 So we know we have a conservation law for momentum. In any circumstances if we have a closed system where all the bodies are interacting are included then the total momentum is going to stay the same. So what? How do we use that? Well it turns out there are certain situations where conservation laws are really useful because they're so robust. As an example here are a pair of black pens. Nothing up my sleeves. All right. Now something complicated is happening. You don't know what it is? You probably never will. In fact, I don't even remember and then I bring these back and the question is what's in my hands? If I show you in this hand, I have precisely one black pen and nothing else. What's in that hand? If you answered the other pen you'd be right and what you used there was a principle called conservation of pens and it was really useful because you knew that there were two pens. I wasn't going to destroy them or create them. So it doesn't matter what complicated thing happened behind my back. You knew exactly that they're going to be two and you could use that fact and in the same way because we have conservation of energy, we now have conservation of momentum, conservation of mass, conservation of charge, any or all of these principles can be used to turn a complicated situation into a simple situation. If the answer doesn't depend on those complications, you can often work out just what you need to from the conservation law itself. So one of the easiest ways to use conservation laws to calculate something is look at the before and after of some complicated situation. Okay, so there's some train tracks and a red car unfortunately is stalled on the train tracks and although a train is coming but fortunately we're coming up behind with our car and we bravely decide to ram their car off the tracks and we're probably going to jam together and we're hoping that we're going to go fast enough to get off the train tracks in time. And so the real question is if I take my car with a certain mass and I slam it into their car, how fast are we going to be going after we hit? So it's going to depend on the masses of the cars. So I'm going to have the mass of my car in one and the mass of their car in two and one thing I could try and do is I could look at all the forces involved. The forces being the force of the front of my car pushing against the back of their car and the Newton's third law force of their car pushing on mine and so I'll be accelerating their car or different bits of it different amounts actually as it crumples and but in general I'll be accelerating their car and they'll be accelerating my car backwards and so we'll end up going at the same speed as we mush together. That's a complicated process and what we could use instead is the idea that if this process is really fast and quite violent then the effect of the friction with the ground is going to be negligible in which case all we need to do is look before the collision and after the collision. In that case the two cars will have a joint velocity and they'll also rather less fortunately have a joint mass which is just going to be M1 plus M2. Now the reason we don't have to track all those forces is that we know that in the absence of external forces or when the external frictional forces can be neglected we know that the momentum is going to be conserved. So the total momentum before the collision is going to be equal to the total momentum after the collision. So before the collision we have one object and another object so we need two momenta to add together and afterwards we have just one thing so we just got one momentum for that. Now non-relativistically the momentum of an object is just its mass times its velocity and the velocity of the second car is initially zero so that's easy and that's very easy to rearrange to find out what the final velocity is. We just simply divide both sides by M1 plus M2 and we're just quickly going to check that the units are correct. So we've got a velocity on one side and here we've got a mass divided by a mass. So a mass plus a mass is a mass. To the top side we've got a mass to the bottom we've got a mass so it's mass divided by mass which gives us no units and then we're multiplying that by a velocity so it's velocity equals velocity so the units are good. Great. Let's check that the limits are good. Okay, so the limit that we are going at zero speed to begin with we're going to be going at zero speed afterwards. That makes sense and the limit that this car is very very heavy. This bottom line is going to be really really big so this final thing is going to be going slower and slower. So if we're trying to knock a truck off the train line we're going to have a lot more trouble than if we're going to try and knock a much smaller car. So that all makes sense.