The kinetic and potential energies of the ball are equal at point III, located 0.5 m above point II, because of the conservation of mechanical energy in the absence of air resistance.
The kinetic and potential energies of the ball will be equal at point III, where the ball is located 0.5 m above point II. Given that the potential energy is zero at point II and air resistance is negligible, energy conservation principles apply. As the ball swings upwards from point II to point IV, it loses kinetic energy and gains potential energy.
At point III, the ball would have lost half of its maximum kinetic energy (which was at point II) and gained an equivalent amount of potential energy (as per the conservation of energy). Therefore, at point III, the kinetic energy (which was maximum at point II) and the potential energy (which started zero at point II and increased to this point) will be equal.
In conclusion, option C, at point III, is where the ball's kinetic and potential energies are equal because this corresponds to the ball having ascended half of the maximum height (assuming a symmetrical path).