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=FΔd=FΔdcosθ\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 v\displaystyle v, kinetic energy is defined as the amount of work needed to accelerate the object from rest to v\displaystyle v. The formula for kinetic energy is

Ek=12mv2\displaystyle E_{\text{k}} = \frac{1}{2} m v^{2}.

A change in kinetic energy represents work being done:

W=ΔEk=12mΔv2\displaystyle W = \Delta{} E_{\text{k}} = \frac{1}{2} m \Delta{} v^{2}.

We can also relate kinetic energy to momentum with

Ek=p22mandp=2mEk\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}}}.