 Okay, so I want to make, show you something about projectile motion. I mean, I know this is tricky stuff. So, I want to show you some cool things, and it's not really an example, but it's just kind of a cool way of looking at it. Okay, so I've already prepared the board. Suppose that I shoot a ball or something, and I have a speed of four meters per second, and I shoot at an angle of 60 degrees above the horizon. Okay? And then every tenth of a second, I take a picture. So, I kind of sketch this out here. I've drawn an x and y axis to scale, and then I have, you can't see. There's a ball at zero seconds, at point one seconds, point two, point three, point four, point five, point six seconds. Okay. So, and it's mostly the scale, but not perfect. I think we'll leave. Okay, so let's look at some stuff, and then maybe this will make sense, maybe it won't. Okay. So first, let me find the displacement vector going from position one to two. Okay? So, if you remember, all right right here, hopefully it won't get in the way. Let me call this delta r equals r two minus r one. So, it's a vector pointing from the first position to the next position. So, here's the first position and there's the next position. So, it's just going to be a vector that looks like this. I'll call this delta r one. Maybe that doesn't make too much sense, but... Okay, that's my first position vector. Displacement, I'm sorry, displacement. Okay. Where am I going with this? Just wait. Just wait, you'll see. Okay, let me do that for the next one too. Delta r two, delta r three. Can you see the red? Let me see. It looks like it's showing up okay enough. Let me draw one more. Delta r four. Okay, so let's look at what happens for each of these displacement vectors during the same amount of time. Let me just take delta r two and break it into an x and a y component. So, this would be delta r two, x. That's the x component of it. And this is delta r two, y. Let me do it for the next one too. In that next tenth of a second, I have delta r three, x, delta r three, y. Okay, now let me show you something cool. Even though it's not completely to scale, look at the length of the x vector during that point ten seconds interval two. It's the same length. So what does that tell us about the velocity in the x direction? I can write, let me just write down the average velocity during time interval two, the average two delta r two over delta t. Okay, so I can rewrite that as the average two x as delta r two x over delta t. So, since they have the same delta position in the x direction, they have the same average velocity in the x direction. So, from this interval to that interval, same average velocity. If you did it for this, guess what you'd find? Right, same, same. The x velocity does not change during this motion because acceleration is just downward in the vertical direction. Okay. Now, if you do it for the y direction, let me do that. Actually, I'm going to do it as a whole vector. Okay, I'm not going to break it into components. Okay, so now let me look at, here is the, if I use units of one seconds in my velocity vectors, this would be the average velocity for unit one, same length, but different units. I know that's confusing. So, let me go ahead and draw that in green. So, this is the average one. This is the average two. And this is three. And this is four. And let me go ahead and draw the next ones. Okay. Now, how do I find the acceleration? Well, remember, acceleration is delta v over delta t. So, what if I want to find the acceleration going from, let's say, the middle of here to the middle of there? I'm going to use the average velocity during that time rule in the middle. That's a little trick, and maybe it's not quite true, but whatever. I think it'll show my point. Okay, so that's going to be equal to, let's say that's equal to v two average, minus v average one over delta t, which is still going to be a tenth of a second. So, I have to do this vector subtraction. So, let me do that with these two vectors. Okay, so in order to subtract vectors graphically, you can draw them starting at the same position and then just draw a vector from the end of two to the end of one. So, I'm going to redraw these two vectors so they're at the same. And in fact, I'm just going to redraw this one. If you move a vector and you don't change the length, you don't change the direction, it's equivalent. So, I'm going to put this vector starting right there. So, it's kind of hard. I need to get the angle right. It's going to be... I'm going to try to do this as precise as possible just to show you that it's not magic. Okay, so that's about 35 centimeters. So, 35 centimeters right there. Oops, I should just draw that in. Okay, so that's v one and that's v two. And so, the change is going to be from here down like that. So, it's going to be delta v one. And then, so that's... When I divide that by my time, I'd get the acceleration during that interval. Okay, so that's going to be proportional to that but in different units. Big whoop. I know what you're saying. I can hear you. Okay. Now, let me do it again for the next interval because this is where it starts getting cool. So, let me again take... I'm going to do again half interval to half interval. So, I'm going to take this vector and move it up there. I guess I should draw that not in green. If I want to represent it as an acceleration, I'll use blue. So, that's the acceleration vector. I'll call it a one even though it's really in between those two. Okay, so now I'm going to draw vector average velocity two up here so I can subtract those two. I'm going to do a measurement so about 29 centimeters. So, if I draw this about 29 centimeters long. One right there. Okay, and then I'm going to get the final minus initial vectors. If I subtract these vectors, I get A. Boom. Look at that. Isn't that awesome? Why is that awesome? Because look at this vector and that vector, the same. They're the same length in the same direction and they're both down. Okay, let's not stop. Let's keep going. So, let's do it again. Actually, let me skip because I don't want to take up too much time. I could do it for this one and this one, but first, notice that the ball is moving up but the acceleration is down. Let's look at these two right here. Excuse me. Okay. So, this vector is 22 centimeters. So, if I move that right there that's a little 21 maybe. That one doesn't look like it works too well. 22. Okay, well, this one's off. But, you know, I think I didn't draw very accurate. It should be a little bit shorter because, yes, the acceleration vector should be the same straight down. That one's a little off. Okay, but even when the ball's the highest point, the acceleration's down. So, in projectile motion no matter in projectile motion we define as things that are moving only due to gravity, the influence of gravity, they have an acceleration vector. You could write it as 0x hat minus 9.8 meters per second squared y hat. That's it. So, that means that in the x direction your velocity is constant in the y direction you have a constant negative acceleration. So, kinematically, we could write this as am I going way over time here? Yeah. I'm going way over time. So, let me just stop it right there and then I'll do another example later.