Final answer:
The increase in the height of a thrown rock does not increase its potential energy.
Step-by-step explanation:
The statement, 'If a rock is thrown into the air, the increase in the height would increase the rock's kinetic energy, and then the increase in the velocity as it falls to the ground would increase its potential energy,' is false.
When a rock is thrown into the air, its kinetic energy is converted to potential energy as it gains height. As it falls back to the ground, its potential energy is converted back to kinetic energy, but it doesn't gain additional potential energy. The total mechanical energy of the rock, which is the sum of its kinetic and potential energy, remains constant throughout the motion.
Therefore, the correct option is False.