# 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 $g\left[\frac{m}{s^2}\right]$. 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 $mg$ is a force (weight) due to gravity and $\left( z_{2}-z_{1} \right )$ 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 $mgz$ 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 $\vec{V}\left[\frac{m}{s}\right]$ and can be evaluated in terms of the work required to change the velocity of the body. The product $\frac{1}{2}mV^{2}$ 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 ds.

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

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.