Conservation of Energy

If a body of mass m is thrown vertically upwards with velocity u at A, it has to do work against the constant force of gravity. When it has risen to point B its reduced velocity is v.

By definition of kinetic energy, loss of k.e. between A and B = work done against gravity. By the defintion of potential energy, gain of p.e. between A and B = work done against gravity.

Loss of k.e = gain in p.e

Therefore 0.5mu^2 - 0.5mv^2 = mgh 

This is called the principle of conservation of mechanical energy:

The total amount of mechanical energy (k.e. + p.e.) which the bodies in an isolated system possess is constant.

It only applies to frictionless motion. Otherwise, in the case of the rising body, work has to be done against friction as well as against gravity and the body gains less p.e. than when friction is absent.

Work done against frictional forces is generally accompanied by a temperature rise. This suggests that we might include in our energy accountancy what is called internal energy. This would then extend the energy conservation principle to non-conservative systems, and we can then say, for example. loss of k.e. = gain of p.e. + gain of internal energy.

The principle of conservation of mechanical energy is a special case of the more general principle of conservation of energy - one of the fundamental laws of science:

Energy may be transformed from one form to another, but it cannot be created or destroyed, i.e. the total energy of a system is constant.