Power

The energy in a given system is not, of course, usually constant in time. Energy is added to a given mass, or taken away, at some rate. We accelerate a car (adding to its mechanical energy). We brake a car (turning its kinetic energy into heat). There are many times when we are given the rate at which energy is added or removed in time, and need to find the total energy added or removed. This rate is called the power.

Power: The rate at which work is done, or energy released into a system.

This latter form lets us express it conveniently for time-varying forces:

$$dW = \vec{F} \cdot d\vec{x} = \vec{F} \cdot \frac{d x}{dt} dt$$ (3.45)

or

$$P = \frac{d W}{dt} = \vec{F} \cdot \vec{v}$$ (3.46)

so that

$$\Delta W = \Delta E_{\rm tot} = \int P dt$$ (3.47)

The units of power are clearly Joules/sec = Watts. Another common unit of power is ``Horsepower'', 1 HP = 746 W. Note that the power of a car together with its drag coefficient determine how fast it can go. When energy is being added by the engine at the same rate at which it is being dissipated by drag and friction, the total mechanical energy of the car will remain constant in time.