This learning sequence is about electrostatic potential. But to start, we're going to think a little bit about mechanics. So let's do a problem from mechanics, where we have a ramp like this that eventually goes flat, and we have a mass on the ramp. So we have mass, m, on this frictionless ramp, it sits at height, h. So if we say it's sitting about right here, and if we compare that to the ground, that's about h, like that. The question is, how fast is it going when it gets to the bottom? What is V final? So we know if we let it go, it's frictionless, we know it's going to be pulled down, and it's going to end with some speed. So there's more than one way to solve this problem. The first way you would probably solve it is with force. You look at this, and you'd probably, maybe draw a free body diagram. Here, I'll just draw it like this. We'll say, well, the mass is going to feel a force pulling it down, mg. You got to break that into components because really it's not this force that's pulling it down the ramp, you want to break that into a component normal to the ramp, and a component along the ramp. So here, you can see there's a right triangle, if we draw this one here, then what we really care about is this component down the ramp, and that component, the sum of the forces for this thing is mg sine Theta, where this is Theta, the angle of the ramp. So if you go through and do the detailed trig on a nicer drawing, you would see this component is sine Theta. It looks like cosine Theta in this right triangle, but that's because this is not Theta, this is 90 minus Theta. This is Theta down here, the way I've written it. The way I've written it, the sum of the forces equals mg sine Theta, and therefore, it also equals, as always, equals ma, Newton's Second Law. So you will look at this and you'd cancel the masses, and you would say the acceleration of this thing is g sine Theta. So that's what it's doing. Then you'd say final velocity, you probably have all these formulas you memorized in the fall semester. So one that I remember, V_final_squared equals V_initial_squared, plus 2ad. This is a useful formula when you have an acceleration over a certain distance. So 2 times the acceleration times the distance that it's going to go. So let's use that to find V final. Then you would get V_final_squared equals this, this is 0, we're starting from rest, so that's just 0, we'll ignore it, equals 2 times the acceleration is g sine Theta, times the distance. So how far is it going to go? Well, it's going to go actually all the way down the hypotenuse of this right triangle. I'm not drawing these real nice, because this is fall semester stuff. So we've got on this hypotenuse, and if this is h, then that is h over sine Theta. So d equals h over sine Theta. So the sine Theta is cancel. What you end up with is that the final velocity, after you take the square root, is the square root of 2gh. So that's how you'd solve that problem with force. But there's another way to solve this problem, that's with energy. You may have noticed, sometimes we solve them with force, and sometimes we solve them with energy. In this case, you simply say, this thing starts with some gravitational potential energy, that it has here relative to what it has when it gets down here, and you may recall that's equal to mgh, how high it is. You know that when it slides all the way down to here, that gravitational potential energy is going to be converted to kinetic energy. So mgh becomes one-half mv, and that's V_final_squared. So just solve for V_final. Let's see. Well, the masses go away, and this two comes over here, and you get the V_final_squared equals 2gh. Therefore, V_final is the square root of 2gh, it's the same. What do you know? So in mechanics, there's two ways to do the problems. You can do them with forces, you can do them with energy. Some problems you can do both. Some problems are much easier to do with one or the other. For example, what if the ramp had a loop in it? If the weight had to go through a little loop to get to the bottom? Then this would become very hard, this wouldn't change at all. It really just cares about the initial and the final heights. So some problems lend themselves to one kind of an analysis, some problem lends themselves to another kind of analysis. So we're going to be thinking about that as we move on to electrostatic potential.
