Power and the Watt
Power is the rate at which work is done or energy is transferred, P = W ÷ t = E ÷ t, measured in watts (W), where 1 W = 1 J/s. In brief: for a force acting on an object moving at constant speed, P = Fv. A machine that does the same work in a shorter time has more power. Larger units are the kilowatt (1 kW = 1,000 W) and megawatt, and electrical energy is billed in kilowatt-hours (1 kWh = 3.6 MJ).
Two people can carry the same load up the same stairs, but the one who is quicker has to supply more power. Engines, motors and lamps are rated by power. This guide defines power and the watt, shows the formulas P = W/t and P = Fv, compares typical power ratings in a table, works through examples and corrects common mistakes.
What is power?
Power measures how quickly energy is transferred: P = W ÷ t, or P = ΔE ÷ t. The SI unit is the watt (W), named after James Watt: 1 W = 1 J/s. A 60 W lamp transfers 60 J of electrical energy every second. Other units: the horsepower (1 hp ≈ 746 W), still used for engines. Average power is the total energy divided by the total time, and instantaneous power is the rate at a particular moment.
Energy, work and power are related but not the same: a large amount of energy can be transferred slowly (low power), or a small amount quickly (high power). The kilowatt-hour is a unit of energy, not power: 1 kWh = 1 kW × 3,600 s = 3.6 × 106 J.
How is power related to force and speed?
For a constant force F along the direction of motion, W = Fs, so P = Fs ÷ t = Fv, where v is the speed. A car engine providing a driving force of 1,500 N at a speed of 20 m/s has a power output of 30 kW. At constant power, a higher speed means a lower force, which is why vehicles use low gears to climb hills, to get a larger force at a lower speed. When an object is lifted at constant speed, the power is P = mgv.
Typical power ratings
| Source or device | Power |
|---|---|
| Resting human body | about 80 to 100 W |
| LED lamp | about 10 W |
| Electric kettle | about 2,000 W |
| Small car engine | about 70 kW |
| Large wind turbine | about 3 MW |
| Large power station | about 1,000 MW |
Worked example: power calculations
- Climbing stairs. A person of mass 70 kg climbs 3.0 m in 5.0 s. Work = mgh = 70 × 9.8 × 3.0 = 2,058 J. P = 2,058 ÷ 5.0 = 412 W. If the climb takes 10 s, the power halves to 206 W, but the work is the same.
- Using P = Fv. A car moves at a constant speed of 25 m/s against a resistive force of 800 N. The engine’s driving force equals the resistance, so P = 800 × 25 = 20,000 W = 20 kW.
- Crane. A crane lifts a 500 kg load at a steady 0.40 m/s. F = mg = 4,900 N, P = Fv = 4,900 × 0.40 = 1,960 W. If the motor is 80 percent efficient, the electrical power needed is 1,960 ÷ 0.80 = 2,450 W.
- Energy and cost. A 2.0 kW heater runs for 3.0 hours: energy = 2.0 × 3.0 = 6.0 kWh = 21.6 MJ.
Where does the watt come from?
James Watt introduced the horsepower in the 1780s to market steam engines, by comparison with the work done by draft horses (about 33,000 foot-pounds per minute). The unit “watt” was adopted by the British Association for the Advancement of Science in 1889 and became an SI unit in 1960.
Which mistakes are common?
Mistake 1: Confusing power and energy
Power is a rate (W). Energy is an amount (J, kWh).
Mistake 2: Writing kWh as a unit of power
The kilowatt is power and the kilowatt-hour is energy.
Mistake 3: Believing that doing work faster requires more work
The work is the same, but the power is higher.
Mistake 4: Using P = Fv when the speed is changing without caution
P = Fv holds at each instant for the instantaneous speed. For average power, use the average speed with a constant force.