Consider a mass m on the end of a spring of force constant k and constrained to move along the horizontal x axis. if we place the origin at the spring's equilibrium position, the potential energy is 2 kx 2 . 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 x max= a and then continues to oscillate about the origin.

(a) write down the equation for conservation of energy and solve it to give the mass's velocity x in terms of the position x and the total energy e.
(b) show that e = 2ka2 , and use this to eliminate e from your expression for x'
(c) solve the result of part (b) to give x as a function of t and show that the mass executes simple harmonic motion with period 27,/m1k.