Total Mechanical Energy

The total mechanical energy is the sum of kinetic and potential energies:

E = K + U.

The law of conservation of total mechanical energy states that the sum of the kinetic energy and potential energy is constant in time. Conservation of total mechanical energy just means

E = Eo

where E0 is E(t=0), the intial energy. This can be rewritten as:

K + U = Ko + Uo or

m v2 + U = m vo2 + Uo.

Remember:

Total mechanical energy conservation only holds for conservative forces (hence the name), such as gravity.

Note that if there is an increase in K, then there must be a corresponding decrease in U.

K + U = Ko + Uo
=> K - Ko = -(U - Uo)
=> $\Delta$K = -$\Delta$U

The method of energy conservation allows the solution of otherwise difficult problems.

Please note:

We have said that the potential energy is only defined up to an additive constant. Here you can see that changes in kinetic energy are only caused by changes in potential energy, and any additive constant in U will cancel out.
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