 So we paused here, right? Let's pick up where we paused. Draw, eraser. Example seven. So we left off example six and our whole concept, our whole idea is that energy is conserved. We said that work was force times distance. It was also the area under a force versus distance graph. It was also the change in connect plus the change in potential. This says what average force is needed to stop a bullet of mass 20 grams of and speed 250 meters per second as it penetrates wood to a distance of 12 centimeters. So now I think they're talking about a force. I think we're going to be using work and the work energy theorem. I think the first thing we're going to ask ourselves here is how much work did the block of wood have to do on the bullet? Let's see. The amount of work is equal to, now I see force times distance, but I don't know the force. So I don't think I can use force times distance and I also see a speed. I think what I'm going to do is use work equals change in potential plus change in kinetic. But also work was what times what? I'm also going to say over here. This is going to be the force times the distance. See, that gives me force in my equation, which is what they want me to find. But it also gives me a way to calculate the amount of work. By the way, do they mention a change in height anywhere here? Pretty sure I can do that. In fact, I think I'm going to have this. Force times distance equals the change in kinetic energy and what's change in anything, Jordan? So if I hear you elucidating correctly, I think you're saying this. Force times distance equals a half mv final squared minus a half mv initial squared. Hey, what's my final velocity in this question? Read it carefully. How do I know? Oh, stop. So this is kind of nice. In fact, the equation I'm going to get, it looks like, is fd equals negative a half mv initial squared. Now, the negative is simply telling me, Jesse, that my force is in the opposite direction of the motion. That this bullet is losing energy, which it is. It's coming to a stop. So I'll sort of ignore the negative. I'll kind of look at it and it reassures me that I've got this set up, right? How would I get the force? Because that's what this question wanted me to find. How do I get the f by itself? Divide. In fact, I think I'm going to get this. f equals negative mv i squared all over 2d. Instead of a 1 half, I just put a 2 in the denominator. Meaghan, that's the same thing, but that looks cleaner to me, yes? Anytime we can get rid of yucky fractions in the front, we'll take it. Do I know the mass? Check. Do I know the initial velocity? Check. Do I know d? Yeah. Well, I should be able to figure out the average force that this would have to exert. That's going to be negative 0.02, that's 20 grams, right? 250 squared all divided by 2 times 0.12 because they gave it to me in centimeters. I'm running out of room. I'm going to put my answer up here. Negative 0.02 times 250 squared divided by bracket 2 times 0.12. You get negative 5200 newtons? Negative 5.21. I'm just going to drop the negative 5.21 times 10 to the third newtons. Is that a lot of force? Can someone divide that by 9.8? How many kilograms is that the same as having dropped on you? About 531 kilograms. It's about three people, four people, died bombing on you. This is why bulletproof vests are so difficult to make. So in a bulletproof vest, does the bullet have 12 centimeters to come to a stop? Uh-oh, hang on, got a late human. Then we'll go back to my wonderful point. As I was saying, so bulletproof vests because this is the mathematics of stopping a bullet. In a bulletproof vest, do you have 12 centimeters to stop a bullet? In fact, how many centimeters do you think you need to stop the bullet in? Probably got about one, maybe two to play with. You know what? I'll say maybe that much if the vest is thick. Maybe you have about that much to play with. You know what? I'm just curious what happens if instead of a 12 centimeters, I put a 2 centimeters, divide that by 9.8, you can still be killed wearing a bulletproof vest just from the impact. Now ideally, if it's hitting you in your rib cage, your ribs are also taking some of that energy because the energy's got to go somewhere. But the other thing that they do with a bulletproof vest is the way they weave the fibers together spreads the energy sideways because energy is a scalar, not a vector, so the direction doesn't matter as long as you can bleed it off somehow. It spreads it sideways through the rest of the weave. In fact, that takes a big chunk of the energy and reduces this number to something much more similar to that number. It makes it survivable. You probably have a broken rib or two, but that is far happier than the alternative. But in the movies, when you see the people wearing a bulletproof vest and they get hit and they kind of shrug it off, no, no, you're down for the count still, but you're alive, which is the idea. Example eight, a 3,000 Newton car at rest. Ooh, I think that's one of those key words that I'm probably going to underline. At the top of an incline, three meters high and 30 meters long is released and rolls down the hill. What is its speed at the bottom of the incline if the average friction force is 200 Newtons? Okay, so now we're taking friction into account. I think I better draw a picture. So there's my incline. And I know that this distance here, how parallel am I? Come on. How about like that? This