Editing Potential energy (section)

== Potential energy for a linear spring ==
{{Main article|Elastic potential energy}}
[[File:Springs 009.jpg|thumb|right|[[Spring (device)|Springs]] are used for storing [[elastic potential energy]]]]
[[File:Longbowmen.jpg|thumb|right|[[Archery]] is one of humankind's oldest applications of elastic potential energy]]

A horizontal spring exerts a force {{math|1='''F''' = (−''kx'', 0, 0)}} that is proportional to its deformation in the axial or ''x''-direction.  The work of this spring on a body moving along the space curve {{math|1='''s'''(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t''))}}, is calculated using its velocity, {{math|1='''v''' = (''v''<sub>x</sub>, ''v''<sub>y</sub>, ''v''<sub>z</sub>)}}, to obtain
<math display="block">W = \int_0^t\mathbf{F}\cdot\mathbf{v}\,dt
= -\int_0^t kx v_\text{x} \, dt
=-\int_0^t k x \frac{dx}{dt}dt
= \int_{x(t_0)}^{x(t)} k x \, dx
= \frac{1}{2} kx^2 </math>
For convenience, consider contact with the spring occurs at {{math|1=''t'' = 0}}, then the integral of the product of the distance ''x'' and the ''x''-velocity, ''xv''<sub>x</sub>, is {{itco|''x''}}<sup>2</sup>/2.

The function 
<math display="block"> U(x) = \frac{1}{2}kx^2 ,</math>
is called the potential energy of a linear spring.

Elastic potential energy is the potential energy of an [[elasticity (physics)|elastic]] object (for example a [[bow (weapon)|bow]] or a catapult) that is deformed under tension or compression (or [[stress (physics)|stressed]] in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the [[electromagnetic force]] between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into [[kinetic energy]].