What is the kinetic energy and velocity of the aluminum...

What is the kinetic energy and velocity of the aluminum sphere in Problem 1.4 at the moment it hits the ground? (Assume that energy is conserved during the fall and that 100$\%$ of the sphere's initial potential energy is converted to kinetic energy by the time impact occurs.)

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (such as friction or air resistance), the total mechanical energy of a system remains constant. In the context of an object falling under gravity, the gravitational potential energy lost by the object is exactly converted into kinetic energy. This concept is crucial for solving problems where energy conversion determines the motion of the object.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. It is typically calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above a reference point. In problems involving free fall, this energy is converted into kinetic energy as the object descends.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is given by the formula KE = ½ mv², where m is the mass of the object and v is its velocity. When energy conservation applies, the kinetic energy at the bottom of a fall is equal to the initial gravitational potential energy at the top.

Velocity Calculation

Calculating the velocity of a falling object involves rearranging the kinetic energy formula after equating kinetic energy to the initial gravitational potential energy. By solving the equation ½ mv² = mgh for v, one obtains the velocity v = ?(2gh). This derivation is essential for determining how fast an object is moving at any point when energy conversion occurs solely due to gravity.

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