Final answer:
The maximum kinetic energy of a simple harmonic oscillator is found at its equilibrium position, which is x = 0, where all the system's energy is kinetic energy.
Step-by-step explanation:
The maximum kinetic energy of a simple harmonic oscillator occurs when it passes through the equilibrium position, at x = 0. At this point, all the energy of the system is in the form of kinetic energy, since the potential energy is zero. Conversely, at the turning points (x = ±A), the oscillator's speed is zero, meaning that all the energy is potential energy, and the kinetic energy is thus at a minimum. In a simple harmonic motion, energy is constantly being exchanged between kinetic and elastic potential energy, but the total energy remains constant throughout the motion.