Editing Power (physics) (section)

==Definition==
Power is the [[Rate (mathematics)|rate]] with respect to time at which work is done or, more generally, the rate of change of total mechanical energy. It is given by:
<math display="block">P = \frac{dE}{dt},</math>
where {{mvar|P}} is power, {{mvar|E}} is the total mechanical energy (sum of kinetic and potential energy), and {{mvar|t}} is time.

For cases where only work is considered, power is also expressed as:
<math display="block">P = \frac{dW}{dt},</math>
where {{mvar|W}} is the work done on the system. However, in systems where potential energy changes without explicit work being done (e.g., changing fields or conservative forces), the total energy definition is more general.

We will now show that the mechanical power generated by a force F on a body moving at the velocity v can be expressed as the product: <math display="block">P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf {v}</math>

If a ''constant'' force '''F''' is applied throughout a [[distance]] '''x''', the work done is defined as <math>W = \mathbf{F} \cdot \mathbf{x}</math>. In this case, power can be written as:
<math display="block">P = \frac{dW}{dt} = \frac{d}{dt} \left(\mathbf{F} \cdot \mathbf{x}\right) = \mathbf{F}\cdot \frac{d\mathbf{x}}{dt} = \mathbf{F} \cdot \mathbf {v}.</math>

If instead the force is ''variable over a three-dimensional curve C'', then the work is expressed in terms of the line integral:
<math display="block">W = \int_C \mathbf{F} \cdot d\mathbf {r}
  = \int_{\Delta t} \mathbf{F} \cdot \frac{d\mathbf {r}}{dt} \  dt
  = \int_{\Delta t} \mathbf{F} \cdot \mathbf {v} \, dt.</math>

From the [[fundamental theorem of calculus]], we know that <math display="block">P = \frac{dW}{dt} = \frac{d}{dt} \int_{\Delta t} \mathbf{F} \cdot \mathbf {v} \, dt = \mathbf{F} \cdot \mathbf {v}.</math> Hence the formula is valid for any general situation.

In older works, power is sometimes called ''activity''.<ref name="Smithsonian Tables">{{cite book|title=Smithsonian Physical Tables|publisher=[[Smithsonian Institution]]|date=1921|editor-first1=Frederick E.|editor-last1=Fowle |edition=7th revised |location=Washington, D.C.|url=https://books.google.com/books?id=tCoJAQAAIAAJ&q=%22Power%20or%20Activity%20is%20the%20time%20rate%20of%20doing%20work%22|oclc=1142734534|archive-url=https://web.archive.org/web/20200423151426/https://www.google.com/books/edition/Smithsonian_Physical_Tables/tCoJAQAAIAAJ?hl=en&gbpv=1&bsq=%22Power%20or%20Activity%20is%20the%20time%20rate%20of%20doing%20work%22 |archive-date=23 April 2020|url-status=live |quote='''Power or Activity''' is the time rate of doing work, or if {{math|''W''}} represents work and {{math|''P''}} power, {{math|1=''P'' = ''dw''/''dt''}}. (p. xxviii) ... ACTIVITY. Power or rate of doing work; unit, the watt. (p. 435)}}</ref><ref name="Heron Motors">{{cite journal |last1=Heron|first1=C. A. |date=1906 |title=Electrical Calculations for Railway Motors |url=https://books.google.com/books?id=b5A4AQAAMAAJ&dq=%22The+activity+of+a+motor+is+the+work+done+per+second%22+%22Where+the+joule+is+employed+as+the+unit+of+work,+the+international+unit+of+activity+is+the+joule-per-second,+or,+as+it+is+commonly+called,+the+watt.%22&pg=PA78 |journal=Purdue Eng. Rev.|issue=2 |pages=77–93 |access-date=23 April 2020 |archive-url=https://web.archive.org/web/20200423142933/https://www.google.com/books/edition/The_Purdue_Engineering_Review/b5A4AQAAMAAJ?hl=en&gbpv=1&dq=%22The+activity+of+a+motor+is+the+work+done+per+second%22+%22Where+the+joule+is+employed+as+the+unit+of+work,+the+international+unit+of+activity+is+the+joule-per-second,+or,+as+it+is+commonly+called,+the+watt.%22&pg=PA78&printsec=frontcover |archive-date=23 April 2020 |url-status=live |quote=The activity of a motor is the work done per second, ... Where the joule is employed as the unit of work, the international unit of activity is the joule-per-second, or, as it is commonly called, the watt. (p. 78)}}</ref><ref name="Nature 1902">{{cite journal|date=1902 |title=Societies and Academies |journal=Nature |volume=66|issue=1700 |pages=118–120 |doi=10.1038/066118b0 |bibcode=1902Natur..66R.118. |quote=If the watt is assumed as unit of activity... |doi-access=free}}</ref>