We wish to slide a ball from rest at the top of an incline of fixed height y to the end of a smooth track of length d. The goal is to design the apparatus so as to minimize the time it takes for the ball to get from the top of the incline to the end of the track. The horizontal length of the incline, x, may be varied from 0 to d, but all other values are fixed.

We neglect air resistance and all frictional forces, and assume that energy is conserved (particularly, assume that at the bottom of the slope there is a smooth juncture). We slide the ball instead of rolling it to avoid the complications of rotational kinematics.

And the answer is...