Physics with Calculus/Mechanics/Energy and Conservation of Energy/Kinetic energy

Kinetic energy is the energy of motion regardless of the direction. Any object (with mass) in motion has kinetic energy. There are several types of kinetic energy. It should be noted that many times several of these types of kinetic energies occur simultaneously.

Translational
This is the energy due to motion from one location to another. It is defined by the integral of momentum with respect to velocity:

$$K = \int \mathbf{p}\,dv = \int m\mathbf{v}\,dv = \frac{1}{2}m\mathbf{v}^2$$

Rotational
This is the energy due to rotation around one point (the center). It is defined by the integral of angular momentum ($$\mathbf{L}$$) with respect to angular velocity ($$\mathbf{\omega}$$):

$$K_r = \int \mathbf{L}\,d\omega = \int I\mathbf{\omega}\,d\omega = \frac{1}{2}I\mathbf{\omega}^2$$