 Conservation of momentum and conservation of energy are fundamental principles in physics and they apply relativity, quantum mechanics and everyday life. So when do you use one rather than the other? So our two conservation laws are conservation of energy and conservation of momentum. Now we're not going to prove this at the moment, but it turns out that the conservation of energy and conservation of momentum come from deeper principles. The conservation of energy actually comes from the idea that the laws of physics stay the same over time. And the conservation of momentum actually comes from the idea that the laws of physics are the same in different places. So those are very deep principles that give us these conservation laws. That link we won't be able to make until university level physics, but for now let's look at just how we use these things. So conservation of energy is particularly useful to use because it's just one number. Energy is a scalar and it has a lot of different forms and so that means you can use it in very complicated situations sometimes. So you might have a roller coaster and if you have someone on this roller coaster and even though as they move along this roller coaster they're going to have forces pointing in different directions due to the roller coaster pushing them away. They're going to have a constant force due to gravity pulling them down. That's going to turn into all sorts of complicated accelerations and so forth. But we know that we can use the conservation of energy because provided the friction between the trolley and the roller coaster is small then that's not going to do any work and so the energy it has at this point is going to be the energy it has at any other point. So if you want to know what speed it's going to say at some other point I just look at the change in the potential energy and that's going to have been turned into kinetic energy. Now conservation of momentum would be a terrible thing to try and use for that problem because momentum is a vector quantity. And in this problem the force of gravity and the force of the roller coaster pushing on it, the normal force will be changing the momentum of this trolley all the time and indeed it will be changing in different directions and we're going to add up all the different changes of momentum in all those different directions as it goes along here to figure out its final velocity. So that would be an incredibly inefficient and hard way of doing this problem. But sometimes momentum is exactly the right kind of law to use and the reason for that is that momentum only has one form. Whereby contrast energy has lots of forms. What that means is if you have say a collision of some kind supposing you have a road and there's a couple of cars and one car bangs into another then the nice thing is that after the collision no matter how mushed up these cars are and no matter how many directions all the different pieces are going we can still use the conservation of momentum because there's only one form of momentum and it's always good to separate your before and after pictures very explicitly in your diagram so I'll do that. Clarity in a diagram really stops you from getting confused and it helps communicate what you're saying to other people as well. Anyway we could have used energy or conservation energy in this problem in some sense because we know that energy is conserved if you have two things smashing together energy doesn't disappear or appear. On the other hand there's a lot of different forms of it so there's the kinetic energy of all these pieces flying off but there's also the potential energy required to change the shape of all those cars and rip bits off. Fundamentally that's an electrical potential energy but it's something we're not tracking and so it's hard to use conservation energy for this problem. Now in a collision like this where their total kinetic energy isn't conserved is called an inelastic collision. So of course everything conserves total energy but not everything conserves kinetic energy. If a particular collision does happen to conserve kinetic energy that's called an elastic collision. So an inelastic collision is one where the energy is changing from one form into a hidden form typically so you might have a ball of putty that falls on the surface and goes splat and takes some energy to deform the ball of putty and so the kinetic energy is not going to be conserved. A rubber ball however is going to deform and then under form again and so it's back in the state it was before which means there's no sort of hidden form energy there and therefore it's got to have all its kinetic energy back again after it bounces. And another thing that's very good at bouncing is not just rubber balls but actually steel balls. This is a Newton's cradle. If you have a steel ball and a couple of these hit each other then what you find is that they don't change shape afterwards. They don't change shape, they don't store energy anyway and so therefore the energy that goes into the collision comes back out of the collision. Now if momentum were conserved there are many ways this could happen. If I've got one ball coming in it could come to a stop and then four of them could go a quarter of the speed. That would conserve momentum. Or if I had one ball coming in I could have two going off at half the speed. So there's all sorts of different ways I could conserve momentum in that collision. However if I'm going to try and conserve kinetic energy then this mass times v squared has got to be equal to the mass times the v squared of the final thing. So if I have four things going at a quarter of the velocity that's going to have a much lower mv squared because the v has to be squared and so I've got a quarter that will turn into a sixteenth. And so when you have just one it turns out the only way you can conserve kinetic energy and momentum at the same time is if you have one bouncing up, exactly one. And similarly if you have two, two bounce up all the way through to if you have four hitting ones is it going to fly off a lot? So in order to conserve momentum and kinetic energy we're going to have four going on. So there's a little bit of jiggle there, these have a little bit of stick and so the collisions are not perfectly elastic but they're very elastic and so that's why you get the kinetic energy being mostly conserved.