01 - Work Energy Theorem

Let us define a property, work, which is the vectorial product of an instantanious change position with its respective force:

$$ W = \int^a_b{dW} = \int^{a}_{b}{\vec{F}\cdot d\vec{r}} = |\vec{F}||dr|\cos{\theta} $$

This can be shown to be equivalent to the change in kinetic energy of a system and can be sub-divided into constituent parts regarding constituent forces:

$$ W = \sum W_{F_i} = \Delta E_K$$

Kinetic Energy:

Let us also define another property, kinetic energy:

$$ E_K = \frac{mv^2}{2}$$

This property is helpful and is conserved. I hope to learn more in-detail about its rigor, but it suffices to know the bases in order to apply it for our use cases.