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ï»¿ >> Hi. Hello class. This is Professor Anderson here. I'd like to talk to you a little bit about kinematic equations of calculus. These are going to be very important when we're considering motion. So, let's -- let's start with kinematic equations in 1-D. And talk about motion. [ Pause ] So, 1-D we're just going to talk about horizontal or vertical motion. We're not going to combine the two just yet. And we're going to make one more important assumption which is that the acceleration is constant. It turns out that if the acceleration is not constant then you can't use the kinematic equations that we're going to talk about. Okay. So, this is an important constraint. So, what do those kinematic equations look like? Well, one of them looks like this. X final equals x initial plus vx initial times t. Plus one-half a sub x, t squared. Okay. How did we get that? Well, we started with a as a constant. We integrated to get velocity. We integrated again to get position. There's another equation which is vx final equals vx initial plus a sub x times t. That is, of course, one integration of a equals constant. And then there's one more clever little kinematic equation. Which is this vxf squared equals vxi squared plus two a sub x. Times xf minus xi. And you can actually get this equation from combining some of the other kinematic equations. And as we're going to learn later on, you can also get this from conservation of energy. Okay. So, let's take a look at a very simple problem. Let's say we do the following. Let's say that we have y as a function of time. Okay. Typically, we say y is vertical, x is horizontal. So, let's worry about vertical for a second. And let's say that we have a particle that has a diagram -- a motion diagram that looks like this. It starts up here at h. It drops down to y equals zero. And it does it in some amount of time. Okay. What sort of function y would properly describe this? So, we could write this function y of time as something like this. [pause] Okay. That negative sign indicates it's curving down. The t squared indicates that it is quadratic. Now, we can't just leave that all by itself because that would say at t equals zero we're starting at y equals zero. But, of course, at t equals zero, we are not starting up at y equals zero. We are starting up at height, h. And so we need to put that right out in front. So,, this equation right here is an adequate description of that curve. All right. And let's see what we can learn from that curve. Okay. So, we have this curve y(t) which is h minus one half gt squared. And let's see what we can learn from that. Let's say we want the velocity. How does the velocity relate to the position? Adrianna, what do you think? How does the velocity relate to the position? [pause] >> (student speaking) derivative. >> The derivative. Correct. It is the derivative of the position. Whoops. Excuse me. It's the derivative of the position. So, v sub y is, in fact, dy, dt. We just need to take a derivative of our function. And we know what our function is. It is h minus one-half gt squared. So, if I take a derivative of h and I take a derivative of one-half gt squared, what do I get? Well, the derivative of h, of course, becomes zero. The derivative of one-half gt squared becomes what? The t comes down in front. I still have the one-half g times t. The exponent becomes a two minus one which is just one. And so we write it like that. And so we get negative gt. So, this is the velocity as a function of time. Okay. Okay. So, if we plot vy as a function of time, what does it look like? Well, at t equals zero, it's going to be zero. So, we're starting there. And then it says it is negative and it increases linearly with time. And so it's going to look something like that. Okay. Negative means the object is going in the negative y direction. It is falling. And it is increasing in magnitude with a slope given by that negative g. All right. That's what vy looks like as a function of time. Okay. What about acceleration? What is acceleration in relation to velocity? >> (student speaking) Derivative. >> Derivative. Acceleration is the derivative of velocity. Right? Velocity is derivative of position. Acceleration is a derivative of velocity. Okay. But we know exactly what that velocity is. We said it is minus gt. So, the derivative of minus gt becomes what? Well, the t just goes away. And so we get minus g. [ Pause ] Now, we kind of knew this already because we set up the problem that way. But this is free-fall due to gravity. And, specifically, it's free-fall due to gravity near the surface of the earth. So, if I take an object -- here's my object and I drop it from a height, h, it falls with an acceleration of minus 9.8 meters per second squared. G is that number. 9.8 meters per second squared. Which means that the speed of this object when it falls is increasing linearly. Okay. As time goes on, the speed just increases linearly. And that means the position is changing quadratically. All right. This is free-fall due gravity. It is, of course, only near the surface of the earth. It changes as we go up in altitude which we are going to see later on. Okay. What about acceleration. What is acceleration in relation to velocity? >> (student speaking) Derivative. >> Derivative. Acceleration is the derivative of velocity. Right? Velocity is the derivative of position. Acceleration is the derivative of velocity. Okay. But we know exactly what that velocity is. We said it is minus gt. So, the derivative of minus gt becomes what? Well, the t just goes away. And so we get minus g. [ Pause ] Now, we kind of knew this already because we set up the problem that way. But this is free-fall due to gravity. And, specifically, it's free-fall due to gravity near the surface of the earth. So, if I take an object -- here's my object and I drop it from a height, h, it falls with an acceleration of minus 9.8 meters per second squared. G is that number. 9.8 meters per second squared. Which means that the speed of this object when it falls is increasing linearly. Okay. As time goes on, the speed just increases linearly. And that means the position is changing quadratically. All right. This is free-fall due to gravity. It is, of course, only near the surface of the earth. It changes as we go up in altitude which we're going to see later on.

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