Definition of KE

For a single particle of mass m moving with a velocity v, the kinetic energy is defined as

[math]\displaystyle{ T \equiv \frac{1}{2} mv^2 }[/math]

Work Energy Theorem

Consider the Kinetic Energy's temporal rate of change assuming that the mass of the particle is constant

[math]\displaystyle{ \frac{dT}{dt} = \frac{m}{2} \frac{d}{dt}v^2= \frac{m}{2} \frac{d}{dt}\vec v \cdot \vec v }[/math]

[math]\displaystyle{ = \frac{m}{2} \left (\vec \dot v \cdot \vec v + \vec v \cdot \vec \dot v \right ) }[/math]

[math]\displaystyle{ = \frac{m}{2} 2 \vec \dot v \cdot \vec v = \vec F \cdot \vec v }[/math]

or

[math]\displaystyle{ dT = \vec F \cdot \vec v dt =\vec F \cdot d\vec r \equiv d W }[/math]

or

[math]\displaystyle{ \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

If we start with the form derived above

[math]\displaystyle{ dT =\vec F \cdot d\vec r }[/math]

The change in the kinetic energy between two points and the corresponding work done as a result are

[math]\displaystyle{ T_2-T_1 = \sum \vec F \cdot d\vec r }[/math]