Definition of KE
For a single particle of mass m moving with a velocity v, the kinetic energy is defined as
- [math]T \equiv \frac{1}{2} mv^2[/math]
Work Energy Theorem
Derivation
Consider the Kinetic Energy's temporal rate of change assuming that the mass of the particle is constant
- [math]\frac{dT}{dt} = \frac{m}{2} \frac{d}{dt}v^2= \frac{m}{2} \frac{d}{dt}\vec v \cdot \vec v[/math]
- [math]= \frac{m}{2} \left (\vec \dot v \cdot \vec v + \vec v \cdot \vec \dot v \right )[/math]
- [math]= \frac{m}{2} 2 \vec \dot v \cdot \vec v = \vec F \cdot \vec v[/math]
or
- [math]dT = \vec F \cdot \vec v dt =\vec F \cdot d\vec r \equiv d W[/math]
or
[math]\Delta T = \Delta W[/math]
- The change in a particle's kinetic energy is equivalent to the work done by the net Force used to move the particle
Forest_UCM_Energy#KE_.26_Work