Solving Spring Force Problems: Showing Maximum Displacement & Velocity

[aboutguava]

I'm just having some trouble with this problem. Any help would really be appreciated. I'm not even sure where to start.

A spring (not ideal) supplies a force given by F = zx^2, where x is measured from the equilibrium position and z is a constant. A mass m is attached to the spring and then released from rest at x = 0 (non-stretched spring)

a) show that the maximum downward displacement of the mass will be (3mg/z)^(1/2)
b) At what displacement will the mass have its maximum velocity

Thank you

 

[Doc Al]

Start by deriving the spring potential energy as a function of displacement x. (Hint: How much work is required to stretch the spring?)