Maximum Height of Ball

A solid ball is revolve toward a ramp. How elevated will it be when it stops and begins to roll back down?

The linear kinetic energy like the ball approaches the ramp is

K.E_.linear =1/2(mv²).

Because the ball rolls without slipping v = ωR.

A Rotational kinetic energy is

K.E_.rotational =1/2(2/5(mR²)) ω²

=1/5(mv²)

So the probable energy at the top of the rise is

P.E. = K.E_.linear + K.E._rotational

mgh =1/2(mv²)+1/5(mv²)

mgh =7/10(mv²)

h = (7/10)*(v²/g).

Therefore the height h doesn't depend on the ramp angle or the mass of the ball. It merely depends on the initial velocity and the acceleration due to gravity. Note that a few simplifications assumed by the solution are that no kinetic energy is lost when the ball strikes the ramp and that the heights are in fact those of the center of gravity of the ball.