Work and Energy

We first define work as the product of the component of the force in the direction of motion times the displacement. If theta is the angle between the force and the displacement, then

To define kinetic energy, start with this equation if motion:

Now, multiply both sides of this equation by the mass divided by 2:

Note that

and

Using these two equations and rearranging, we get:

We define the first term on the right to be the final kinetic energy and the second term to be the initial kinetic energy. (The above result assumes that the altitude of the object did not change.)

Work equals the change in kinetic energy.

The work done in lifting an object of mass from a vertical distance above a reference point , to a vertical distance :

Remove the parenthesises:

We define the first term on the right to be the final potential energy and the second term to be the initial potential energy:

Work equals the change in potential energy

In the more general case where both the speed and the altitude change:

For the special case were there is no work done:

The total initial energy must equal

the total final energy.