Three identical solid spheres move doen through three inclined planes A,B and C all same dimensions. A is without friction B is undergoing pure rolling and C is rolling with slipping. Compare the kinetic energies `E_(A),E_(B)` and `E_(C)` at the bottom.

Explanation

If these objects slide down a frictionless ramp from the same height, they would reach the bottom in equal time. The masses nor their densities matter. The potential energy is mgh. The K.E of these objects `1/2mv^(2)`. Setting these two equal mgh `=1/2mv^(2),v^(2)=2gh`. The final velocity and thus travel time does not depend on mass.
Rolling changes things, now part of the potential energy turn into rotatiional kinetic energy. The object contains the most rotational energy has the least translational K.E is `1/2 cmv^(2)` where c is coefficient governing the type of rolling object. Now potential energy is distributed between both rotational andkinetic energy according to the equation `mgh=1/2m(c+1)v^(2)` again mas sis irrelevant and `v^(2)=2gh(c+1)`. The larger the value of c the slower the final velocity and the longer it takes for the object to react the bottom.