91Ó°ÊÓ

Mechanical energy is the sum of both kinetic energy and potential energy in a system. In this specific hill scenario, the mechanical energy of objects like the basketball or the can of juice revolves around how their energy transforms as they roll down the hill.
Since friction is ignored in this exercise, the mechanical energy becomes conserved. That means:

• The total energy remains the same at the start and at the bottom.
• Potential energy starts high at the top and converts into kinetic energy at the bottom.

In simple terms, as an object rolls down the hill, all of its stored energy (potential energy) becomes motion energy (kinetic energy). Therefore, the sum of kinetic and potential energy stays constant for both the basketball and the can of juice. This principle of conservation of mechanical energy is key in solving such physics problems.

Kinetic energy is the energy of motion. Any time an object is moving, it has kinetic energy. The faster it moves, the more kinetic energy it possesses. This is mathematically expressed by the formula:\[KE = \frac{1}{2}mv^2\]
where:

• \(m\) is the mass of the object
• \(v\) is its velocity.

The key to understanding this part of the exercise is realizing that as the basketball or the juice can rolls down the hill, potential energy is converted into kinetic energy. At the bottom, all the energy is in the form of kinetic energy. This situation shows how the velocity of an object connects directly to its energy. Hence, once you calculate potential energy at the top, determining the kinetic energy and thus the speed of the object at the bottom becomes straightforward.

Potential energy can be thought of as stored energy that has the potential to do work. When an object is elevated at a height, like at the top of the hill in our problem, it has gravitational potential energy. The formula for this energy is:\[PE = mgh\]
where:

• \(m\) is the mass of the object,
• \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\) on Earth),
• \(h\) is the height above the ground.

This form of energy is all about position. The higher the object is lifted, the more potential energy it holds. For the hill example, the potential energy at the top is completely converted to kinetic energy at the bottom. By using the critical idea that potential energy translates to kinetic energy without loss, we can determine details like speed at the base or the estimation of the hill's height, as shown in the solved steps.