 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 laws 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 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 a 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 the momentum is so interesting 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 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 change your momentum of particle b We write like that And we know what that is because we know that the change of momentum It's just given by the impulse and so if we apply these forces for a certain amount of time The change of momentum for particle a is just the force on that particle, which is f1 times that time And the change of momentum of 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 of momentum particle b is non-zero But the change of momentum of each of them exactly cancel and so the total momentum of the system is conserved