Final answer:
The box, sliding without friction, would reach the bottom of an incline sooner than a rolling disk because it has a higher final velocity due to its entire gravitational potential energy converting into translational kinetic energy. The respective times to reach the bottom depend on the final velocities, which are different for sliding and rolling objects.
Step-by-step explanation:
The time it takes for both a box and a disk to reach the bottom of an incline can be determined based on their modes of motion. The disk, being a solid cylinder, will roll down the incline without slipping, a situation for which the conservation of energy gives a final velocity v = (4gh/3)^1/2. However, if the object simply slides down without friction and without rolling, then the final velocity is given by v = (2gh)^1/2, which is 22% greater than the rolling case. Therefore, the sliding object reaches the bottom sooner because its motion involves only translational kinetic energy, whereas the rolling disk also involves rotational kinetic energy, which takes away some of the energy that would otherwise be used for translation.
To find out exactly how much sooner the box reaches the bottom compared to the disk, we need to calculate the time it takes for each to reach the bottom and then subtract the two. The time can be calculated using the kinematic equation t = (2h/g)^1/2/v, where h is the height of the incline and g is the gravitational acceleration. By substituting their respective final velocities into this equation, you can find the difference in time.