 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 in one. 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. Now 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 well 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 and 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 the 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 realize 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 to 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. Now once again we do a quick check of units. We've got mass divided by mass, so that gives us nothing. And we've got velocity equals velocity. So that works well. And it's fairly obviously the same on the bottom line. Let's also check limits. There are lots of limits we can pick. So to start with let's assume that the first mass isn't moving, which means the second mass just goes straight up into it. And then we'd expect them to go straight up. So if we look down at our equations down here, if the first mass wasn't moving, that would be zero. So this would be zero. And so this thing here would not be moving sideways, which is correct. And it would be moving straight up instead. However, if this mass say was zero, we'd expect no matter how fast it was going, it wouldn't really change the trajectory of the first particle. So if we go down here and set mass two to zero, then what we get is that this is zero, because the top line there is zero. And so we'd find that it would be going only in the horizontal direction. And indeed it'd be going the same velocity it always was. So in other words, if this had zero mass, even when they stuck together, the velocity of the first mass would be unchanged, like a tiny insect running into a truck. And that's exactly what our equations say is the limits look good. And if we wanted to work out the angle of which these two masses move after they stick together, then we can easily work that out in terms of the components because what we have up here is the y component of velocity. And the x component of the velocity. And so the tan of theta is going to be the ratio of those two, which is just going to be... And so you can see that the tan of that angle is just going to be the ratio of the momenta of the two particles.