distance here is 30 meters and this distance here is 3 meters. Okay, let's try writing a work energy equation here. So I'm going to put an equal sign and this equal sign is before and after. So before at the top of the hill, what types of energy does this little vehicle possess? Potential. So I'm going to say potential energy initial. Has it gotten the kinetic energy? No. Is there any other type? I don't think so. In fact, what I'm going to do is I'm going to move this equal sign right to here. Now after what types of energy does this car possess? Well, there is going to be a final amount of potential energy, but I think because that's ground level, that's going to vanish. I'll write that though just so I know when I'm studying later on. I thought of it. Is it going to have a speed at the bottom of the hill? So it's going to have some kinetic energy. Are they talking about friction at the bottom of the hill as well? Okay, so we can think of this two ways. We can either say whatever is extra is the work done by friction or heat or Tyler, we could have put this on this side and said we're going to start out with this much energy, but we're going to lose energy to heat and I would have put a minus over here. But minusing something on this side, Megan, is the same as plusing it over here. So let's see what we got. Oh, and I should put a little F right there, shouldn't I? We're going to have MGH initial equals a half MV final squared plus what is the work done by friction? Now it's heat, but work is what times what? Force times distance. Which force are we talking about for heat? Ah, friction times distance. Hey, does the mass cancel this time? Nope. What are we trying to find? Speed at the bottom? Well, let's get this by itself. Let's minus this over. So we would have MGH initial minus whatever you lose to heat. That equals a half MV final squared. How would I get the V by itself? Divide by what and what? Divide by a half and divide by M, okay. V final squared is going to be MGH initial minus the force of friction times D all divided by 0.5. There's my one half, M. Is that okay, Si? Do I know the mass? Oh, and by the way, this gives me the squared. What will I do with this answer when I'm done? Do I know the mass? Not directly. They didn't tell me the mass of the car. You know what they told me? What do we call the Newton measurement? They told me the weight of the car. How do I go from weight to mass? Take this number and do what? Divide by 9.8. Okay, so you know what? The mass is going to be 3000 divided by 9.8. Try that again, Mr. Dewey. Divided by 9.8. Come on. 306.1. I'm going to make a little note over here on the margin. M equals 306.1 kilograms, so now I'm going to start to plug numbers in. 306.1. Well, wait a minute. I see MG here. You know what MG is going to be? 3000. Why don't I just write the 3000? So 3000 times the height. What's the height? 3 minus force of friction. What did it say the average force of friction was? 200. Now, which way does friction act? I think it acts along the ramp. So which distance am I going to use here? I got to use the 30. Now, how could I ramp this up, no pun intended? How could I make this a bit more difficult? I've seen them give you this and the angle theta, and you've had to do the trig to find the hypotenuse of the triangle. Okay, I've seen that. Or I've seen them give you this and the angle, and you had to do the trig to find the height. Okay, fair enough. It adds a nice one little extra. Okay, a little bit of trig. But yeah, they told us this time. 30 divided by 0.5 times 306.1. And once again, I'm brilliantly running out of room. My space organization hasn't been great. 300, 3,000 times 3 minus 200 times 30. Oh, 3,000. Divided by 0.5 times 306.1. Close bracket. And I get 19.601. Someone want to double check my math, is that right? Yeah, that's 9,000 minus 6,000 is 3,000. Divided by 1.5. That's the same as multiplying by 2. Okay. Oh, but that's V squared. Now what? 4.43. So here's a nice combined heat friction work energy question. Is that okay the way we approach that? Justin, we're good. Again, the reason I like energy so much is not just because the law of conservation of energy dominates the universe. Because it's a scalar, it's a very powerful tool. The math is nowhere near. Did you see any signs or cosigns anywhere in my equations just now? It's a scalar. Who cares? Direction doesn't matter. It's not too many physics quantities that have that. A technical comment on ramps. The normal force, which is a force on every ramp, it does no work because it's 90 degrees to the direction of motion. And so if the ramp is frictionless, the energy of the system will be conserved. It'll just be potential at the top equals kinetic at the bottom. And if it's frictionless, you don't care about what's going on in between. The same argument also applies to tension in a pendulum chord. So if you have a frictionless rope in our magic physics world that Evan doesn't really exist. If you have a frictionless rope, you'll also have constant energy. Tarzan. Tarzan has mass 85 kilograms and he runs at 6 meters per second. He grabs onto a vine and he swings up until he stops. So what we're imagining here is we have a rope hanging from a vine. Tarzan comes running along as fast as he can, grabs it, swings up until he stops. And then he swings back down, but we're not going to worry about that. So write a work energy equation. Have they mentioned