Consider a mass m on the end of a spring of force constant k and constrained to move along the horizontal x axis. let’s place the origin at the spring’s equilibrium position. the force is given via hooke’s law as fx(x) = −kx.
a) find the potential energy of the system as a function of k and x.

b) suppose that at time t = 0 the mass is sitting at the origin and is given a sudden kick to the right so that it moves out to a maximum displacement at xmax = a and then continues to oscillate. use conservation of energy to solve for the velocity of the mass in terms of the position x and the energy e. then show that e = 0.5ka2 , and use this to eliminate e from your expression for the velocity.

c) use your result from part b) to find the position x as a function of t and show that the mass executes a simple harmonic motion with period 2π(m/k)^1/2.