I'm going to do this two ways. For any of you that have already taken calculus, you know what I'm going to do. I'm going to use the derivative first and the integral, or antiderivative second. If you haven't had calculus yet, and you don't know what I'm talking about, then I suggest that you leave this page and go to the example problems because I'm not going to take time to explain the derivative and integral. In other words, you won''t know what I'm talking about. You'll be confused. VERY CONFUSED.

Okay, so here we go. First off, we have to assume one equation, and that is:

Yes, yes, I know. This is one of the equations I said you would be deriving. That's why I'm going to do this with the integral also. Just hang in there. I'll make this quick. When you take the derivative (with respect to t) of this equation, you get dx/dt=V_i+at, which is your first equation. You can then take the derivative again and get dv/dt=a, acceleration. There, I'm done. See, that was painless. Now we can go the integral, which is a little more practical since you don't assume that big long equation at the very beginning.

First off, we're going to take the integral with respect to t. The only equation we have to assume is f(t)=a. So when we take the integral of that, we get f(t)=at+C. Through experiments and all sorts of fancy stuff, we could determine that C=V_i and f(t)=V_f. I don't want to do that because I'm lazy and I don't know how. Just know that that is what it is.

Now we can use this last equation we derived and get the distance equation, or equation number 2. If you know how to use the integral, you will get f(t)=1/2at²+v_it+C. Again, use experiments just like Newton did and you will determine that f(t)=x_f and C=x_i.

Well, there you have it. Every conceivable way to get the equations of motion. You now know as much if not more than I do. Good luck and don't forget to try a few example problems.