Analyzing the mechanical energy of the system, we have that:
When the object is abandoned at position x = A, the mechanical energy of the system is equal to the stored potential elastic energy, as there is no movement and hence kinetic energy. Like this:
When it reaches position x = -A, again the object will be momentarily stationary (v = 0), having its mechanical energy equal to the potential elastic energy of the system.
At the point where x = 0, the opposite phenomenon will occur to the maximum elongation, where:
Thus we can conclude that at position x = 0, the maximum velocity of the spring-mass system occurs, since all mechanical energy is the result of this velocity.
For all other points in the system:
Since there is no energy dissipation in this model, all mechanical energy is conserved during the movement of a horizontal spring-mass oscillator.