Mechanical work is the energy transferred by a force F over a displacement d. Work is a scalar quantity (since it’s the dot product of two vectors) and we can derive the theorem very easily.

Multiplying both sides of “eff-equals-em-ey” by a tiny displacement in the (let’s say) x direction, we get:

We know from the equations of motion,

Substituting for we get,

Now we see that work here is just the change in kinetic energy. When we include non-conservative (NC) friction (as true physical systems do) we get some definitions for work:

The Work and Energy Theorem
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Here, U is the potential energy and E_k is the kinetic energy. Using the definition of the potential energy above, we can find the U associated with any force by integrating. The table below shows a few examples: