Question

In which case does a ball reach a greater height when sliding up a frictionless ramp and rolling without slipping up another frictionless ramp?

a) Sliding up the ramp.
b) Rolling up the ramp.
c) Both cases result in the same height.
d) It depends on the ball's initial velocity.

Answer 1

Both a sliding ball and a rolling ball on frictionless ramps will achieve the same height due to energy conservation; however, the rolling ball will take longer to reach that height because of its rotational motion.

Step-by-step explanation:

In determining which case allows a ball to reach a greater height when traveling up a frictionless ramp, we must consider energy conservation and how the initial kinetic energy is transformed. When a ball slides without friction, all of the initial kinetic energy is translational. However, when a ball rolls without slipping up the same ramp, part of its initial kinetic energy is rotational in addition to translational. Because a rolling ball has to distribute its kinetic energy into both rotational and translational forms, it theoretically will not reach as great a height as the sliding ball, which puts all its energy into translational kinetic energy. Nevertheless, in a perfectly frictionless scenario, since no energy is lost, both balls should reach the same height, which is equal to the height they would have if they started from rest at the top. The only difference is that the rolling ball will take a longer time to reach its maximum height due to the simultaneous rotation.