The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.

Suppose that a body undergoes displacement Δ*x *under the action of a conservative force *F*. Then from the WE theorem we have,

If the force is conservative, the potential energy function *V*(*x*) can be defined such that

The above equations imply that which means that *K + V, *the sum of the kinetic and potential energies of the body is constant.

Over the whole path, *x _{i} *to

*x*this means that

_{f},The quantity *K *+*V*(*x*), is called the total mechanical energy of the system. Individually the kinetic energy *K *and the potential energy *V*(*x*) may vary from point to point, but the sum is a constant.

The total mechanical energies E_{0}, E_{h}, and E_{H} of the ball at the indicated heights zero (ground level), h and H, are

a result that was obtained for a freely falling body.

Further,

which implies,

and is a familiar result from kinematics.

At the height H, the energy is purely potential. It is partially covered to kinetic at height h and is fully kinetic at ground level. This illustrates the **conservation of mechanical energy**.

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