A solid sphere and a hoop have the same radius R and the same mass M. They are each released from rest from the same height H at the top of an...

A solid sphere and a hoop have the same radius R and the same mass M. They are each

released from rest from the same height H at the top of an inclined plane. They roll

without slipping to a point of height h then leave the ramp at an angle (Theta). They reach a

maximum height Y above the ground and land a horizontal distance D from the launch

point.

a) Derive an expression for the angular speed of an object at the apex of flight. Simplify as

much as possible. Hint: Are there external torques on a projectile?

b) Derive an expression for the distance D. Simplify as much as possible. Which object

travels further? Explain this in terms of energy conservation.