SPEAKER 1: Let's look at a more mathematically sophisticated way to do 2D kinematics. We're going to do it in terms of the position vector r, so an r with a vector hat, which equals the displacement from the origin. Remember when someone asked earlier, why is position a vector? Because position is really displacement. It's just displacement from the origin. R does not mean radius. It has nothing to do with a circle, or circular motion, or a sphere, anything with a radius. As you go further in physics, you'll find r is just the general vector used to describe position, when it could be along x or y or anywhere in space. So let's look at what we really mean by that. Let's draw a couple of r's. Well, it's simply the vector from the origin-- here's the origin, to a position. Our initial one is right there. There is ri. I'll make that bigger. Let's consider two positions, where it started and where it ended, r initial, r final. So for r final, let's say-- really, it can be any of these. Let's just pick another point along the motion. Let's say here. Now, I'm going to draw. There's r final, drawing it nice and heavy, so you can see the-- whoops. That's r final. In general, there's nothing mysterious about these. R really is just xi hat plus yj hat. It's just where you are in the x and y-axis, just drawn as a vector from the origin. But I want to show you how this actually make sense, when you put it together, to get the displacement, or to get delta r. Right? Let's look at these two positions. Now, let's draw them on top of each other here. Here is r initial, and here is r final. But, we want to get the change delta r. So, delta r is what? Delta r is the displacement r final minus r initial, just like delta x was the displacement x final minus x initial, right? If we want to vector add these, but subtract them, you would say rf, and then you would add ri, but you would flip it over and make it go negative. So, ri is basically going to be like that. That's negative ri. Then, vector add those, and it looks like this. That is delta r. That's rf minus ri. There it is, right there, delta r. You look at it, and what does it look like? It looks like the vector here. Delta r is along the path. It's just the vector going along the path. It's the delta x. It's to say, it's just a generalized x and y displacement vector, for a change in position. So, we didn't move from the origin to these two points, as we move from this place to this place. Delta r is just a way to get a vector that goes from the initial position to the final position. If you wanted to, you could write it out in all of its glorious components. Let's do that, just to make you feel better, just for fun, if this seems weird and unusual. You could write this as, for the x components, it's x final minus x initial i hat plus y final minus y initial j hat. Just like the position vector is just xi hat and yj hat. It's just a more compact way to write things. It's nice because, now we can just use the 1D equations with r hat, if we want to. So, we can say, just like 1D. And now, we can say things like, well, the 2D-- if we have the velocity anywhere in a two dimensional plane, the average velocity is delta r over delta t. The instantaneous velocity in a plane v is dr dt. So, those are some good kinematics equations, or definitions, that you can use to derive kinematics equations. In this example, then, if we wanted to write out the kinematics, you would just apply the kinematics you know from 1D into the different components. So, what we would do-- the main thing you're trying to do, as we move forward into trajectories, if you can write the equation for r, you're done. You can solve any problem. So, let's do that. Let's write the vector equation for r. R is a position at any time, right? The x component, we just think about x kinematics. This thing is moving, its velocity has an x component vx, and a y component, vy. So, we say, in the x, we have constant velocity motion with no initial displacement. It's x initial plus vxt, x initial is 0. It's just vx times t, i hat. That's the x component of r, the position. And in the y, it does have an initial position. Y initial plus the y component of velocity, times t, like that. Now remember, those components are like this. This is xy. We can draw the velocity like that. It has an x component and a y component. That's what vx and vy are. Once you have this, you can do lots of things. What's the velocity vector? Well, that's easy. Just take the derivative. The derivative of this, vx times t, the derivative of that is just vx i hat. What's the derivative of this? Derivative of that, with respect to time 0, vy j hat. That had better be the velocity vector. In this case, it's very straight forward. What's the acceleration vector? Take another derivative. 0 plus 0. There is no acceleration, right? This was uniform motion. So, get used to using r. I'm going to show you how if you can find r, you're done. You can solve any problem.