# Kinetic and Potential Energy

### Potential Energy

The various forms of energy of interest to us are introduced in terms of a body having a mass m [kg]. This body can be solid, liquid, gas, or a system containing all the phases of matter. The various forms of energy include potential, kinetic and internal energy. Potential energy (**PE**) is associated with the elevation of the body and can be evaluated in terms of the work done to lift the body from one datum level to another under a constant acceleration due to gravity [latex]g\left[\frac{m}{s^2}\right][/latex]. Accordingly,

$$\Delta \text{PE} = \text{PE}_{2}-\text{PE}_{1}= m \times \Delta pe=\int_{z_1}^{z_2} mg dz.$$ If mass of the body of interest is constant, we may write

$$\Delta \text{PE} = mg\left( z_{2}-z_{1} \right ).$$ Note that [latex]mg[/latex] is a force (weight) due to gravity and [latex]\left( z_{2}-z_{1} \right )[/latex] is a distance. Force times distance is work and thus potential energy and work share the same units, Joules [J] in the SI system. The product [latex]mgz[/latex] is the gravitational potential energy, where *z *is measured with respect to some datum, typically the ground. Since mass is required to calculate potential energy, it is an *.*

### Kinetic Energy

Kinetic energy (**KE**) of a body is associated with its velocity [latex]\vec{V}\left[\frac{m}{s}\right][/latex] and can be evaluated in terms of the work required to change the velocity of the body. The product [latex]\frac{1}{2}mV^{2}[/latex] is the kinetic energy of a body, where velocity *V* is measured with respect to a datum, typically a motionless reference. So long as the mass of a body does not change from one state to another, we may express the change in kinetic energy as

$$\Delta \text{KE} = \text{KE}_{2}-\text{KE}_{1}= m \times \Delta ke= \frac{1}{2} m \left (V_{2}^{2}-V_{1}^{2} \right ).$$

The work associated with a change in kinetic energy can be expressed in terms of a force vector **F** and displacement vector *d***s. **

$$\int_{{s_1}}^{s_{2}} {\bf F} \cdot d{\bf s}$$

This force, regardless of direction, will be supplied by a body’s interactions with its surroundings, such as air drag, gravity, or a human hand. The energy that is imparted to a body through work to increase its velocity is stored as kinetic energy. Kinetic energy is is reduced with a body does work on its surroundings to reduce velocity. A ball moving air as it travels (air drag) is an example of a ball doing work on its surroundings.

An extensive property is a physical quantity whose value is proportional to the size of the system it describes, or to the quantity of matter in the system. For example, the mass of a sample is an extensive quantity; it depends on the amount of substance. The related intensive quantity is density which is independent of the amount. The density of water is approximately 1g/mL whether you consider a drop of water or a swimming pool, but the mass is different in the two cases.

Dividing one extensive property by another extensive property generally gives an intensive valueâ€”for example: mass (extensive) divided by volume (extensive) gives density (intensive).

Insert citation here: https://en.wikipedia.org/wiki/Intensive_and_extensive_properties