Final answer:
The hollow sphere, having a smaller moment of inertia compared to the hollow cylinder, will roll to a higher vertical height on the incline when given the same initial velocity. This is due to a more efficient transfer of kinetic energy to potential energy for the hollow sphere.
Step-by-step explanation:
The problem at hand involves conservation of energy applied to rolling bodies and utilizes the concept of rotational inertia. A hollow cylinder and a hollow sphere, both of the same mass and radius, are given the same initial velocity and are allowed to roll up an incline. The key difference lies in their moments of inertia; a hollow sphere has a smaller moment of inertia than a hollow cylinder. This difference affects how far each object will roll up the incline against gravity before coming to a stop.
When the cylinder, which has a larger moment of inertia, rolls up the incline to a vertical height of 1.0 m, it has converted its initial kinetic energy to gravitational potential energy. Since the hollow sphere has a smaller moment of inertia, for the same initial kinetic energy, it will be able to convert more of this energy into potential energy before coming to a stop. This implies that it will reach a higher vertical height than the hollow cylinder on the same incline with the same initial conditions.