Power:

The time rate of doing work on a particle is called instantaneous power. Power is represented by the P as

\( P=\frac{dW}{dt} \),

where W is the work done by the force.

If \( \vec{F} \) be the force acting on the particle and \( d\vec{r} \) be the displacement of the particle, then

\( dW=\vec{F}\cdot{d}\vec{r} \)

since only the component of \( \vec{F} \) in the direction of the displacement \( d\vec{r} \) is effective in producing the motion of the particle.

Now, power\( P=\displaystyle{\frac{\vec{F}\cdot{d\vec{r}}}{dt}=\vec{F}\cdot\frac{d\vec{r}}{dt}} \).    

Here, \( \displaystyle{\frac{d\vec{r}}{dt}}=\vec{v} \) is the velocity of the particle, So power is, \( P=\vec{F}\cdot{\vec{v}} \).

Average Power:

If \( W \) amount of work is done by the force on a particle during the time interval \( t \), then the average power of the force acting on the particle is defined by the ratio of the work done by the force to the total time t. It is denoted by

\( P_{av}=\frac{W}{t} \).

Power is a scalar quantity.

Unit of power is \( Joule/sec \) or \( Watt \).

Power is also measured in \( Horse\ Power \) and \( 1\ H.P.=746\ watt \).