 So now let's take a look at some special cases of polynomials for motion in one dimension. Before we get into the special cases, let's just briefly talk about what polynomial order is. If I've got a polynomial, I can look at all the different powers in that equation. And I find the one that's got the highest power. In this case, it's t to the sixth. That's going to tell me what order my equation is. It's a sixth order equation. If my terms are out of order, it's not going to be the last one. But when I look through this one, I see that 7 is my highest. And so this is a seventh order equation. Well, the first special case we want to look at is the zero with order, which means there's no terms that have a power of t. Here's some examples of zero with order polynomials. They have no dependence on t, so they're just individual values for the position. If I take one of those examples and remember that I can find the velocity and the acceleration using derivatives, I find that the velocity with respect to time of just a constant 2 meters is zero. And the acceleration is the derivative of the velocity with respect to time, and that's also zero. Graphically, my position was just 2 meters, didn't change with time. And the slope of that graph is zero, so my velocity was always zero. And the slope of zero is just zero, so again, my acceleration is zero. We often refer to this zero with order equation as a constant position equation. Now, a little bit more interesting is the first order. In this case, I've got a power of t to 1, or just plain t. And here's an example of one of those equations where I've got a t, but no higher power. Now, remember, a lot of the textbooks sort of strip out the units there, just to make it a little bit easier to see that I've got a term with t and no other time dependence in that equation. You can have it in several different formats. You don't have to have both terms here. You can have just the t term. You can have negative signs. They can be in different orders as long as t is the highest order you have. Starting with that example for the position equation, again, I can find my velocity and my acceleration by looking at the derivatives. My first derivative of the position with respect to time for this equation is just two meters per second. If you're having trouble seeing that, sometimes you can go through and temporarily strip out the units just to look at it in its form so you can see that the derivative of 2t is just t, and the derivative of 3 is zero. Looking at the next derivative, my acceleration, I see I'm back to zero because I have only two meters per second with no time dependence there. If I plot these things, what I see is that my position was of the form of a straight line. My velocity is the slope of that line, so it's a constant value here. And because this is a horizontal line and has no slope or slope of zero, the acceleration is still sitting at just zero. So we call these first order equations constant velocity situations. Then we get to second order, our last special case we're going to take a look at. And that's got my t squared, so the 2 is the highest power. And here's an example of a full second order equation. Again, if you look at it without the units, it's a little easier to see that that's a t squared. But we want to keep track of what those units are. Here's some other examples of things that would also be second order equations. And again, I don't have to have all the terms. They can be in different orders. If I take that larger position with respect to time equation, again, you might want to see it as the stripped out version without the units in there. I can, again, find my velocity and my acceleration by taking the derivatives with respect to time. And if you're having troubles doing that, go back and review the earlier videos. But in this case, I've got 2 meters per second plus 6 meters per second squared time. In this case, I've just got 6 meters per second squared. Again, I can take these three equations and plot them. Our second order equation looks like a part of a parabola. The slope is constantly changing on this, so my velocity is not constant. But it is a straight line. And because it's a straight line, it has a slope. And so I've got a value for the acceleration. But that second order equation has constant acceleration, just not constant velocity or position. So that's the last of the special cases we're going to consider for now. We'll take much more notice of the constant velocity equations in the next few lectures.