Power
Just as we need to define the rate of change of the position or that of the velocity to describe motion fully, we sometimes need to quantify the rate at which work is being done or energy is being exchanged. The corresponding physical quantity is called power.
Power
Power is the time rate of energy transfer.
Average power is defined as
\[\overline{P} := \frac{\Delta E}{\Delta t},\]
where \(\Delta E\) is the energy exchanged in time \(\Delta t.\)
If energy is transferred in the form of work, the formula for average power is
\[\overline{P} := \frac{\Delta W}{\Delta t},\]
where \(\Delta W\) is the work done in time \(\Delta t.\)
The instantaneous power (or simply power) is average power in the limit as \(\Delta t\) approaches zero:
\[P := \frac{\mathrm{d}E}{\mathrm{d}t}\ \mathrm{or}\ P := \frac{\mathrm{d}W}{\mathrm{d}t}.\]
Power is a scalar quantity.
The SI unit of power is the joule per second, also known as the watt:
\(\left[P\right] = 1\,\frac{\textrm{J}}{\textrm{s}} = 1\,\textrm{W}.\)
When the force is constant, the work can be written as
\[W = \mathbf{F} \cdot \Delta \mathbf{r}.\]
In this case, the power can be expressed in the form
\[P = \lim\limits_{\Delta t \to 0}\frac{\mathbf{F} \cdot \Delta \mathbf{r}}{\Delta t} = \mathbf{F} \cdot \lim\limits_{\Delta t \to 0}\frac{\Delta \mathbf{r}}{\Delta t} = \mathbf{F} \cdot \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \mathbf{F} \cdot \mathbf{v}.\]