 Okay, so I want to go ahead and do a short example related to conservation of energy. So generally speaking, if we have energy, we have, or a simple example, we can say we have a mass, right? And say this mass is held up at some distance above the ground, and then at that position, we have potential energy, right? Now if we release this mass and we let it fall some distance, such that it's now down here and it's moving with a velocity, so it's moving at some velocity v, now we would say it has kinetic energy. Now if it's still not sitting on the ground, right, if it's moving and it's above the ground, it also has potential energy, right? So it has some potential energy, but it's a different potential energy than what we had when it was up high at rest. And the difference between these two is that this change in position as it changes gets lower closer to the earth, but increases in velocity, we have the work of gravity being done on this system. So gravity applies a force to our mass, moves that mass by a certain distance, that's work, right? That's our definition of work, f times d. So that's gravity doing work on our system. So let's look at an example. So suppose we have a car and that car is traveling on an inclined plane, so a hill, right? And I'm going to draw a rudimentary car just for simplicity and it moves from one position to the next, forgotten space for my wheels. And basically the idea for the setup for this problem is we have position one over here where the car is moving at 60 miles per hour. And now we're going to start coasting up this hill and we're going to let off the gas, right? So we're not putting any more energy into the system from the engine, right? We're just letting the car coast up this hill. And let's go ahead and define the angle of this at 10 degrees, so it's a 10 degree slope. And our V2 over here is unknown, so we don't know after we've traveled up this hill, let's say a distance of one mile, we want to know what's happening, right? So basically this is kind of how we can set the problem up for understanding from a conservation of energy standpoint. We have a known position one, a less known position two, and we can figure out the states between those two things, right? We can say there's one state at position one and the state at position two. So we can summarize all of that and say, well, what we know is at one, we have velocity equals 60 miles per hour, which isn't really a normal unit for when we're doing problems like this. So let's go ahead and convert that, 5,280 feet per mile. One hour is 3,600 seconds. And if we do that conversion, multiply those numbers out, we'd get 88 feet per second. Now that's more normal units that we might work in in American engineering units. So at this state, if we know that this object is moving, we can say, well, oops, it's got kinetic energy, right? And we know the equation for that is one half mv squared. I did forget to note that, let's say we give the car a weight of 4,000 pounds and we can use that then for our one half mv squared and plugging that in, we get 4,000 pounds. Of course, we need mass, so we need to convert from weight to mass. We can use the acceleration of gravity to do that, which is 32.2 in English American units and multiply this by its velocity of 88 feet per second squared. Now if we go ahead and calculate all of that out, we're going to get 4,881,000 foot pounds. And great, that's the energy that the car has when it enters our system at state one. Now at state two, we know considerably less about the system. We don't know its velocity, we don't know its kinetic energy, give this little subscripts just to signify that there's two different positions, and we don't really know what's going on, right? But we know that work is being done on the system and what kind of work is that? Well, it's what we just talked about. It's the work of gravity, right? This car is moving from one low position to a high position. There's gravity doing work on it and that work is slowing it down, right? So we have work from position one to position two, which we can go ahead and figure out. Now we might need a free body diagram to better understand this. So I have my car, probably should draw that at a better angle. It's oriented at that 10 degree angle. It's got a, man, I could draw straight lines, that would be awesome. Still not straight, that's fine. We have a force of the weight acting down. We have a normal force acting perpendicular to the road surface. And if we kind of figure out our angles here, we could draw a vertical or excuse me, a line normal to that, which would have that same 10 degree angle. So what that means is that the load that's acting against the direction of the car is going to be a portion of this weight, right? So we have work acting against the motion of the car at an angle, if I can find my mouse at an angle of 4,000 at an angle of sine 10 degrees, right? And it's acting over a distance D, which is the change in position from one to two. Now the reason we've used sine 10 here is because we're trying to get only the portion that's in the direction of the motion of the car. The car is moving this way, right up that ramp. So we want to project the weight onto that surface, which by this angle, so katoa opposite is opposite of 10 is this parallel direction. So we can say, great, the portion of the weight that's acting against the direction of motion is sine 10. And we put that negative sign in front of it because that portion of the work is acting down this way while the car is moving to the right. So the work acts to the left, car moves to the right. That's negative, right? It's opposing the motion, which is just a sign convention. So if I go ahead and write this in here then, we have, stop pushing my buttons on my pen, 4,000 sine 10 times 5,280 feet, which is our one mile change in distance. If I use my calculator and multiply this all out, I get negative 3667 450 foot-pounds. Now looking at that, we should see a problem, right? And what is that problem? Obviously, I don't have your feedback to see if you're answering my question, but I should see a problem in that I come in with 481,000 foot-pounds of kinetic energy and then the work on the car due to gravity from one to two is negative 3.67 million foot-pounds. And that's a problem because it's bigger, right? So if I add one kinetic energy at position one plus the work done, negative 3.7 million foot-pounds, it's going to be a negative number, right? And we can't have negative energy. So what does that mean? It means the car stops before it gets to that one mile that we've set for it. It never actually gets there. So that doesn't really do anything for us. So we can go ahead and say, well, what we've learned is that the kinetic energy actually goes to zero. So our kinetic energy, Ke2, is equal to zero. And it doesn't occur at that mile. It occurs somewhere earlier, but we can figure out where it happens. So we can go ahead and say, well, the kinetic energy at position two is going to be equal to the kinetic energy at position one plus the work from position one to position two. And we redefine position two just to mean wherever the car stops, right? So if we put in our numbers, we have that 481,000 plus the work from position one to position two. Now we need to redefine this, right? Because previously we substituted in D of one mile, but we've learned that that's not true, right? D never gets to one mile. So instead, we have our same equation minus 4,000 sine 10 times D, where D is our unknown because we can substitute in zero for kinetic energy two. So if I go ahead and solve this equation, somehow gotten out of drawing mode, if I go ahead and solve this equation for D, I get 692 and a half feet, which is equal to approximately 0.13 miles. So the car gets nowhere near my one mile distance, right? It stops early at just over a tenth of a mile. Now it's important to note that we've neglected friction in this. Friction is always negative energy. It always acts against what we're doing. So in reality, the car probably wouldn't make it that 0.13 miles either. It would stop early. But in general, this is one way that we can use conservation of energy to solve problems in terms of how we understand things from one set of criteria to a second state of criteria and use that conservation of energy balance to figure out what's happening.