yep this graph is for the potential energy for gravity and for elastic we're graphing this function so it's going to be a constant boy that's a very good question is then just going to be a constant did I say energy was conserved is it going to be a constant what did I say was going to be conserved because there's no outside work being done the w term is zero I said that's a situation of conservation of energy would be conserved energy would be a constant so isn't this just a constant then oh it's not why not what did I say was constant I didn't say this was constant I said this plus kinetic energy was constant these two are going to change together but their total is going to change because some of that is going to kinetic energy as he speeds up and then it's going to come back from the kinetic energy back into these two terms as he slows down and comes to a stop finally then there is no kinetic energy it's all come back into here and then he's going to take his bounce and he's going to bounce there forever because nobody can get to him so we're going to start here this is indeed the constant total energy level we'll never go above that because we don't have anywhere for energy to come to come into the problem as he falls as y decreases as he falls towards this arbitrarily chosen origin what happens to our energy curve our potential energy curve what happens to what we're trying to graph here what happens to this decreases he's falling what's the curve look like so I can sketch it in for this part here I need to graph that because that's that part there that's a y greater than zero part what's that equation look like that's a straight line g is a constant this mx plus b what's b the intercept there is a one at y equals zero this term becomes zero so we know it's a straight line from here to here isn't it straight line positive slope or negative slope it's positive slope only we're not going out that way we're coming back but that's still a positive slope so there's this potential energy curve for the trip down to y equals zero just arbitrarily chosen y equals zero point if we pick some other point that's just going to shift this line somewhere else but the curve shape isn't going to change at all is it comfortable with that? it's got positive slope mg see that's a positive slope even though what we're really doing is coming this way down the graph because he's falling then what does the graph do well now we he falls past our origin spot continues down into the negative y region which is this equation see equation sense since we're out of time and you need something to do for the weekend otherwise you're going to end up in jail somehow you have to look at your bedroom speeding or something most likely I saw you with that thing out damn Joey you saw that? I ducked into my office Joey didn't have anywhere to go your job is to finish the curve you know he's going to go a little bit farther from position and then he's going to bounce from there so he's going to go a little bit farther in fact we already know how much farther what was it? 2.7 meters we know he's going to go another 2.7 meters you find out what that curve does in between for Monday what? maybe I was joking I never try to mislead students right Alan? not the cake for now