Very simple problem. Derive the equation 1/2mv^2 for kinetic energy.
The Attempt at a Solution
∫m (dv/dt) dx
Now, here is where I get confused. Here is the supposed solution that I don't understand.
m (1/2v2) +c
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
= 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?