$P = F times v$
Explanation
Power is defined as the time rate at which work is done or energy is transferred. In mechanics, instantaneous power can be expressed in terms of force and velocity.
Derivation
Mathematically, power ($P$) is the derivative of work ($W$) with respect to time ($t$):
$$P = \frac{dW}{dt}$$
Work done by a constant force ($F$) acting on an object causing a displacement ($s$) is given by:
$$W = F \cdot s$$
Substituting this into the power equation:
$$P = \frac{d(F \cdot s)}{dt}$$
Since the force $F$ is constant, it can be taken out of the derivative:
$$P = F \cdot \frac{ds}{dt}$$
The rate of change of displacement with respect to time ($\frac{ds}{dt}$) is velocity ($v$). Therefore:
$$P = F \times v$$
This relationship holds true when the force and velocity are in the same direction. In vector notation, it is the dot product $P = \vec{F} \cdot \vec{v}$.
Key Takeaway: Instantaneous power is the product of the force applied to an object and the velocity of the object in the direction of the force ($P = Fv$).