 So lesson two, conservation of energy, the law of conservation of energy as it's often called, which means it's a pretty important one. In fact, this is one here that I use as a physics nerd almost more than any other thing. Whenever I'm talking with somebody and we're having a discussion about whether or not something can be done, almost always I fall back on the law of conservation of energy. I was marking tests yesterday and I was marking tests while watching TV and Superman Returns was on and there was a scene where Superman is hovering in midair and suddenly he accelerates and goes shooting to catch something and I said, you know, even the flying part, if that was possible, the acceleration from zero to a big speed is not because he's changing his kinetic energy and that energy got to come from somewhere and he didn't somehow get tired. So all the superhero stuff, if you're wondering, even if the science could somehow work whether it's possible, no, it violates the law of conservation of energy a couple years ago when the X-Men movies were coming out. Somebody asked me, what about Wolverine? You know what? The energy for his body to heal itself would have to come from somewhere. He'd be eating nonstop 24-7 if something was, law of conservation of energy. It says this, the total energy of a closed system is neither increased nor decreased in any process. Energy can be transformed from one form to another, but the total remains constant. As an equation, we say this, put your pencils down. I'm going to show you the original and then I'll give you the better version. The original says this, the total change in energy is zero. Don't write that one down. The secondary equation says that means that whatever potential energy you gain, you must lose that much kinetic energy. Okay, I guess you can write that one down, but I don't use that one very often, but I use it once in a while. In other words, if you're gaining potential energy, you're slowing down. When I throw something in the air, is it getting higher? Yes. Is it gaining potential energy? Yes. What happens to its speed? It slows down. The one that we usually say is this. The amount of kinetic energy at the beginning and the amount of potential energy at the beginning has to be the same as the amount of, let's be consistent, Mr. Dewey. Kinetic energy at the end and the amount of potential energy at the end. You can get this line by writing this as final minus initial, by writing this as final minus initial in brackets with a negative in front of it, and when you do a bit of rearranging, you get the second one. But here's what we're saying, however much energy you have at the beginning, that's how much energy you have to have at the end. Or if we include friction, kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final, and anything that's missing must have gone into heat, the work done by friction. Have a little note here. It says, Mr. Dewey has a good demo for this. I do have a couple of good demos for this. Let me pause the video recording. So a projectile is launched straight up into the air. If it reaches a final height of 93 meters, what was its initial velocity? And it says, use energy to solve this. Now we could solve this with kinematics, and d equals v i t, d equals 93, final velocity is zero. You know what? We can solve this with conservation of energy by going kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final. Jordan, are any of these zero now that you're paying attention? Let's see. How high are we starting out at? No, no, how high are we starting out at? How fast are we going at the top for a split second? For a split second you come to a stop at the top? As long as we're being launched straight up, does the question say straight up? So this equation simplifies to a half m v initial squared that equals m g h final. Is that okay? Sorry, what? I heard someone say something, I thought. Masses cancel. That's kind of nice, because they didn't tell me the mass. I'm a little worried for a second there. And what are they asking me to find? The initial velocity. It looks like that's going to be 2 g h square root of, yeah? That there looks an awful lot like v initial squared equals 2 a d, it's v final squared equals v initial squared plus 2 a d, that's also where that comes from. Let's crunch the numbers, what do we get? Square root of 2 times 9.8, why didn't you put negative 9.8 scalar? Energy is a scalar, thank you very much, we don't care about direction. The height of 93, what was its initial velocity? Sorry, 42.7 meters per second. Anyone else? Is that right? I'm going to say energies, that's probably an easier approach than using kinematics from unit one. Now that worked because it was going straight up. Example two, a 350 kilogram roller coaster car is traveling at 22 meters per second. Energy is what we use an awful lot for amusement parks, especially roller coasters. They're a classic example of conservation of energy, did I mention that I was in California on the weekend? Okay. 350 kilogram roller coaster car is traveling at 22 meters per second. If the car starts out at a height of 12 meters, how fast