Final answer:
The question explores the conservation of mechanical energy for rolling objects and calculates the height reached using initial velocity, with energy split into translational and rotational kinetic energies.
Step-by-step explanation:
The question pertains to rolling motion in the context of physics and is concerned with finding the height that different objects reach when they roll up an incline without slipping, given their initial velocity. For example, a ball treated as a spherical shell with an initial velocity is analyzed to determine the height it attains when it rolls up a hill without slipping.
Here's how to calculate the height reached by the rolling object: The total mechanical energy (kinetic plus potential energy) of the object is conserved. The initial kinetic energy is transformed into gravitational potential energy as the object ascends, leading to a reduction in its velocity due to gravity. If no slipping occurs, the kinetic energy is divided into translational and rotational energy. The conservation of energy principle allows us to equate the initial kinetic energy with the potential energy at the height achieved.