And now another kind of velocity instantaneous velocity. And instantaneous velocity, we will describe that with a cursive v, but we're not going to put a vg. There's no average subscript. When you see it with just the v by itself, with a vector, that means instantaneous velocity. And really, the difference is it's for very short interval delta t. And if you know you're calculus, you know I mean very short interval, like zero. Okay? So let's think about this. Let's start with the demo. So it's the same old example too. It's just how sliding along a 1d track, uniform motion back to the simple case. So let's go make our plot again of what that looked like. There's time plus t, there's plus x position, and it's a line, right? Uniform motion. So what is vav? Let's first let's go back and talk about average velocity again and see what was that equal to delta x over delta t. So we're going to go back to that idea. You had t initial, t final, and the way you found the average velocity is you took these up, and then you went over, and we said, okay, it's kind of like the slope. That's a had delta x displacement over here, and delta t down there. So what we're going to do is conceptually, we're going to let delta t be very small. Okay? And what we're going to do then, delta t be very small instead of this big delta t, let's put t initial here, and let's put t final there. That goes up there, that goes up there, that goes over, and that goes over. And you can see what's going to happen. This got smaller, but this got smaller, right? They both got smaller. The ratio is going to be the same because it's uniform motion. Any two delta t's you pick. Any delta t you pick, you're going to get the same average velocity, whether it's big or down here, or up here, or really small. So if we let it be really small, we'll get the velocity at an instant, because we're going to get the same value, no matter how small it gets, I could go here. Here's tf. Oh, if you can only see how small that is, zoom in. There you go. Even for that small of a delta t, we're going to get the same velocity. So what this means is mathematically, let's see how we would say this. Cully. What does this mean? Mathematically speaking, this is the derivative. So this is calculus-based physics. So we will talk about concepts from calculus, although you can kind of get by if this is the first time you're seeing them. But basically, this is the simple idea of the derivative. And what we mean is, as delta t goes to zero, as we make it really thin, then basically the v average becomes, it goes towards this thing we just call v, the v instantaneous. Just like as I was drawing here, as we shrunk down delta t towards zero, we get not the average velocity over some region, but the velocity at an instant. Instantaneous velocity. Mathematically, then, we don't write the deltas. The deltas mean a big separation. When you shrink it down, we just call it d. It becomes dx dt. So that is a formula for the instantaneous velocity. It's the derivative of the position as a function of time. And it's a vector because the position, remember, is also a vector. It's the displacement from the origin. So this is our formula. If you know calculus and you're given the position as a function of time, you know it's a derivative. You know the mechanics of taking a derivative. If you don't know calculus, don't worry about it. We'll teach you a little bit about how to do it. But basically, there's a mathematical procedure to find dx dt when you know x. There's a function of time. Also, point out, it's still the slope. It's still, what it means is, it's a slope at a certain point. If you have a calculus, you're saying, well, then why all the complication? Why not just call it the slope? I'll get used to it. You'll find out. There'll be lots of complication. So that's instantaneous velocity. Let's now look at some more velocities.