will the car be traveling when it gets to ground level? Because energy is a scalar, Justin, I don't care whether the roller coaster looks like this, or like this, or even like that. Because energy is a scalar, all it's concerned with is your before and your after. So roller coasters usually are curvy, that's why energy is such a nice tool to analyze them. To try and do it with forces, your angle is changing all the time, you'd be doing all sorts of cosines and sines along the way. We're going to say this, kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final. Sally, what did you say kinetic energy was equal to? No, the equation, kiddo. You just looked it up. This is going to be a half mv initial squared plus mgh initial equals one half mv final squared plus mgh final. Are any of those zero? Let's see. Is our initial speed zero? No. Is our initial height zero? No. Is our final speed zero? No, in fact that's what they're asking us to find, isn't it? Is our final height zero? Ah, this is nicer. Oh, and not only that. What? Justin, do you have to step on a scale before you go on a roller coaster? This is why. Otherwise, they would have to carefully weigh everybody and balance them all off to make certain that the roller coaster wouldn't come to a stop midway through the coaster and they'd have to get everybody off. So yeah, as it turns out, because there's an m in everything, the mass is canceled. By the way, does this question mention friction or anything like that or heat? If it did, I would have plus heat right here, but it didn't. You know what else I'm going to do? I don't like the fractions. I'm going to multiply everything by, why, Gordon? Well, what's two times a half cancels? What's two times a half? Now here the two doesn't cancel, but I'm going to suggest to you that this equation looks way nicer. Yeah? Oh, how about I get the v final by itself, because that's what it's asking me to find. Square root? The v final is going to be big square root of v initial squared plus two gh initial. It's going to be big square root of v initial, 22 squared plus two times 9.8 times 12. Let's see, 22 squared plus two times 9.8 times 12 square root, 26.8? Does that make sense? If I got an answer less than 22, I'd be nervous, because I think roller coasters speed up as they get lower. Do they not? But if I also got an answer of like 80 meters per second, I'd be going, I don't think so. So 26.8, that seems okay, meters per second, turn the page. An 18 kilogram object is dropped from a height of 40 meters, and it strikes the ground with a speed of 25 meters per second. How much heat energy was produced during the fall? Tyler, is this question mentioning heat? So I'm going to tweak my conservation of energy equation. This is going to be kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final plus whatever is missing must have gone into heat. Sally, what did you say the equation for kinetic energy was? So this is going to be a half MVI squared plus MGHI equals one half MVF squared plus MGHF plus heat. Are any of these zero? Let's see. Dropped, what does that tell me about my initial speed? If we drop an object, we must have been holding it still, right? So gone. Is my initial height zero? Nope. Is my final speed zero? Well, after impact it is, but I think this is talking about right at impact. Oh, in fact, it tells me my final speed. So no, it's not zero. Is my final height zero? Ah, take that. In fact, if I'm looking at this, Tyler, it seems to me that the heat is going to be MGH initial minus a half MV final squared. Connor, is that okay? Right? Minus the half MV squared over. Do the masses cancel this time? Tyler says no. He's correct. Why not, Tyler? Heat does not contain an M minute anywhere, does it? So Justin, I lied a little bit. Maybe when we're including friction, the mass does matter, although do you think they make roller coasters almost frictionless? They roll really smooth. So that's why even though the mass does make a tiny bit of difference, they build a safety margin into the ride and they make it as frictionless as possible. So I fibbed a little. This is going to be M18 G9.8 40 minus a half 1825 squared. Mr. Duc, what do you mean by heating up the air? Is there not air resistance in the real world? And isn't that actually air friction? Yeah, this would be the work done by the air and it would have gone primarily into heat. Some of it into sound because when you drop something, you can sometimes hear it whistling. You can hear the noise. So fair enough. But most of it into heat. What do we get? You get 1431, the nodding, so 1.43 times 10 to the third unit's heat joules. And if you don't think that heat can be generated during impact, you've forgotten the chrome spheres that I just smacked together. I can still smell actually the smell of burning paper in the air. Example four, how much work is needed to slow down a 1200 kilogram vehicle from 80 kilometers per hour to 50 kilometers per hour? Okay, Brendan, work is what times what? Are they mentioning a force or a distance anywhere in this question? No. And Meaghan forces also what under a force versus distance graph, it's