# Work & kinetic energy

A force is said to do work when it acts on a body and results in a displacement in the direction of the force. It is a scalar quantity defined by

$\displaystyle W = \vec{F} \cdot \Delta{} \vec{d} = F \Delta{} d \cos{\theta}$,

and it is measured in joules (J). Work can be zero in three situations: when force is zero, when displacement is zero, or when force and displacement are perpendicular. When the angle between force and distance exceeds 90º, work becomes negative. On a force-position graph, the area under the curve represents work.

Kinetic energy is energy due to motion. For an object moving at speed $\displaystyle v$, kinetic energy is defined as the amount of work needed to accelerate the object from rest to $\displaystyle v$. The formula for kinetic energy is

$\displaystyle E_{\text{k}} = \frac{1}{2} m v^{2}$.

A change in kinetic energy represents work being done:

$\displaystyle W = \Delta{} E_{\text{k}} = \frac{1}{2} m \Delta{} v^{2}$.

We can also relate kinetic energy to momentum with

$\displaystyle E_{\text{k}} = \frac{p^{2}}{2 m} \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad p = \sqrt{2 m E_{\text{k}}}$.