Anderson Video - Kinematic Equations from Calculus

Professor Anderson
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>> Hello class. Professor Anderson here. Let's talk about the kinematic equations and where they come from in terms of calculus. So if we start with an acceleration and we say that acceleration is constant, how do we get to velocity? Well, velocity turns out to just be the integral of acceleration with respect to time. And if a is, in fact, a constant, it comes out of the integral. If I integrate dt, all I get is t. And then whenever I do and integral, I always have to add a constant of integration. And we will tell you what that constant is in a second. All right. What about position? Position x is the integral of velocity times -- over time. And we know exactly what velocity is now. It is at, dt. But we have this constant of integration here to worry about, so we have to include that. Let's call that thing an initial velocity Vi. So what we should really have here is not just at, we should have at plus V sub i. And now when we integrate this, what do we get? Well, if I integrate at, a is a constant. But the integral of t becomes t squared over 2 so I get 1/2 at squared. Vi is initial velocity and that's a constant so we just add Vi times t. And then we have to add another constant of integration. And that constant of integration we are now going to call X sub i. And so look what happens. This is how the kinematic equations developed. We have this equation for V which becomes the following: Vf equals Vi plus a times t. And we have this last equation for x which becomes Xf equals Xi plus Vxi times t plus 1/2 a sub x t squared. So the kinematic equations, you don't really have to remember all them if you just remember how to do integrals. If you know how to integrate, all you have to do is start with a and you can get to two of our kinematic equations.
>> Hello class. Professor Anderson here. Let's talk about the kinematic equations and where they come from in terms of calculus. So if we start with an acceleration and we say that acceleration is constant, how do we get to velocity? Well, velocity turns out to just be the integral of acceleration with respect to time. And if a is, in fact, a constant, it comes out of the integral. If I integrate dt, all I get is t. And then whenever I do and integral, I always have to add a constant of integration. And we will tell you what that constant is in a second. All right. What about position? Position x is the integral of velocity times -- over time. And we know exactly what velocity is now. It is at, dt. But we have this constant of integration here to worry about, so we have to include that. Let's call that thing an initial velocity Vi. So what we should really have here is not just at, we should have at plus V sub i. And now when we integrate this, what do we get? Well, if I integrate at, a is a constant. But the integral of t becomes t squared over 2 so I get 1/2 at squared. Vi is initial velocity and that's a constant so we just add Vi times t. And then we have to add another constant of integration. And that constant of integration we are now going to call X sub i. And so look what happens. This is how the kinematic equations developed. We have this equation for V which becomes the following: Vf equals Vi plus a times t. And we have this last equation for x which becomes Xf equals Xi plus Vxi times t plus 1/2 a sub x t squared. So the kinematic equations, you don't really have to remember all them if you just remember how to do integrals. If you know how to integrate, all you have to do is start with a and you can get to two of our kinematic equations.