Final answer:
The rolling ball reaches a greater height on a frictionless ramp compared to the sliding ball because it has both translational and rotational kinetic energy, which when conserved, allows it to ascend higher due to the additional rotational energy being converted to potential energy.
Step-by-step explanation:
The question at hand explores the concepts of kinematics and rotational dynamics in the context of objects moving on an inclined plane. In the given scenario, we have two different situations where a ball moves up a frictionless ramp. In the first situation, the ball slides up the ramp; in the second, it rolls without slipping up another ramp of the same slope angle, both starting with the same initial velocity.
The key difference between sliding and rolling involves the conservation of energy. When the ball slides up the ramp, all of its kinetic energy is in the form of translational kinetic energy (translational kinetic energy). However, when the ball rolls without slipping, its kinetic energy is a combination of translational and rotational kinetic energy (rotational kinetic energy). Since both balls have the same initial translational kinetic energy and the rolling ball has additional rotational kinetic energy, the rolling ball has more total kinetic energy. This additional energy will allow it to reach a greater height on the ramp.
In the case of the rolling ball, the conservation of total mechanical energy allows us to conclude that since no external forces are doing work (the ramp is frictionless), the initial mechanical energy is conserved. The increase in mechanical energy for the rolling case comes from the initial rotational kinetic energy, which is then converted to gravitational potential energy as the ball rises up the ramp. Therefore, the rolling ball reaches a greater height compared to the sliding ball because it possesses additional energy in the form of rotation.