the area. Have they given me a graph here? You know what other, our third definition of work, we're going to use the work energy theorem. We're going to say work equals the change in potential plus the change in kinetic. Although Meaghan looked carefully at this question, is it mentioning a change in height anywhere? I don't think there is any change in potential. It's going to be change in kinetic. Oh, and what's change in anything? Okay, so this is going to be a half MV final squared minus a half M. Jordan, what's change in anything? I can't believe I picked Meaghan for that one, which is going to be a half MV final squared minus a half MV initial squared. Are any of those zero? No. Although, what do you notice about our velocities? Kilometers per hour, how do I go from kilometers per hour to meters per second? Is that why I actually did a little example of being in class last day that we had to do that with the crash? Oh, okay. You know what, Ryan? I'm going to do all my calculator, because this is straight plug and chug now. I'm going to say that, oh, does the mass is canceled, by the way? There's masses on this side, but is there a mass in here? Nope. So the work is going to be a half times 1200, and, Ryan, I'm just going to go 50 divided by 3.6 squared, and that way I don't have to worry about rounding off or anything, minus a half times 1200, 80 divided by 3.6 squared. Now, some of you are going, hey, wait a minute, Mr. Duc, why did you put the 50 first, because look, that is the final velocity, isn't it? I think it is. In fact, I think we're going to get a negative answer, which makes sense, because we're moving forwards, but which way is the force acting backwards? It's in the opposite direction of motion, or we're losing energy. That's really what negative work means. The car is losing energy. Of course it is. It's slowing down. 0.5 times 1200 times bracket, 50 divided by 3.6 plus bracket, squared, minus 0.5 times 1200 times bracket, 80 divided by 3.6 plus bracket, squared, does that look right? Half mv squared minus a half mv squared. Of 1.8 times 10 to the 1, 2, 3, 4, 5. Which force does the work? Friction. Hey, where do you think this energy is transferred to? Do brakes heat up? Yeah. Now, this is a bit incorrect now. You may have heard the term with some of the hybrid vehicles, the term regenerative braking, what they're actually doing is instead of having that energy get transferred to heat, they're pulling that back into the batteries. When you brake, you start to recharge your batteries. Brilliant. Because that's energy. It's got to go somewhere. Why not use it? Oh, and that's a fair chunk of energy. That's half the energy that it took to piggyback somebody up a mountainside. Would it tire you out to piggyback somebody up a mountainside? Remember we did that question last day, and it goes around 350,000 joules? So that's a lot of energy. We can do something with that. Absolutely. How far are we getting here? I don't know if we'll get through the whole lesson. We'll see. Got too many toys to show you. A 12.5 kilogram object is pushed across a level floor with a coefficient of friction of 0.35 for a distance of 6.4 meters by an applied force of 58 Newtons. I think I'd better draw a free body diagram. Over here. What are the forces acting on this block, Gordon? Get the obvious ones. The mg down, and it says a nice level floor, so the normal force is going to be straight up, and I think it's the same size as mg. What other forces? Friction would be slowing it down, and then there is an applied force, and I'm pretty sure the applied force is winning because it says it's being pushed across the floor. It's either tied or it's winning. So how much work is done by the applied force? Work is force times distance. Do I know the applied force? Do I know the distance? The applied force did 58 times 6.4 371 joules of work. How much work is done by friction? Well, the work done by friction is going to be friction times distance. What is what times what? I don't know the normal force. This is going to be mu mg times the distance. It's going to be 0.35 times 12.5 times 9.8 times 6.4. How much work does friction do? 234? Technically I'm wrong in the opposite direction of motion, or we're losing energy to friction. What's the net overall amount of work done on this object? The net work is going to be the work done by the applied force combined with the work done by friction, and because friction is negative, really that's the same as just going applied minus friction, if you want to think of it that way. That's going to be 371 plus negative 274. This is what's left over to be transferred into the object. What do you get? I wrote a 74 here, Mr. Dewey, it should be a 71. 103? 96.3? 