heat here? No? Then I think it's going to be the potential energy initial plus the kinetic energy initial equals the potential energy final plus the kinetic energy final. Put your pencils down for one second once you've written that down. See, here's what we're really saying. If the energy is conserved, we're saying the change in energy is zero. Don't write that down. What we're saying is that means that the change in potential plus the change in kinetic has to be zero. You didn't lose or gain any energy. Which really means the change in kinetic is equal to negative change in potential. Jordan, what's change in anything? Ke final minus Ke initial equals negative Pe final minus Pe initial. Jordan, can you see that this would become a negative right here? But I could plus it over next to the final and I would have Ke final and Pe final both positive on the same line. And can you see that this is a minus minus, which would make the initial what? A plus and I could plus this initial over here and I'd have both positives on this side. That's where that equation comes from. It's saying you're not losing any energy. I find it though quicker just to jump right to here, but there's the fancy derivation. Now that I've done that, hold on. So now pick up your pens. Are any of these zero? We're going to assume to make the math. The other reason we like energy is we can make a zero as a height, Jesse, wherever it's most convenient. Let's make Tarzan's head where he's grabbing the vine. Let's make that a height of zero. Let's say he's starting on the ground. Oh, and there's one more that's zero. The final kinetic energy because the very, very top of his swing. How fast is he going for a split second? Megan, think about this. We've got a swing. We've got an arc. If we tried to include direction, this would be very difficult. We need to know the angle between the rope and the vertical. In fact, we need to know the stuff that we did last unit with forces. We draw a free body die. Here, my equation looks like this. A half mv initial squared equals mgh final. Oh, did they need to tell me Tarzan's mass? As it turns out, nope. So why did they? Well, sometimes they'll give you extra information that you don't need. That happens quite a bit. Because in the real world, that's really what happens. What are we trying to find? By the way, there's my work energy equation. I guess now I'm already answering part b. How high can he swing? So let's get the h by itself. h final is going to be vi squared all over 2g. Jordan, instead of a 1 half on the front, I just put a 2 in the denominator because that's the same as multiplying by 1 half. It looks cleaner. The g would also end up in the denominator. What a nice clean equation. If we had tried solving this using forces, Justin, I guarantee nowhere near as nice. But this is, I'm going to argue, really plug and chug. You could almost do this in your head. Let's see, vi 6 squared divided by 2 times 9.8. 6 squared is 36. 2 times 9.8 is 20, almost. This is roughly 36 divided by 20. That's roughly 1.8. It's going to be close to 1.8 if I do the arithmetic in my head. Let's see how close I am. 6 squared divided by bracket 2 times 9.8. Oh, there you go. I was accurate to two sig figs in my head. 1.84 meters. Example 10. If this ramp is frictionless, hey, this is a roller coaster. Write a work energy equation and find the speed of the 1,000 kilogram roller coaster at the bottom of the ramp. Okay. I like this question. I like this question. I like, you know what? It's an amusement park ride. I love this question. I love this question. I really love this question. And again, just one more time, Jesse, so you all get the point. The fact that the track curves, because energy is a scalar, don't worry about it. It says write a work energy equation, kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final plus heat. Oh, no friction. Are any of these zero when I do this and you don't talk any louder? That helps me not at all. I think I agree with that. This time, kinetic energy initial. Nope. I guess it has some speed coming over that top hill first. So I think we're going to get this. A half m v initial squared plus m gh initial equals a half m v final squared. What do you see? Mass is canceled. That's always nice. And I'm going to do one more thing because I hate those one halves. I'm going to multiply by two, multiply by two, multiply by two. That's not illegal, right? I've done the same thing to both sides of the equation, but what's nice about that, Brandon, is that's going to cancel and that's going to cancel. I get a much cleaner equation. It's just that middle two doesn't cancel, but I'd rather have one two than two halves. I get this. Vi squared plus two g hi equals vf squared. What are they asking me to find, Jesse, in part A? Oh, final speed? Now I would have no problem, see this equation? Could I tell you how fast it was traveling at the bottom and instead say, how high did the hill have to be? Another nice twist on this. In fact, probably that's what they would be doing in the design of a roller coaster. How fast do we want to go? Let's get it to 80 kilometers an hour. How high do we have to be able to hill that? Or I could tell you how high the hill was, how fast it was at the bottom, and I could say how fast was it traveling at the top of the hill? Okay. I think that's the only ways I could mix and match that. Oh, and we got the vf by itself already. How do they get rid of the square, Jesse? Okay, so if I hear you correctly, you're saying v final is going to be great big square root of what was v initial? What was it? I've scrolled down, sorry. Two. Plus two times 9.8 times, oh, the height I still got, 10. Am I going to use that 17.5, by the way? Why did they throw it in there? Extra info. What do you get? Don't all rush for your calculators at once or anything? 