# Instantaneous Power , Average Power

Power :  Power is defined as the rate of work done.

$\displaystyle P = \frac{dW}{dt}$

This is called Instantaneous Power .

$\displaystyle dW = \vec{F}.\vec{dx}$

$\displaystyle P_{ins} = \frac{\vec{F}.\vec{dx}}{dt}$

$\displaystyle P_{ins} = \vec{F}.\vec{v}$

If the force is variable, we calculate the average power as

$\displaystyle P_{avg} = \frac{\int_{0}^{t} P dt}{\int_{0}^{t} dt}$

$\displaystyle Average Power = \frac{work \; done}{time}$

$\displaystyle P_{avg} = \frac{W}{t}= \frac{m v^2}{2 t}$

$\displaystyle P_{avg} = \frac{1}{2}mv\frac{v}{t} = \frac{1}{2}\vec{F}.\vec{v}$

$\displaystyle P_{avg} = \frac{1}{2}P_{ins}$

Power can also be expressed as the rate of change of kinetic energy.

Let a body of mass m move with a velocity v. The kinetic energy of the body is

$\displaystyle K = \frac{1}{2}m v^2$

$\displaystyle \frac{dk}{dt} = \frac{1}{2}\frac{d}{dt}(m v^2)$

$\displaystyle = m v. \frac{dv}{dt}$

$\displaystyle = m \frac{dv}{dt} .v$

= Fext . v

= P

Therefore, $\displaystyle P = \frac{dK}{dt}$

Solved Example : A particle is projected with a speed v at an angle θ with the horizontal. Find the mean power delivered by gravity during the ascent of the particle.

Solution : The magnitude of mean power for

$\displaystyle P_m = \frac{1}{t_0} \int_{0}^{t_0} \vec{m g}.\vec{v’_y} dt$

$\displaystyle P_m = \frac{1}{t_0} \int_{0}^{t_0} (-m g v’_y) dt$

Where v’y = vy – gt

$\displaystyle P_m = \frac{1}{t_0} \int_{0}^{t_0} mg (v_y – gt ) dt \quad (numerically)$

Since vy = t0/g

$\displaystyle P_m = \frac{m g v_y}{2} = \frac{m g vsin\theta}{2}$

Exercise : Two bodies of masses m1 and m2 (m2 > m1) are connected by a light inextensible string which passes through a smooth fixed pulley. What is the instantaneous power delivered by an external agent to pull m1 with constant (a) velocity v (b) acceleration a at any instant t Exercise : A small body of mass m is located on a horizontal plane at the point O. The body acquires a horizontal velocity vo . Find the mean power developed by the friction force during the motion, if the coefficient of friction μ = 0.27, m = 1.0 kg and vo = 1.5 m/s