Energy
Overview
Work
Work is defined as the
dot product
of the force and the distance through which a particle moves.
{% 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 %}
{% \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} %}
{% 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 %}
Kinetic Energy
Define the kinetic energy of a particle as
{% KE = \frac{1}{2} m v^2 = \frac{1}{2} m \vec{v} \cdot \vec{v} %}
{% m \vec{v} \cdot \frac{d \vec{v}}{dt} = \frac{d}{dt} (\frac{1}{2} m v^2)%}
This implies that the work is the change in the kinetic energy.
{% W = \Delta KE %}
Topics
