Final answer:
The sphere that slides down an incline with no friction reaches the ground first because it converts all its potential energy into translational kinetic energy, unlike the rolling sphere which also converts some energy into rotational kinetic energy.
Step-by-step explanation:
When considering two spheres, one that rolls without slipping and one that slides down an incline with no friction, the sphere that rolls will reach the ground later than the one that slides.
This happens because the rolling sphere has rotational kinetic energy in addition to translational kinetic energy, and since they both have the same potential energy at the start (due to having the same mass and being at the same height), the sphere that rolls converts part of this potential energy into rotational energy, meaning that it will have less translational kinetic energy and thus will move slower down the ramp.
The detail of this result is explained by considering the conservation of energy. For the sliding sphere, all of its potential energy is converted into translational kinetic energy, while for the rolling sphere, the potential energy is split between translational kinetic energy and rotational kinetic energy, which is determined by the object's moment of inertia and angular velocity.
Finally, since rolling involves friction (which is what prevents slipping), there is energy dedicated to rotating the sphere, whereas in the frictionless case, all the energy goes to linear motion. This results in a lower final velocity for rolling down the ramp and hence, it takes more time for the rolling object to reach the bottom.