What is the principle of conservation of mechanical energy?

The principle of conservation of mechanical energy states that the total mechanical energy of a system remains constant.

Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE) of a system. The principle of conservation of mechanical energy states that the total mechanical energy of a system remains constant as long as there are no external forces acting on the system. This means that the sum of KE and PE at any point in time will always be equal to the sum of KE and PE at any other point in time.

Mathematically, this can be expressed as:

KE + PE = constant

where KE = 1/2mv^2 (kinetic energy) and PE = mgh (potential energy), where m is the mass of the object, v is its velocity, g is the acceleration due to gravity, and h is the height of the object above a reference point.

For example, consider a ball being thrown upwards. At the point of release, the ball has only KE. As it moves upwards, its KE decreases and its PE increases until it reaches its maximum height, where it has only PE. As it falls back down, its PE decreases and its KE increases until it reaches the point of release again, where it has only KE. At all points in time, the sum of KE and PE remains constant.

The principle of conservation of mechanical energy is a powerful tool in solving problems involving motion and energy. It allows us to simplify complex problems by considering only the initial and final states of a system, without having to consider the intermediate steps.