Guys, occasionally in some problems, you have to calculate things like speeds and energies, but instead of using your energy conservation equations, you're going to have to use these things called potential energy graphs. I'm going to show you how to use and sort of read these potential energy graphs. We're going to work out this problem together. Let's check this out. The whole idea here is that potential energy graphs will graph the potential energy of an object on the y-axis versus the position of that object on the x-axis. These can actually tell us some pretty interesting things about the motion of an object without having to get into your crazy work energy equations. Let's go ahead and get to the problem here. We have a marble that is following this potential energy graph, and we're going to release the marble from rest at x equals 1. I'm going to locate x equals 1 on my diagram, figure out where that point is. I'm going to call this point A, and we're releasing this marble right here with a speed of 0. In this first part, we want to calculate the total mechanical energy of the marble. We know exactly how to do that. The mechanical energy, right at point A, is just going to be k + u. So that's just going to be the kinetic at A plus the potential at A. So how do we calculate this? Well, remember kinetic is always related to speed, and we just said that the speed at A is going to be 0. So there actually is no kinetic energy here at point A. So all of the mechanical energy is really just going to be the potential energy. Now, what I want to point out here is that this potential energy isn't necessarily gravitational potential. It's not spring potential energy. It's kind of just some vague potential energy that I don't really know what it's from. So we actually don't use an equation to calculate this. We're just going to look at this. We're just going to look at the value on the graph at point A to figure this out. So here at point A, the y-value is equal to 8 joules, and that's our potential energy. So, really, our total mechanical energy is actually equal to 8 joules. That's the answer.

So what ends up happening is that in general, the mechanical energy at any point is going to be the k + u at that point. Now you only have to solve for this once in a problem because we know that mechanical energy is always going to be conserved if the work done by non-conservative forces is 0, and that's always going to be the case in these problems. So what I like to do is I like to draw a little horizontal line once I've calculated the mechanical energy, and I say that this is the mechanical energy of this object throughout the entire motion. It always has to equal 8 joules. This marble, no matter where it is on the graph, always has to have 8 joules of mechanical energy. Let's take a look at part b now. Part b, we're asked to calculate the kinetic energy at x equals 3. So x equals 3 is right over here, so I'm going to call this part, point B. And before we actually go ahead and calculate this, I kind of want to use a roller coaster analogy. I like to think of this marble as kind of like a roller coaster cart that's traveling on some tracks. The potential energy graph is basically the roller coaster track. So as we're going from A to B, we're going to be going downhill, and therefore, we're going to gain some speed and therefore some kinetic energy. And that's what I want to figure out. So how do I figure out k_{B}? Well, remember, the whole idea with these problems is that the mechanical energy is conserved, so that I can say that the mechanical energy at B is just going to be k + u. This is going to be k_{B}+u_{B}. Now we actually know what the mechanical energy is because we said it always has to equal 8 joules no matter what. So we know this is 8, so all we have to do is just figure out what the potential energy at B in order to figure out what k_{B} is. So we've got this 8 joules. This equals k_{B} + 2. So you have 8 minus 2 equals k_{B}, and therefore you get 6 joules. Going back to our roller coaster analogy, this makes sense. You're going from A, and as you're going from A down to B, you're losing potential energy and therefore you have to gain kinetic energy for your total energy to be 8. So really, the kinetic energy is just going to be the difference between the mechanical energy and the potential at any specific points.

So one way I like to visualize this is by basically looking at the potential energy graph. Here at part A, all of my mechanical potential. Here at part B, I know that I've lost potential and gained some kinetics. So what happens here is that my potential energy at part B is still equal to 2 joules, and the kinetic energy is really just going to be the difference between where I am on the graph and my line of 8 joules. So this vertical line here really just represents my kinetic energy, and I know that this is equal to 6. Let's move on to part c now. Part c we're we're supposed to figure out the speed of the marble at x equals 4. So here I'm back up to x equals 4. So this is going to be my point C, and this is actually very straightforward. We want to figure out the speed at v_{C}. We're really just going to use our roller coaster analogy. As we're going downhill, we're picking up speed, but then if you're going uphill, you're actually going to lose that speed again. So what ends up happening is, if I started from rest here at point A, and then I'm basically back up to the same heights, if you will, then the speed here at 0 at C also has to be 0. You can't end up going any higher than the initial height from which you started from unless you actually had some initial energy or initial speed, which you didn't in this case. So your speed here at C is going to be 0. And again, this makes sense because basically you can't go anywhere above this 8 joules of energy. Alright? So actually, this brings me to an important conceptual point: you're going to have a speed here at 0, and you're going to have a speed here at 0, And basically, what's going to happen is at this point you're going to go downhill, and this point you're going to go downhill again. So without any additional energy that's added into this problem or removed, the objects are always going to remain stuck underneath this line. They're always going to remain stuck underneath my mechanical energy line, between these two points right here, my v_{C} or my A and my C. So these are actually called turning points because there are places where the marble is just going to keep turning around forever. Unless it's given some additional energy, it's never going to be able to escape this little sort of well that it's fallen into.

Alright. So now, let's move on to part d. Part d, we're going to figure out without touching the marble again, right, without actually inputting any energy into the system, can it ever reach x equals 5? So x equals 5 is right here, so I'm going to call this point right here point D. So what we know here is that the mechanical energy for this marble throughout the entire problem has always been 8 joules. If we look at the energy that you would require to be at part d, we look across, we look horizontally, and the potential energy you would need is 10 joules. So what ends up happening here is that your mechanical energy, your 8 joules of mechanical energy will always be less than the 10 joules of energy you would need. So the answer to this problem is actually no. You could never actually reach part D. One way I like to think about this also is if you have zero velocity at speed here at part C, and you turned around and went back down the hill again, there's no way you could actually continue upwards and actually arrive at part D. Alright. So that's it for this one, guys. Let me know if you have any questions.