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

$W=\overrightarrow{F}\cdot \mathrm{\Delta }\overrightarrow{d}=F\mathrm{\Delta }d\cos \theta \text{,}$

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 $v\text{,}$ kinetic energy is defined as the amount of work needed to accelerate the object from rest to $v\text{.}$ The formula for kinetic energy is

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

A change in kinetic energy represents work being done:

$W=\mathrm{\Delta }{E}_{\text{k}}=\frac{1}{2}m\mathrm{\Delta }{v}^2\text{.}$

We can also relate kinetic energy to momentum with

$E_{\text{k}}=\frac{p^2}{2m}$ and $p=\sqrt{2m{E}_{\text{k}}}\text{.}$