 Let's have a look at a situation where things are moving in more than one spatial dimension. Supposing I have one mass heading in one direction with some velocity, we'll give it velocity v1 and we'll say it has mass m1 and we have a different particle and I'm going to call this particle 2. It's going to have velocity v2 and it's going to have a mass m2. I know I'm not being very imaginative but it's much harder to mess up when you have such a boring naming system like that and those two particles are going to collide with one another and when they collide they're going to stick together and so I've got a before diagram and an after diagram and I suppose these velocities are all vectors so I should give them effect notation. One of the really handy things about drawing diagrams is it lets you mark the symbols. So for example I gave that v1 and v2 and called this having mass m2 and this having mass m1 and the diagram makes it very clear to me what is what and that's a very useful property of doing this and so I need to put my mass for this thing and that's obviously going to be the two things stuck together from conservation of mass in 1 plus m2 and if I want to find out for example how fast this is going in what direction then I'm going to need a conservation law because tracking the individual forces involved in those things sticking together is going to be very messy and so what I can talk about is I can talk about the conservation momentum so I know that the momentum before and the momentum before is made up of two pieces because we have two particles the momentum before is equal to the momentum afterwards I'll call that p final now momentum before is made up of the mass and the velocity and again this is because we are working non-relativistically here okay now the important thing to realise about this equation here or indeed about this equation here are that these are vector equations well there's a couple of ways of looking at that first of all you could say well the magnitudes of both sides have to be the same and the directions of both sides have to be the same or one of the more useful ways of looking at vectors is you can break them down into their components and so what we could say is that the x component of these two vectors added together has to be equal to the x component of that vector or we could say that the y components of those two vectors added together has to equal the y component of that vector or the z and if I'm going to be talking about x y then I should have an x and y in my diagram so I know which is which and now I can start to do this so we'll start with the x component now up here this velocity v1 is going exactly in the x direction and so the x component of it is going to be its full magnitude of v1 and up here the x component of this one is actually zero because v2 is going in the y direction and so we've got an equation for the x component of v3 now let's do the y component and now the y component of this one is zero and the y component of this one is precisely v2 I make sure that this one is going in that direction and that's where I've defined my y and so I've got to make sure that I don't get a minus sign from that but in this case v2 is going in the positive y direction and here I have an equation of motion for the y component of v3 and so I have two unknowns two equations I can work them out