Relation between power and velocity

To connect power directly to force and velocity, start from the definition of work done by a force $\vec F$ for a displacement $d\vec r$:

Differentiate both sides with respect to time (equivalently, write $dW/dt$ by multiplying and dividing by $dt$ inside the integral) and use $d\vec r/dt = \vec v$ (the instantaneous velocity):

Since the left side is, by definition, the instantaneous power:

$\boxed{P = \vec F\cdot\vec v}$ …