The total mechanical energy of the system at any time t is the sum of the kinetic energy of motion of the ball and the elastic potential energy stored in the spring:
[tex]E=K+U= \frac{1}{2}mv^2+ \frac{1}{2}kx^2 [/tex]
where m is the mass of the ball, v its speed, k the spring constant and x the displacement of the spring with respect its rest position.
Since it is a harmonic motion, kinetic energy is continuously converted into elastic potential energy and vice-versa.
When the spring is at its maximum displacement, the elastic potential energy is maximum (because the displacement x is maximum) while the kinetic energy is zero (because the velocity of the ball is zero), so in this situation we have:
[tex]E=U_{max}= \frac{1}{2}k(x_{max})^2 [/tex]
Instead, when the spring crosses its rest position, the elastic potential energy is zero (because x=0) and therefore the kinetic energy is at maximum (and so, the ball is at its maximum speed):
[tex]E=K_{max}= \frac{1}{2}m(v_{max})^2 [/tex]
Since the total energy E is always conserved, the maximum elastic potential energy should be equal to the maximum kinetic energy, and so we can find the value of the maximum speed of the ball:
[tex]U_{max}=K_{max}[/tex]
[tex] \frac{1}{2}k(x_{max})^2 = \frac{1}{2}m(v_{max})^2 [/tex]
[tex]v_{max}= \sqrt{ \frac{k x_{max}^2}{m} }= \sqrt{ \frac{(12 N/m)(0.20 m)^2}{0.4 kg} }=1.1 m/s [/tex]
