Section 7.6 in Matter and Interactions (4th edition)

## Power: The Rate of Energy Change

Until now, you have only considered that energy changes from one state to another, but not the rate at which that change can occur. The rate at which energy changes is called **power**. **In these notes, you will read about the definition of power.**

### Climbing Stairs

Consider climbing a flight of stairs to your apartment or dorm room. If you climb that flight at a leisurely pace (say, climbing a flight in a minute or two), you might not really notice your chemical energy expenditure.

On the other hand, if you run up the stairs (climbing the flight in a few seconds), you will definitely notice your chemical energy expenditure. Your heart will be pumping harder; you might be breathing a bit harder, and your internal body temperature will go up quickly.

This rate of energy expenditure that you notice is related to your power output.

### Power

**Power is energy change per unit time.** It is often associated with the work done by a force (or the net force). Thus the typical way to represent the average power is,

$$P_{avg} = \dfrac{W}{\Delta t} = \dfrac{\vec{F}\cdot\Delta \vec{r}}{\Delta t}$$

where this force could be an individual force or the net force. That is, you can determine the power due to the work by a single force or by the net force.

The units of power are Joules-per-second (J/s) or Watts (W) named after James Watt, the developer of the steam engine. Incidentally, Watt's steam engine is a major contributor to the Industrial Revolution.

#### Instantaneous Power

Using the average power, you can consider taking smaller and smaller steps in space. These small steps could become so small that you are considering the instantaneous power due to a force,

$$P = \dfrac{\vec{F}\cdot d\vec{r}}{dt} = \vec{F}\cdot\dfrac{d\vec{r}}{dt} = \vec{F}\cdot\vec{v}$$

The resulting dot product of the force (or net force) and the instantaneous velocity gives the instantaneous power for that force.