Consider a thin hoop and a uniform cylinder of equal mass and radius competing in a race down a ramp. 

a. Write a general formula for the conservation of energy for both items.

b. Determine the velocity of the hoop at the bottom of the ramp using energy conservation. 

c. Determine the velocity of the cylinder at the bottom of the ramp using energy conservation. 

d. What physical variables are the final velocities independent of? How do you know?