Power

Power answers a question work alone does not: not merely how much work is done, or how much energy is transferred, but how quickly. Two machines might do exactly the same amount of work lifting the same load through the same height, yet one may take a few seconds and the other several minutes -- the first is delivering far more power than the second, even though the total work done, and the total energy transferred, are identical in both cases.

Average power delivered over a time interval $t$, during which a total work $W$ is done, is defined as

Instantaneous power, the power being delivered at one particular instant, is defined, for a force $\vec F$ acting on a body moving with instantaneous velocity $\vec v$, as the dot product

where $\theta$ is the angle between the force and the instantaneous velocity at that instant. This follows directly from the definition of work: over a very short time interval $dt$, the small displacement is $d\vec s = \vec v\,dt$, so the small work done is $dW = \vec F \cdot d\vec s = (\vec F \cdot \vec v)\,dt$, and dividing by $dt$ gives instantaneous power as $\vec F \cdot \vec v$ directly.

Power is a scalar quantity (a dot product of two vectors), with SI unit the watt ($\text{W}$), where $1\ \text{W} = 1\ \text{J/s}$: a device delivers a power of one watt if it does one joule of work every second. Larger, more practical units built from the watt include the kilowatt ($1\ \text{kW} = 1000\ \text{W}$) and, historically, the horsepower ($1\ \text{hp} \approx 746\ \text{W}$), still sometimes used to rate vehicle and motor output. …