A ball is released from a vertical height of 40 cm. It rolls down a "perfectly frictionless" ramp and up similar ramp. What vertical height on the second ramp will the ball reach before it starts to roll back down?

Well, first of all, if the ramps are perfectly frictionless, then the ball

won't roll ... it'll slide. Kind of like a bowling ball does at the beginning

of a shiny, highly-waxed lane. You need friction against the ball to roll it.

But in any case, no matter what the slopes of the ramps are, even

if they have different slopes, the ball will rise on the second ramp

to the same height from which it was released on the first ramp.