Potential energy is associated with forces and is stored within an object. It depends upon the position of the object relative to its surroundings. An object raised off the ground has gravitational potential energy because of its position relative to the surface of the earth.This energy represents the ability to do work because, if the object is released, it will fall under the force of gravity and do work upon what ever it lands on. For instance,dropping a rock on a nai lwill do work on the nail by driving it into the ground.

Suppose an object is moving in a straight line at velocity v₀. To increase the velocity of the object up to v₁, a constant force F_net would need to be applied to the object. The work W done on an object by a constant force F is defined as the product of the magnitude of the displacement d multiplied by the component of the force parallel to the displacement, F_||

W = F_||d. (Equation 1)

In the case of the moving object, if the force is applied in the direction parallel to the motion of the object, then the net work is simply equal to the net force times the distance traveled:

W = F_netd. (Equation 2)

From kinematics, it is known that the final velocity of an object under constant acceleration is:

v₁² = v₀² + 2ad. (Equation 3)

Applying Newton's second law, F_net = ma, and solving for the acceleration in Equation 3 gives:

W_net = F_net d = mad = md(v₁² - v₀² )/(2d) = (v₁² - v₀² )/2. (Equation 4)

Equivalently:

W_net = ½ m v₁²-½ m v₀². (Equation 5)

If translational kinetic energy is defined as KE = ½ mv², then this is just the work-energy principle: the net work done on a system is equal to the change in the kinetic energy of the system.

Now consider gravitational potential energy. If an object starting from a height h falls from rest under the influence of gravity, the final velocity of the object can be found using Equation 3

v² = 2gh. (Equation 6)

After falling from height h, the object has kinetic energy equal to ½ mv² = ½ m(2gh) = mgh. This is the amount of work the object can do after falling a vertical distance h and is defined as the gravitational potential energy, PE:

PE = mgh, (Equation 7)

where g is the gravitational acceleration. The higher the object is placed above the ground, the more gravitational potential energy it has. Gravity is acting, or doing work, on the object, so in this scenario, W_net = mgh. From the work-energy principle, it is known that this gravitational potential energy should then be equal to the change in kinetic energy:

½ mv² = mgh. (Equation 8)

Sample calculated values of the initial potential energy at various heights are listed in the PE column of Table 1, found using Equation 7. The final velocities measured from the experiment are also in the table. The translational kinetic energy is calculated using these measured values of the final velocity. According to the work-energy theorem, the KE and PE columns in the table should be equal, and they nearly are. The discrepancies in the two values simply come from errors in the measurements taken throughout the experiment, where a percent difference of around 10% can be expected from this type of experiment.

Note that as the initial height is increased, the final velocity also increases at a rate that is proportional to the square root of the height increase (c.f. Equation 6). The potential energy of the system also increases with increased height. Furthermore, note that the cart with the increased mass (the last three rows in Table 1) has both higher potential energy and kinetic energy when compared to the lower-mass cart (first three rows), but the final velocities of this cart are the same as for the lower-mass cart. This makes sense because the final velocity is only a function of the height (Equation 6).

Table 1: Results.

Applications of the work-energy principle are ubiquitous. Roller coasters are a good example of this energy transfer. They pull you up to a great height and drop you down a steep incline. All the potential energy that you gain at the top of the incline is then converted to kinetic energy for the rest of the ride. The coasters are also massive, which adds to the potential energy. Skydivers use this principle as well. They ride in an airplane that does work on the system to bring them to a height of around 13,000 feet. Their initial velocity in the vertical direction is nearly zero just before they jump out, and they quickly reach terminal velocity (because of air resistance) after jumping. Firing a gun also converts potential energy to kinetic. The gunpowder in the ammunition has a lot of stored chemical potential energy. When it is ignited, it does work on the bullet, which exits the muzzle with a tremendous amount of kinetic energy.

The work-energy principle has been derived in this experiment. Using a glider on an inclined air track, the work done by gravitational force has been experimentally verified to equal the change in the kinetic energy of the system.