Difference between (mv^2)/2 and mv.

What is the difference between kinetic energy and momentum?

((mv^2)/2)/mv = (mv^2)/2mv = mv/2

Why is kinetic energy momentum squared and divided by 2?

W = ∫ F dr
  = ∫ ma dr
  = ∫ m dv/dt dr
  = ∫ m dr/dt dv
  = ∫ m v dv
  = 1/2 m v^2

P = ∫ F dt
  = ∫ ma dt
  = ∫ m dv/dt dt
  = ∫ m dv
  = mv