Power is just the time derivative of energy, like velocity is the time derivative of position. Both batteries and capacitors store energy, which they can release at some given peak and average power. Batteries tend to focus more on energy density (energy stored per weight), while capacitors tend to focus more on power density (rate of energy stored or released per weight).

They store energy but there's a limit to how quickly you can get that energy out, hence a limit to power. And I suppose you can kinda talk about them "storing power" in the sense that power is what you might be interested in getting out of them.

At least, that's my understanding of it.

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