Editing Force (section)

=== Kinematic integrals ===
{{main|Impulse (physics)|l1=Impulse|Mechanical work|Power (physics)}}
Forces can be used to define a number of physical concepts by [[integration (calculus)|integrating]] with respect to [[kinematics|kinematic variables]]. For example, integrating with respect to time gives the definition of [[Impulse (physics)|impulse]]:<ref>{{Cite book
|title=Engineering Mechanics
|first1=Russell C.
|last1=Hibbeler
|publisher=Pearson Prentice Hall
|year=2010
|edition=12th
|isbn=978-0-13-607791-6
|page=222
}}</ref>
<math display="block">\mathbf{J}=\int_{t_1}^{t_2}{\mathbf{F} \, \mathrm{d}t},</math>
which by Newton's second law must be equivalent to the change in momentum (yielding the [[Impulse momentum theorem]]).

Similarly, integrating with respect to position gives a definition for the [[work (physics)|work done]] by a force:<ref name=FeynmanVol1/>{{rp|((13-3))}}
<math display="block" qid=Q42213>W= \int_{\mathbf{x}_1}^{\mathbf{x}_2} {\mathbf{F} \cdot {\mathrm{d}\mathbf{x}}},</math>
which is equivalent to changes in [[kinetic energy]] (yielding the [[work energy theorem]]).<ref name=FeynmanVol1/>{{rp|((13-3))}}

[[Power (physics)|Power]] ''P'' is the rate of change d''W''/d''t'' of the work ''W'', as the [[trajectory]] is extended by a position change <math> d\mathbf{x}</math> in a time interval d''t'':<ref name=FeynmanVol1/>{{rp|((13-2))}}
<math display="block"> \mathrm{d}W = \frac{\mathrm{d}W}{\mathrm{d}\mathbf{x}} \cdot \mathrm{d}\mathbf{x} = \mathbf{F} \cdot \mathrm{d}\mathbf{x},</math>
so
<math display="block">P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}W}{\mathrm{d}\mathbf{x}} \cdot \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{F} \cdot \mathbf{v},
</math>
with <math qid=Q11465>\mathbf{v} = \mathrm{d}\mathbf{x}/\mathrm{d}t</math> the [[velocity]].