Why is kinetic energy a cos² function for U(x)=4x² with E=7J?

[dmayers94]

Homework Statement

A potential energy function is given by U(x) = 4x^2 and I found the total energy to be 7 J. They ask: Graph position vs. time and kinetic energy vs. time using this information and assume that the particle has an initial position of 0.

The Attempt at a Solution

I got the first part correct. It is just a sine curve because the object oscillates and is bounded by the total mechanical energy of 7 J. The second graph, KE vs. time, is a cos^2 function, but I'm not sure why. I know that as the potential energy increases, position increases, and when potential energy is at a max, KE is 0. The shape makes sense to me when I compare it to the position graph, but how did they come up with cos^2 mathematically? Why not cos(2x) instead?

 

[BruceW]

You need to work back from the main principles, to get the reason for using a cos^2 function. What is the main principle between energy, potential energy and kinetic energy? And what is the equation for this relationship?