KAHIBARO
Table of Contents
Meaning of average power
Average power tells us how fast work is done, or how fast energy is transferred, over a time interval. While work measures the total amount of energy transferred, average power measures the rate of that transfer.
If an amount of work $W$ is done during a time interval $\Delta t$, then the average power is
$$
P_{\text{avg}} = \frac{W}{\Delta t}
$$
This idea is very useful when the rate of doing work is not constant. Instead of asking how fast energy is transferred at one exact moment, average power asks for the overall rate during a finite period of time.
The formula for average power is
$$
P_{\text{avg}} = \frac{W}{\Delta t}
$$
Average power is total work done divided by total time taken.
Average power and energy transfer
Because work is a transfer of energy, average power can also be written in terms of energy:
$$
P_{\text{avg}} = \frac{\Delta E}{\Delta t}
$$
Here, $\Delta E$ is the amount of energy transferred or transformed in the interval $\Delta t$. This makes power an important idea in many physical situations, not only when a force moves an object, but also whenever energy changes form.
For example, if a machine transfers $5000 \, \text{J}$ of energy in $10 \, \text{s}$, then its average power is
$$
P_{\text{avg}} = \frac{5000}{10} = 500 \, \text{W}
$$
Unit of average power
The SI unit of power is the watt, abbreviated as $\text{W}$. One watt means one joule of work done per second:
$$
1 \, \text{W} = 1 \, \text{J/s}
$$
This unit shows clearly that power is a rate.
| Quantity | Symbol | SI unit |
|---|---|---|
| Work | $W$ | joule, $\text{J}$ |
| Time interval | $\Delta t$ | second, $\text{s}$ |
| Average power | $P_{\text{avg}}$ | watt, $\text{W}$ |
$$
1 \, \text{W} = 1 \, \text{J/s}
$$
Power is measured in watts, not joules.
Interpreting average power
A large average power means that a large amount of work is done in a short time. A small average power means that the same work takes longer, or less work is done in the same time.
Suppose two people both carry out $1000 \, \text{J}$ of work. If one person does it in $5 \, \text{s}$ and the other in $20 \, \text{s}$, then their average powers are
$$
P_{\text{avg,1}} = \frac{1000}{5} = 200 \, \text{W}
$$
$$
P_{\text{avg,2}} = \frac{1000}{20} = 50 \, \text{W}
$$
The first person has the greater average power because the work is done more quickly.
Example with lifting an object
Consider lifting a box upward. If the work done against gravity is $240 \, \text{J}$ and this takes $4.0 \, \text{s}$, then
$$
P_{\text{avg}} = \frac{240}{4.0} = 60 \, \text{W}
$$
This means energy is transferred at an average rate of $60$ joules every second during the lift.
If another person lifts the same box through the same height in $2.0 \, \text{s}$, then the work is the same but the average power doubles:
$$
P_{\text{avg}} = \frac{240}{2.0} = 120 \, \text{W}
$$
Average power versus instantaneous power
Average power is calculated over a time interval. It does not describe what happens at each moment during that interval. If the force or speed changes during motion, the power may vary from moment to moment, even though the average power over the whole interval has one value.
For example, when climbing stairs, a person may speed up or slow down. The average power depends only on the total work done and the total time.
Average power uses a finite time interval:
$$
P_{\text{avg}} = \frac{W}{\Delta t}
$$
It is not necessarily the power at every instant during the motion.
Common situations
Average power appears in many everyday and physical situations. A motor that lifts loads, an athlete running up stairs, and a machine transferring energy all have an average power that depends on total energy transfer and total time.
| Situation | Total work or energy transfer | Average power meaning |
|---|---|---|
| Lifting an object | Gain in gravitational energy | How quickly the lift is done |
| Engine or motor | Mechanical work output | Rate of useful work over time |
| Electrical device | Energy used | Rate of energy consumption over time |
Key idea
Average power connects energy and time. It tells us not just how much work is done, but how quickly it is done.
The central idea of this chapter is
$$
P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}
$$
Average power is the rate of doing work or transferring energy over a time interval.