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Isn't power defined by: $$P=\frac{dE}{dt}$$ (?)

I see in all websites that power is the amount of work done per time. Is it represented only like this? Why?

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$P=\frac{dE}{dt}$ is correct.

Work and energy are so closely related that it is very easy to get them mixed up. The relationship between work and energy is this: the increase in kinetic energy of an object is equal to the amount of work done on it. Work is specifically tied to applying a force to an object over a distance, but colloquially we tend to ignore this distinction.

In theory, power should be the derivative of energy, as you wrote, but in practice work and energy are so similar and so easy to mix up that people will say "power is work done per time" and generally be close enough to the right meaning.

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  • $\begingroup$ Derivative of kinetic energy or total energy? $\endgroup$ Sep 18, 2020 at 23:22
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    $\begingroup$ @AntoniosSarikas Depends on what you are doing, as my answer points out, but generally speaking total energy. Consider this: if you do electrolysis on water, the electrical energy is being transferred to (mostly) chemical energy. If we define power as the derivative of kinetic energy, the electrical system has no power, which is clearly not the most useful definition. $\endgroup$
    – Cort Ammon
    Sep 19, 2020 at 20:30
  • $\begingroup$ So can we say that it "depends" right? Because when an object falls down from a height then if we take the derivative of total energy with respect to time as the power it is zero whereas if we think power as rate of change of kinetic (or work done to the object) then it is clearly different than zero. $\endgroup$ Sep 26, 2020 at 13:42
  • $\begingroup$ @AntoniosSarikas "It depends," but there is typically an accepted answer depending on the phrasing. We usually try clarify which power we are looking for based on the adjectives. Remember that the $E$ in the equation stands not just for "energy," but the specific energy that we are talking about at the time. If the meaning of that $E$ is ambiguous, then the meaning of $P$ is ambiguous. If $E$ is well defined, then $P$ is well defined as well. $\endgroup$
    – Cort Ammon
    Sep 26, 2020 at 17:35
  • $\begingroup$ To build a contrived example, consider a battery powered device that plays music. We let it drop from a tall height. Our discussion of power in the system will have to disambiguate whether you want to include the power being dissipated by the speaker or not. If we are discussing entropy and the second order of thermodynamics, we probably intend to include the power of the battery being used to drive the speaker. If' we're talking about kinetic weapons, we probably choose to ignore those chemical effects. $\endgroup$
    – Cort Ammon
    Sep 26, 2020 at 17:37

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