Mechanical energy refers energy that depends on the position or motion of masses, not other types of energy associated with heat, light, electricity, etc. We divide mechanical energy into to types: potential energy (PE) and kinetic energy (KE). Potential energy is energy that depends on the position or location of the mass, and kinetic energy is energy associated with the velocity of the mass. Both types of energy are defined by the amount of work needed to produce a given situation, so in general the work done on a system will be equal to the change in its potential and/or kinetic energy:

W = DKE + DPE

where, as usual, the symbol “D” means “the change in”, which will be positive if the quantity increases and negative if the quantity decreases.

## Potential Energy

Potential energy refers to energy that can be used to perform work at some later time. In other words, it is stored energy. We can store chemical energy (gasoline), electrical energy (batteries), or other kinds of energy, but right now we are only concerned with mechanical energy.

Mechanical potential energy always involves changing the location or position of a mass. For example, to store energy in a compressed spring you must move the end of the spring to a new location. In general,

 The amount of potential energy stored in a configuration is equal to the amount of work done to put the system in that configuration.

Notice that this definition doesn’t say anything about what configuration the system was in to begin with. To this extent, potential energy is an arbitrary number because it depends on what you want call “zero.” In most practical situations, though, there is an obvious reference point to define as zero potential energy. With the compressed spring, for instance, we would say that its zero-position is when it is at its relaxed length, and its potential energy is the work done in compressing it from its relaxed state.

### Gravitational Potential Energy

Gravitational potential energy is energy that a mass has because of its altitude. Lifting a mass up to a given height takes work, and this energy can be stored indefinitely. When the mass is allowed to fall back to its original height, this energy is recovered and can be used to do work. This is the principle of a hydroelectric dam: nature does the work of lifting the water up into the mountains, we retain it behind a dam until we are ready to use it, and then we let the water fall through a turbine, which is used to generate electricity.

To calculate gravitational potential energy we need to calculate force multiplied by distance. The average force needed to lift a mass is exactly equal to its weight, if the mass ends up with zero velocity (if it had velocity then it would also have some kinetic energy, which is discussed in the next section). Thus, the work to lift a mass m a height h is

PE = (Force)(distance)

PE = (mg)(h)

PE = mgh

As mentioned before, the origin is arbitrary. You could be lifting the mass from the ground, from the floor, or from a tabletop. It does not matter, as long as the zero-point is clearly specified.

### Potential Energy is Path-Independent

What if the mass in the previous example was not lifted straight up? It could have been pushed up a ramp, or it may have first been lifted to a height above h and then lowered back down. It turns out that this doesn’t matter. The work done against gravity would still come out to be exactly mgh, regardless of the path the mass followed. Of course, following a different path might cause some extra work to be done against friction, but we don’t count that. When calculating gravitational potential energy, you only consider the work done against the gravitational force.

• Potential energy depends only on the final configuration of the system, not on how it got there.

## Kinetic Energy

A mass m traveling at a velocity v has a kinetic energy given by the formula It makes sense that the energy depends on both mass and velocity. Imagine throwing rocks against the wall. The size of the dent in the wall would be an indicator of how much energy the rock was carrying. The dent would be bigger if you used a heavier rock, or if you threw it harder.

The fact that kinetic energy depends on the square of the velocity is significant. It means that a car going 60 mph has four times as much kinetic energy as a car going 30 mph, because if the speed is doubled, then v2 gives a factor of four. This is seen in the fact that braking distances get much longer at higher speeds—to stop the car you need to bleed off all the kinetic energy.

### Deriving the Kinetic Energy Formula

Suppose we have an object of mass m starting from rest. We apply a constant force F, until the mass reaches a velocity v.

From Newton’s second law, the acceleration will be a = F/m. The acceleration will be constant because F is constant, and so we can use the equations for constant acceleration, specifically In this case the mass is starting from rest, so v0 = 0, and for simplicity we will represent the interval xx0 with d. Now the formula looks like If we use the substitution a = F/m, we get or But we recognize that Fd is the work done, which by definition is equal to the kinetic energy given to the mass, and so In the previous derivation, we did not have to require that the mass start from rest. We would find that any change in kinetic energy is accompanied by an equivalent amount of work.

Proof: let a force F act on mass m for a distance d. Then a = F/m and   W = KK0

Or W = DK

The W refers to work done on the mass by some external force, so that an increase in kinetic energy means that a positive amount of work was done on the mass, and a decrease in kinetic energy means that a negative amount of work was done on the mass.