Power

Energy transfer per time

Power is the rate at which work is done or energy is transferred. If an amount of work $W$ is done over time interval $Delta t$, average power is

P_{avg}=rac{W}{Delta t}

Instantaneous power is the time derivative of energy transfer:

P=rac{dW}{dt}

Power is measured in watts, where

$1 W=1 J/s$

A more powerful machine does not necessarily do more total work; it does work faster.

Power from force and velocity

If a force acts on an object moving with velocity ec{v}, instantaneous power is

P=ec{F}cdotec{v}

This follows because

dW=ec{F}cdot dec{r}

and

rac{dec{r}}{dt}=ec{v}

so

P=rac{dW}{dt}=ec{F}cdotrac{dec{r}}{dt}=ec{F}cdotec{v}

Only the component of force along velocity transfers energy.

Positive, negative, and zero power

If force and velocity point in the same direction, power is positive. The force increases the object's mechanical energy. If force and velocity point in opposite directions, power is negative. The force removes mechanical energy. If force is perpendicular to velocity, power is zero.

For example, friction often has negative power because it acts opposite motion. A centripetal force in uniform circular motion has zero power because it is perpendicular to velocity.

Lifting at constant speed

Suppose a motor lifts a mass $m$ upward at constant speed $v$. The upward force must balance weight, so

$F=mg$

The power supplied is

$P=Fv=mgv$

This shows that lifting a heavier object or lifting at a higher speed requires more power.

Vehicles and engines

A vehicle traveling at steady speed must still use power because resistive forces such as air drag and rolling resistance remove energy. At constant speed, the engine's power output balances the rate at which resistive forces do negative work.

If the total resistive force is $F_{res}$ and speed is $v$, the required power is approximately

$P=F_{res}v$

Air drag often increases strongly with speed, so the power required at high speeds can be much larger than expected.

Efficiency

Real machines waste some input energy as thermal energy, sound, vibration, or other forms. Efficiency is

eta=rac{useful output energy}{input energy}

or, for power,

eta=rac{useful output power}{input power}

Efficiency is usually expressed as a percentage. No real machine is 100 percent efficient.

Power and energy bills

Electric energy use is often billed in kilowatt-hours. A kilowatt-hour is a unit of energy, not power:

$1 kWh = 1000 W imes 3600 s = 3.6 imes 10^6 J$

A device with high power used briefly may consume less energy than a low-power device used for many hours.

The big idea

Power measures how quickly energy is transferred or work is done. It connects energy methods to time. The formula P=ec{F}cdotec{v} shows that power depends on both force and motion direction. Power is central to engines, motors, human performance, electricity, machines, and energy efficiency.