14.1. Yeah. Faster than when we started, so that makes sense. It probably says I didn't do a sloppy calculator mistake and not something totally unrealistic, like 80 meters per second, but I don't think roller coasters go that fast. Oh, you're really cool. It says if the speed at the bottom of the ramp is 12 meters per second, write a work energy equation and find the energy loss due to friction. So it's going to be almost identical. Kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final, but whatever's extra must have been the heat. I'll call it work done by friction since they use the word friction. Are any of these zero? How would I get the work done by friction by itself? What would I do with this number? Minus it. For some reason, I see a lot of kids want to divide it over, and I think it was because last unit we were almost always dividing over, but these equations are pluses, so it's easier. It's minusing it, right? So I'm going to say this then. The work done by friction is going to be a half m v initial squared plus m gh initial minus a half m v final squared. Do I know the mass? Oh, do the masses cancel this time, Jesse? Why not? You're right. I'll come. Is there a mass here? Is there a mass here? Is there a mass here? Is there a mass here? So why not? No mass in heat. Common mistake number three or four. So you're going to see a roller coaster question on your test, and I'm probably going to give you at least one that has friction somewhere along the way because, oh, I can be a little more accurate like the real world. A lot of kids cancel out the mass because they're so used to canceling it out. Now what we did say yesterday, Sally, was as long as I can get this really small, in other words, as long as my roller coaster is really well greased up and lubricated, like very, very low friction, our roller coaster is fairly low friction the way they make it now. Yeah, then as long as it's really small and for all intents and purposes, the masses cancel enough that we still don't have to step on a scale. Hey, let's crunch the numbers. The work done by friction is equal to .5, what did I say the mass was? A thousand, oh, that's a nice number. Two squared plus 1,000 times 9.8 times 10 minus .5 times 1,000 times, what did I say, 12 squared. What'd you get, Tyler? 2,800, anybody else? By the way, I think you've found by now most energies are in the thousands, right? That's why I did so many of those on the first day and Mosquito Pushable, I was trying to give you a feel for what's a reasonable answer. Is it 2,800, anybody else? Yep, 2,800. Keep that number on your calculator, by the way. 2,800, oh, so 2.8 times 10 to the what, fourth? Joules. Actually, that makes way more sense to me because I would think it's a long roller coaster and moves a lot of heat. Don't close your books yet, we're going to add a part C. Find the average force of friction. Now, I'm saying average force of friction because with a curvy track like that, I assume this roller coaster is slowing down and speeding up along the way and that would probably, Justin, that seems to me, change, the friction would get bigger and it would get smaller. I'm going to find the average force of friction. Now, how can I do that? What did we just find, Tyler, in part B? Specifically, can you say in terms of work, the work done by what? What does the W stand for? What does that stand for? In terms of work, the work done by what? Ah, very good. And work is what times what? Ah, if I want to do part C, I think what you're saying is the work done by friction is equal to the force of friction times the distance that friction acted for. Let's get the force of friction by itself. I'll pick on you, Tyler, because you're on a roll. How would I get the force of friction by itself? Oh, this is actually not going to be too bad. So if I hear you correctly, you're saying the average friction force is going to be the work divided by the distance. What was the work done by friction? 28,000, 2.8 times 10 to the fourth. Now, careful. Which distance will I use from my diagram? The track. That's why they gave it to us. Okay. So what was the average force of friction? And hopefully you kept that on your calculator. Divide by 17.5. Sorry. 1.6 times 10 to the third. Force, units. Newtons. Newtons, right? Not joules. Okay. Oh, by the way, this is a curvy track. If it was a straight track and you knew angle theta, you could find the normal force, because it would be mg perpendicular. I could actually ask you to calculate the coefficient of friction, because you would know friction. You would know the normal force. You can find mu. This friction is mu times the normal force. So mu would be friction divided by the normal force. There's all sorts of ways that I can take this, which I think is about a five-mark question and add for two extra marks a nice little, hey, let's do something with the information. Okay. Is that all right? Now, I already gave you the homework yesterday. I said try 1, 2, 3, 4, skip 5, 6, 7, and 8.