>> All right, let's say you're riding your bicycle and you're moving along at speed VI and you go through this dip and then you go up a hill and let's calculate what the speed is at the top of the hill. And pretend that you're coasting the whole way, okay, you're not pedaling, there's no air resistance. How can we attack this problem? Well, let's give you some givens first. Let's say that your initial speed, VI, is 10 meters per second. Reasonable speed for a bicycle. Let's say that your initial height is 7 meters and your final height is 11 meters. Okay? And we're not going to give you the mass. Hopefully we won't even need that. Let's see. All right, conservation of energy, what does it tell us? It tells us whatever energy we have initially has to be there in the final picture. Initially we, of course, have kinetic energy. But we're at some height, YI, so we have some initial gravitational potential energy. In the final picture we have kinetic energy and we have gravitational potential energy. We're up at height YF. Conservation of energy is really nice because you can, in fact, skip that whole middle section. Whatever energy is there initially has to be there finally and you can put that final point wherever you like. Okay, we know what kinetic energy is. It's one half MV squared. Initially we're going at VI. Gravitational potential energy is MGY initial. Kinetic energy finally is one half MV squared and now it's VF. And gravitational is MGY. All right, let's solve this thing for VF. That's what we're looking for. To do that we first cross out the Ms. If I divide both sides by M all of those go away. Now let's multiply everything by 2 to get rid of that one half. So, we have VI squared -- whoops. So we have VI squared plus 2G Y initial, equals VF squared plus 2GY final. And now we can rearrange some terms to isolate VF. VF squared is going to be VI squared, plus 2GY initial and then I need to move this one over to the other side. So, that becomes minus 2GY final. And now we can simplify and take the square root. And what do we have? We have VI squared plus -- I have some common factors here, so I can factor out a 2G. I have a Y initial and I have a Y final. And we're going to take the square root of that whole thing and now we have some numbers we can plug in and try it. Okay, so we have the square root of VI squared. VI is 10. And then we had 2G times Y initial, which is 7, minus Y final, which is 11. We know that G is 9.8. We can plug all these numbers in and see what we get. I get the square root of 21.6, which is 4.65. And the units are meters per second. So, that makes sense. You ended up going slower than you started initially, but that's because you went up this hill. You ended up at a higher altitude that you started at. Now, there's something sort of interesting right here, right? We have a 7 minus 11, that's a negative number, okay? And if that negative number keeps getting bigger eventually that negative number, multiplying by 2G is going to be as big as 10 squared. And that is the limit of how high you could possible go. When V final equals zero, that's as high as you can possibly go up that hill. If you get the square root of a negative number here, that means that you can't go that high. So, if for some reason on your homework say, "Those numbers are indicating a square root of a negative number," that just means that you can't make it all the way up the hill. All right. Cheers.