Power
Power is the rate at which work is done or energy is transferred, equal to work divided by time (P = W/t) and measured in watts.
Formula
P = \frac{W}{t}
Definition
Power is how fast work gets done: two people can do the same job, but the one who finishes quicker used more power. It is the rate at which work is done or energy is transferred, $P = W/t$, with $W$ in joules and $t$ in seconds, and its SI unit is the watt ($1\ \text{W} = 1\ \text{J/s}$); one kilowatt is $1{,}000\ \text{W}$, and one horsepower is about $746\ \text{W}$. Instantaneous power is $P = dW/dt = \vec{F}\cdot\vec{v}$, the dot product of force and velocity, and for electrical circuits $P = IV = I^2R = V^2/R$. Average power over an interval is total energy transferred divided by elapsed time. Power is a scalar, and its units reduce to $\text{kg}\cdot\text{m}^2/\text{s}^3$.
Example
Running up a flight of stairs takes more power than walking up the same stairs; the work is the same, but running does it in less time. A $50\ \text{kg}$ student who climbs $4\ \text{m}$ of stairs in $5\ \text{s}$ does $50 \times 9.8 \times 4 = 1{,}960\ \text{J}$ of work against gravity, so the power is $1960 \div 5 = 392\ \text{W}$, about half a horsepower. A car cruising at $30\ \text{m/s}$ against $600\ \text{N}$ of total drag and friction needs $P = Fv = 600 \times 30 = 18{,}000\ \text{W}$, or $18\ \text{kW}$, just to hold speed. At $60\ \text{m/s}$, aerodynamic drag roughly quadruples, so the power demand rises by about eight times.
Key Insight
A light bulb's wattage is its power: a $100$-watt bulb uses energy faster than a $60$-watt bulb, which is why it is brighter and hotter. Your electric bill charges for energy, not power; a kilowatt-hour is $1{,}000\ \text{W}$ running for one hour, which is $3{,}600{,}000\ \text{J}$. Because $P = Fv$, an engine with fixed power delivers less force as speed increases. That is why cars accelerate hard from a stop and sluggishly at highway speed.