Conservation of Mechanical Energy

Let's take the average human's day, for example, to demonstrate the law of conservation of energy. Most humans have to do tasks during their day which require mental attention and even physical exertion. When they do these tasks for long periods of time, they'll start to feel tired and not have as much energy. They can gain it back when they eat and sleep so that the next day, they can repeat these tasks again in a similar fashion. This is the basic principle behind the conservation of energy. This law states that energy can't be created or destroyed, but only converted. Another way of saying this is that a closed system's energy doesn't change its value, it simply changes form.

This same principle is explained through the first law of thermodynamics

The more mathematical definition of the law of conservation of mechanical energy is:

This states that the sum of kinetic and potential energies of an object is constant at all times.

A conservative force is simply a force that acts such that the total work the force does on an object depends only its final and initial states, not its path. An example of a conservative force is gravity: if you move an object up and down repeatedly such that its final height equals its initial height, gravity did 0 total work on the object because the changes in gravitational potential energy that are negative(object moving down) cancel out the changes that are positive(object moving up). Another way to look at this mathematically is that if a force is conservative, it follows this law:

Non-conservative forces are pretty much any force that doesn't follow this law or adhere to the statement made in the first sentence of the first paragraph above.