Mechanical energy is the energy associated with an object's motion and its position. In other words, an object's mechanical energy is the sum of its kinetic energy and its potential energy.

Principle of the Conservation of Mechanical Energy

As long as the only forces acting are conservative forces, the total mechanical energy is constant.

ΔU + ΔK = 0

U_i + K_i = U_f + K_f

This is just our master equation with the work term set to zero.

Mechanical energy can be lost when non-conservative forces act (like friction), but the energy merely gets transformed into other kinds of energy.

The Law of the Conservation of Energy

Energy can neither be created nor destroyed, just transformed from one kind to another.

Mechanical energy that is lost or gained generally comes from work done by non-conservative forces. Call this W_nc:

ΔU + ΔK = W_nc

U_i + K_i + W_nc = U_f + K_f

A law is far more fundamental than a principle. Laws apply all the time. Principles have fine print.

If you slide a book across a table, it comes to rest. What happened to the kinetic energy?

Friction is very good at turning kinetic energy into thermal energy.

Two identical balls are released from rest at the same time. Ball A travels down a straight incline to the floor, while ball B travels down a steeper slope and then takes an almost horizontal path to the bottom of ball A's incline.

Which ball wins the race?

1. Ball A
2. Ball B
3. It's a tie

The one thing that should be equal for the two balls is the speed at the bottom. They both experience the same change in potential energy, so their final kinetic energies should be equal.

Ball A travels the shortest path, but it picks up speed gradually all the way down the incline. In contrast, ball B drops quickly and reaches top speed quickly. It then travels at this top speed for the rest of the trip. B wins the race because its larger average speed more than makes up for the extra distance traveled.

A heavy weight is dropped from a height of 1.2 m on a soda can, crushing it.

Is mechanical energy conserved in this process?

1. Yes
2. No

No. The potential energy decreases and the kinetic energy stays the same (zero before and after), so overall there's a loss of mechanical energy.

Is energy conserved in this process?

1. Yes
2. No

It had better be! If not we're breaking one of the fundamental laws of physics. The mechanical energy lost must have been transformed into other forms of energy (heat, sound, etc.). The total energy in the system should be the same.