The short answer is that it has to do with the integration of the momentum term in the derivation of the kinetic energy.
The more formal answer is:
Strangely enough I only thought about this while I was reading on a completely different topic and noticed that the equation given for the potential energy stored in a spring is E = 1/2kx^2. I occurred to me that this expression looked surprisingly similar to the expression for kinetic energy. It all became clear to me when I realized that the momentum is a linear function of the velocity (p=mv) as is the force a linear function of the displacement (F=kx).
The more formal answer is:
Strangely enough I only thought about this while I was reading on a completely different topic and noticed that the equation given for the potential energy stored in a spring is E = 1/2kx^2. I occurred to me that this expression looked surprisingly similar to the expression for kinetic energy. It all became clear to me when I realized that the momentum is a linear function of the velocity (p=mv) as is the force a linear function of the displacement (F=kx).