Overview
Formulation
Work is defined as the dot product of the force and the distance through which a particle moves.
{% \displaystyle W = \int_a^b \vec{F}(\vec{s}) \cdot d \vec{s} = \int_a^b \vec{F}(\vec{s}) \cdot \vec{v} dt = m \int_a^b \vec{v} \cdot \frac{d \vec{v}}{dt} dt %}
Given the following identity
{% \frac{d}{dt} v^2 = \frac{d}{dt}\vec{v} \cdot \vec{v} = \frac{d \vec{v}}{dt} \cdot \vec{v} + \vec{v} \cdot \frac{d \vec{v}}{dt} = 2 \vec{v} \cdot \frac{d \vec{v}}{dt} %}
Work can then be shown to be
{% \displaystyle W = \int_a^b \vec{F}(\vec{s}) \cdot d \vec{s} = m \int_a^b \vec{v} \cdot \frac{d \vec{v}}{dt} dt
= \frac{m}{2} [ v_f^2 - v_i^2 ]
%}