Final answer:
When an object starts from rest, its initial velocity is 0 m/s. To determine the final velocity of a hoop rolling down a hill, we use energy conservation, equating the hoop's initial potential energy to its final kinetic energy, which includes both translational and rotational components.
Step-by-step explanation:
When an object of mass 5kg starts from rest, its initial velocity is 0 m/s. This is because the term 'starting from rest' in physics means that the object's initial state of motion is such that it has no velocity. In your specific question asking about the final velocity of a hoop rolling without slipping down a 5.00-meter-high hill, we can use the principles of conservation of mechanical energy for a rolling object. Since the hoop starts from rest, all its initial mechanical energy is in the form of gravitational potential energy, which then converts to kinetic energy as the hoop rolls down the hill.
To find the final velocity of the hoop when it reaches the bottom of the hill, we can set the initial potential energy equal to the final kinetic energy. The kinetic energy of a rolling object is the sum of its translational kinetic energy (1/2 m v^2) and its rotational kinetic energy (1/2 I ω^2), where m is the mass, v is the linear velocity, I is the moment of inertia, and ω is the angular velocity. For a hoop of mass m and radius r, the moment of inertia I is given by mr^2, and by the condition of rolling without slipping, v = rω. This leads to a final kinetic energy expression of (1/2 m v^2 + 1/2 I ω^2) = (1/2 m v^2 + 1/2 mr^2 ω^2/v^2) for the hoop.