96.8? Oh, you kept the exact values? That's what's extra. Where's that going? That's going into kinetic energy. The object's speeding up. How much, that's how much energy there is to speed it up, a half MV squared. In fact, I could figure out what V was, assuming we started from zero. If I knew it started from rest, this is my change in kinetic. And Jordan, what's change in anything? I could go kinetic energy final minus kinetic energy initial, if the kinetic energy initial was zero, I could find in this question as a part D, how fast is the object traveling after 6.4 meters? I could figure that out. Oh, apparently I thought that was a great question, because when I look at part D, that I was going, swear I would have asked that, how would I have missed that? Assuming the object started from rest, what will the final speed of the object be, after 6.4 meters? So we said 96.8 equals the change in kinetic. 96.8 equals one half MV final squared, minus a half MV initial, but Justin, what is the initial? Zero. Is that okay? All right. It looks like the final is going to be 2 times 96.8 divided by the mass, all square rooted. What was the mass? I turned the page. 12.5? Okay. So this is going to be 2 times 96.8 divided by 12.5 square rooted, 3.93 meters per second. Megan, my hockey player, this is what you're doing when you're skating on the ice. You exert a force bigger than friction. You do work against friction on the ice. Whatever energy is left over after friction goes into changing your kinetic energy and causing you to speed up. Although it's a bit more complicated than that because you can't keep speeding up forever and ever and ever and ever and ever, you do plateau and top out. We'll talk about that a bit later. Then E says, calculate the final velocity of the object after 6.4 meters. There should be a meters there. I won't put it in. Meters. Using kinematics, okay. I would go v final equals, I don't know. What was v initial? Zero. Right? From rest. What was the distance that we traveled, Gordon? 6.4. You know, if I knew the acceleration, I could go vf squared equals vi squared plus 2ad. How could I figure out the acceleration? Winner minus loser. Right? Just like last unit. We would go, well, my free body diagram, who's winning? Who's losing? Winner minus loser equals, there's only one mass, so ma. What was the force applied? 58 minus, friction is what times what? Mu times the normal force. I don't know the normal force. Oh, it's going to be minus mu mg all divided by the mass. That equals the acceleration. It's going to be bracket 58 minus, what was mu, 0.35? Yeah? 0.35 times 12.5 times 9.8, closed bracket, divided by 12.5. I get an acceleration. Do you guys get 1.13? Yeah? And I would bring that over to here, 1.13. And now I can go vf squared equals vi squared plus 2ad, vi is zero. Vf is going to be the square root of 2ad. 2 times 1.13 times 6.4 square root. And I get a slight, did I type this in right? Sorry? Is my acceleration wrong? Oh, I know why, because I put a 12.8 in there, didn't I? And that should be a 12.5. OK, so the acceleration, 1.21, that's what I was asking. People weren't giving me feedback. 1.21 even, so let's try that again. 2 times 1.21 times 6.4, this one should be bang on. I'll bet you square root, yeah, 3.93. Look at d, look at e, look at d, look at e. Which one was easier? Energy is a much nicer one to do arithmetic with, I got to say. It is. How far are my, I think, let me see. We've got 20 minutes, let's go a few more. I think we're going to finish this one. A spring in a toy gun requires an average force of 1.2 newtons to compress it or pull it back a distance of 3 centimeters. This one's a bit longer than 3 centimeters, but you get the idea. If you're exerting a force over a distance, you must be doing work on it. You're adding energy to the system. How much stored energy is there? Let's see. The work is going to be force times distance, which is going to be 1.2 times 0.03. How many joules of energy is stored in here? 0.036 joules. That's stored elastic energy. Potential, B, if all of this is transferred to a 10-gram bullet, how fast will the bullet move as it leaves the gun if we ignore friction? I think what we're saying here is all of your potential is transformed into kinetic. Reason I can say that is I'm going to assume what's my final potential energy, 0, and what's my initial kinetic energy starting from rest, 0. So I'm saying all the stored energy goes into shooting the gun. That's not quite true. Can you hear it go out? There's sound energy and heat energy, but we're keeping it simple. So we're going to have this 0.036. Can you put on the appropriate block, Dylan? Thank you. Equals 1.5 MV squared. Let's get the V by itself. V is going to be 2 times 0.036 divided by the mass, which is 10 grams, which is 0.01 square root of 2 times 0.036 divided by 0.01 square root 2.7 meters per second. So traveling about from, put on the appropriate block, please. Traveling about from me to Brendan in one second, I think that's a little slow. I think I made these numbers up. Jesse, I should probably retweet these, because it seems to me goes a little faster than that, right? Little. Need a little more kinetic energy, I think, for that one. That's a weird color, right? OK. Here's your first start of homework you can start here. First bit of homework you can start here. Tarzan, number one. Number two, number three, number four, six, seven is good. Eight is good. This is not all due tonight, because I see you guys tomorrow. Let me press pause for a second here.