The potential energy of a spring is:  U = ½kx²

A block connected to a horizontal spring sits on a frictionless table. The block is moved to compress the spring and is released from rest. What happens to the energy initially stored in the spring? What is the maximum speed of the block?

To solve, use D0EL   (energy conservation).

Step 1: Define/draw system and coordinates. (see transparency)

Initial position i and final position f (equilibrium position).

Step 2: Define potential energy zero.

Define the zero of the potential energy at the equilibrium point.

Step 3: Energy conservation statement.

U_i + K_i = U_f + K_f

Initial kinetic energy K_i = 0 because the particle is released from rest.
Final potential energy U_f = 0 by definition.

Conclusion: U_i = K_f   or   kx_i² = ½ mv_f²   →   v_f = 2 x_i (k/m)^½

The total energy remains constant and merely transforms back and forth between potential and kinetic.