Problem $3.$

Rolling off a cliff.

Relevant learning goals:

$-$ Understand the motion of objects that are rolling without slipping on a smooth surface.

$-$ Analyze problems using the conservation of mechanical energy involving rotating objects.

Subskills: $1$

A solid uniform ball rolls without slipping up a hill. At the top of the hill, it is moving horizontally, and then it goes over the vertical cliff.

$($a$)$ How far from the base of the cliff does the ball land, and what is its speed just before it lands?

$($b$)$ Notice that when the ball lands, it has a greater translational speed than when it was at the bottom of the hill. Does this mean that the ball somehow gained energy? Explain!