Two identical balls roll across the top of a table on parallel paths. One of the balls has to roll down into and up out of a dip in the table. The other ball rolls on the flat all the time. Which ball gets to the far side of the table first, and why?.

i think i remember now...

assuming smooth 'dip' and there is no loss of energy....

the 'roll' of the ball is to be maintained. So the initial energy is sum of linear and angular motion.. 1/2 mv^2 + 1/2 Iw^2 (where I is angular inertia of ball (mr^2 for shell, i think its mr^2 /3 for filled sphere) and w is angular velocity (v/r) )

a small potential energy will be added to this of form mgh (where h is depth of dip from surface level). so both linear and angular velocity will increase.

So possibly assuming a certain shape (maybe a half circle) of 'dip' one could find the difference in the time taken.