 In this video, we are going to calculate the work required to x-ray the object that initially is addressed at velocity 0 to a velocity final of v. So let's start by writing down what we know. We know there is a link between the final velocity and the initial velocity from kinematics. We know that the final velocity is the initial velocity plus acceleration times time. We also know the displacement delta s is the initial velocity times time plus one-half times the acceleration times time squared. Then, what else do we know? We know from Newton's laws of motion, from dynamics, we have the f net is n a. And finally, from the energy chapter, we have that the mechanical work is fourth times the displacement times the cosine of the angle. From kinematics, we actually know way more. We have another equation and I forgot to write that it's the final square is the initial square plus two a and changing displacement. Now why don't you just write that one down? Because looking at our work equations, we need to find a way to calculate the force. Well that we can do from dynamics. We need a way to calculate s, the distance it will take us to accelerate the object to our final speed. And I could calculate it with my first two equations but often have time in there. We have no information about time. So that one here could be an interesting equation and we could use it to calculate our distance. So let's do that. Let's start with force. So f net is n a. I am pulling that way, I am accelerating that way. So my acceleration or my f net, which is that only force here that I have in forward direction is n a. Then my displacement. Now I am taking my equation from kinematics. So let's call this equation two. My final square is v initial which is a zero plus two times a times delta s. So we get that delta s is v final which is v. So v squared over two a. What's the angle between the direction of the force? The angle is zero. So let's combine. Work is the force. So mass times acceleration times displacement. So times v squared over two a and n times cosine of zero. Now cosine of zero is one. Dexuration goes away and when I rewrite it I get m v squared over two. Now maybe that doesn't look similar or familiar yet but if I rewrite it like this. Have you seen this before? It's of course you have. That's the formula for kinetic energy. When you get energy is one half and v squared. And it has to be that formula because it just did. If you think back at final energy is initial energy plus work. So we added the work of one half and v squared. Initially we had velocity of zero so no kinetic energy as well at all. And final was the energy made of well of kinetic energy final. So we have just proved the formula for kinetic energy that it must be one half and v squared.