Any given spring anchored at one end has what is called a “spring constant,” k. This constant linearly relates the spring’s restoring force to the distance it is distended. The end has what is called an equilibrium point, its position when the spring has no stresses on it. After a mass attached to the free end of the spring is released, it oscillates back and forth. Its kinetic energy and potential energy stay constant. As the mass passes through the equilibrium point, the kinetic energy reaches its maximum. You can calculate the kinetic energy at any point based on the spring’s potential energy when initially released.
Determine the spring’s initial potential energy. From calculus, the formula is (0.5)kx^2, where x^2 is the square of the initial displacement of the end of the spring. The kinetic and potential energy at any point will sum to this value.
Identify the spring’s maximum kinetic energy, at the equilibrium point, as equal to the initial potential energy.
Calculate the kinetic energy at any other point of displacement, X, by subtracting the potential energy at that point from the initial potential energy: KE = (0.5)kx^2 - (0.5)kX^2.
For example, if k=2 Newtons per centimeter and the initial displacement from the equilibrium point was 3 centimeters, then the kinetic energy at 2 centimeters of displacement is (0.5)2_3^2 - (0.5)2_2^2 = 5 Newton-meters.