1. The problem statement, all variables and given/known data Very simple problem. Derive the equation 1/2mv^2 for kinetic energy. 2. Relevant equations 1/2mv2. 3. The attempt at a solution ∫f(x) dx ∫ma dx ∫m (dv/dt) dx m∫(dv/dt) dx Now, here is where I get confused. Here is the supposed solution that I don't understand. m∫(dx/dt)dv m∫v dv m (1/2v2) +c =1/2mv2 Yet, this begs two questions. 1. Why am I able to move the derivative from (dv/dt)dx to (dx/dt)dv? 2. I don't understand the concept. Why is the work done (well, after considering Kf=ki + w) equivalent to the intergral of a force with respect to time? I don't really understand this concept. Can anyone explain this to me? For example, the following equation for a spring. kx is the force of a spring, so to find the work done by a spring, we do: k ∫x dx k(1/2x2) = 1/2kx2 - 1/2kx2 Of course, this is the familiar equation for work done by a spring. But wouldn't we need to multiply this equation by x (to get F*d), so actually 1/2kx3?