Final answer:
The statement about converting between potential and kinetic energy as a rock is thrown is false. Objects with the same momentum but varying mass differ in kinetic energy, and those with the same kinetic energy differ in momentum. Energy can transform between kinetic and potential forms without external work.
Step-by-step explanation:
Understanding Kinetic and Potential Energy
Regarding the physics problem presented, the statement that an increase in height would increase the rock's kinetic energy is false. When a rock is thrown into the air, its kinetic energy is converted to potential energy as it rises. Thus, the potential energy increases with height, not kinetic energy. Conversely, as the rock falls, its potential energy is converted back to kinetic energy, resulting in an increase in velocity and kinetic energy, not potential energy.
Now, for objects with the same momentum, the one with smaller mass has a higher kinetic energy (K = p²/2m). And for objects with the same kinetic energy, the one with the larger mass has a higher momentum (p = √(2mK)). This is due to the relationships defined by the equations of motion in classical mechanics.
Finally, the sum of kinetic and potential energy can change without work being done, such as when gravitational force causes an object to fall, converting potential energy to kinetic energy without external work.