 So Newton's second law tells us exactly how things are going to accelerate when you apply forces to them and Newton's third law tells us that forces always occur due to interactions between two objects and that the two forces between the two objects are always equal and opposite. But why is that true? Why are these forces always equal opposite? It didn't have to be that way. One force could have been bigger than the other and they could have both on average shot off in one direction. It turns out that the answer to that led us to discover a very important foundational principle that's true throughout relativity and quantum mechanics and indeed areas where Newton's law is break down and that principle was about the conservation of something called momentum. If you look at Newton's second law where the force equals the mass times the acceleration and I've written the force and the acceleration as vectors because they have not just magnitude but also direction then you see that the way that the velocity changes depends on the mass and if we write for a small amount of time, a very small amount of time we can write the acceleration as a change in the velocity and which is a vector because it's got direction over that small change in time and if we multiply both sides by delta t and we know that the mass doesn't change here so the mass times the change in the velocity is the same thing as the change in the mass times the velocity. Then we can make two new definitions. The first is this the force times the time this is called an impulse and the second is this the mass times the velocity which is known as the momentum and there's a very simple physical picture for these quantities. The impulse is how hard a shove you give something so you can either give something a really large force for a short amount of time or a smaller force for a longer time and still give it the same total amount of impulse the same total amount of change to its momentum and the momentum is basically just how hard something is to stop. If something has a really large momentum you need either a really large force or a really long time to try and stop it and momentum turns out to be a really important very fundamental quantity and it's usually given the symbol P and Newton's second law is even simpler if you write it in terms of momentum so the force is just that for a small amount of time and so we can see that the force is just defined as the rate of change of momentum. So these two forms of Newton's second law are the same and this one does have one less symbol in it I guess but that's not why momentum is so interesting and important. The reason momentum is so interesting and important is due to Newton's third law. If we have two objects A and B and they apply a force on each other and the only forces that exist to where you got two objects and they apply a force on each other according to Newton's laws then those two forces have to add up to zero. So Newton's third law says that this force F1 is equal and opposite to F2 which means that in vector terms if we add them up they exactly cancel. So while this interaction is occurring and these two forces exist then particle A is going to experience a force and it's going to accelerate in that direction and some momentum is going to change in that direction. So it's going to get a change of momentum and B likewise is going to accelerate in the direction of the force and so it's going to get a change of momentum in the direction of the force and if we add up those two changes in momentum so once again we use the capital delta to denote a change so the change of momentum of particle A right like that and the change of momentum of particle B we write like that. And we know what that is because we know that the change in momentum is just given by the impulse. And so if we apply these forces for a certain amount of time, the change in momentum for particle A is just the force on that particle which is F1 times that time. And the change in momentum for particle B is just the force acting on particle B times that time. And of course we can factorize the time out of that. And we know that if you sum up those two forces because of Newton's third law they have to add up to zero. So in other words the change in momentum of particle A is non-zero and the change in momentum particle B is non-zero but the change in momentum of each of them exactly cancel. And so the total momentum of the system is conserved. So the total momentum of this system was conserved. And remember we didn't know anything about these two forces, why they were there? Only that they obeyed Newton's third law and all forces do obey Newton's third law. So no matter what A and B were we knew that the momentum was conserved. And in fact that's true for any closed system. What do I mean by a closed system? Well supposing we ignored particle B and we just looked at particle A, obviously that momentum is not conserved because this momentum here, the change in momentum for particle A is non-zero, it's going to change. But when you look at this force you say well I know that particle A must be interacting with something. That's the only way forces come about. So I go hunting for that and when I find particle B you say all right now I've got all the particles that are interacting. And so a closed system is where you've included all the bits that are interacting with each other. That's not interacting with anything outside the system. And then if you look at that the total momentum of any closed system has to be conserved. And this is a really really fundamental principle. It's true in quantum mechanics, it's true in relativity. You have to change the form of momentum slightly for relativity but there is still a thing called momentum and it's still a special thing that's conserved for any closed system.