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## Homework Statement

Very simple problem. Derive the equation 1/2mv^2 for kinetic energy.

## Homework Equations

1/2mv

^{2}.

## 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 (1/2v

^{2}) +c

=1/2mv

^{2}

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(1/2x

^{2})

= 1/2kx

^{2}- 1/2kx

^{2}

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 1/2kx

^{3}?