Time from Work and Power Calculator

Definition: power, work, and time

Average power is the quotient of work and the time in which it was done: P = W/t. The three quantities of this definition can be computed from one another: work from power and time (W = P·t), power from work and time (P = W/t), and — as on this page — time from work and power (t = W/P). Work is understood broadly here as energy transferred: the mechanical work of a force, heat delivered to water, electrical energy stored in a battery.

The formula t = W/P says that time is inversely proportional to power. A device twice as powerful does the same work in half the time; one ten times weaker takes ten times longer. This is a hyperbolic relation, which the chart under the result draws. The unit follows from the definition: a joule divided by a watt, that is by a joule per second, gives a second.

Historical note: from Watt’s horse to the kilowatt-hour

The concept of power as “work over time” took shape in the 18th century together with the steam engine. James Watt, comparing his engines with horses at mill wheels, defined horsepower around 1782 as 33,000 pounds lifted one foot per minute — precisely work divided by time. The question “how long will the engine take to pump the water out of the mine” was the first practical application of t = W/P.

The unit of power was named the watt at the 1882 meeting of the British Association for the Advancement of Science, and in 1889 the International Electrical Congress adopted it. With electrification came the kilowatt-hour: the work done by a power of 1 kW in one hour, that is 3.6 MJ. Electrical energy is billed in it, which is why the result panel also shows the entered work in kWh — so that charging or cooking time can be compared with the bill.

Derivation and step-by-step calculation

Step 1. From the definition of average power, P = W/t. Step 2. Multiply both sides by t and divide by P: t = W/P. Assumption: the power is constant over the whole interval; when it varies, W is the integral ∫P dt and t must be found from the equation ∫₀ᵗ P(τ) dτ = W. Step 3. For W = 300,000 J and P = 2,000 W: t = 300,000 / 2,000 = 150 s = 2 min 30 s. The same work in kilowatt-hours is 300,000 / 3,600,000 = 0.0833 kWh.

Step 4. When the work is not given directly, it must be computed first. Lifting a mass m to a height h: W = m·g·h, for 800 kg and 30 m — 235,440 J. Heating a mass m of water by ΔT: Q = c·m·ΔT, for 1 kg and 80 K — 4,186·80 = 334,880 J. Accelerating a mass m to speed v: W = ½mv², for 1,500 kg and 27.8 m/s — about 580 kJ. Only such a result is then divided by the power; the neighboring calculators (potential energy, heat, kinetic energy) compute these works separately.

Assumptions: constant power and efficiency

The first assumption is constant power. An electric motor at start-up, a kettle just after switching on, or a charger in its final phase do not run at constant power. Charging a lithium-ion battery proceeds at constant current only to about 80 % of capacity, after which the power drops, so the real time to a full charge is longer than W/P computed from the rated power — usually by 20–40 %.

The second assumption is 100 % efficiency: all of the device’s power goes into useful work. A kettle loses a few percent of heat to the surroundings, an elevator motor has an efficiency of 70–90 %, a phone charger 80–90 %, and a combustion engine only 25–40 %. If you know the efficiency η, divide the calculator’s result by η, or enter the product P·η in the power field. For a 2,000 W kettle at 90 % efficiency the time to boil a liter of water grows from 167 s to 186 s.

Work as energy: mechanical, thermal, electrical

The joule is the common unit of all kinds of energy, so the same formula works for mechanical work, heat, and electrical energy. Mechanical work: force times distance, W = F·s, or the change of potential energy m·g·h and kinetic energy ½mv². Heat: Q = c·m·ΔT, where for water c = 4,186 J/(kg·K); boiling a liter of water from 20 °C takes about 335 kJ, and evaporating it another 2.26 MJ. Electrical energy: voltage times charge, U·Q, in practice given in Wh or kWh.

Conversion factors used on this page: 1 kWh = 3.6 MJ, 1 Wh = 3,600 J, 1 kcal = 4,184 J, 1 ft·lb = 1.3558 J. A 4,000 mAh phone battery at 3.8 V holds about 15 Wh = 54 kJ; an electric car battery 60 kWh = 216 MJ; a resting human gives off about 100 W, so 100 kcal (418 kJ) is “burned” in a little over an hour, and at 400 W of effort in seventeen minutes.

Orders of magnitude of time: from microseconds to years

The same formula spans an enormous range of scales. A camera flash releases about 10 J in 1 ms, that is at 10 kW. A 2 kW kettle boils a liter of water in under 3 minutes. A 12 kW elevator lifts 800 kg to the tenth floor in 20 s. An 11 kW wall charger fills a 60 kWh battery in 5.5 hours, and a 1.4 kW household outlet in 43 hours. A 10 W LED bulb uses 1 kWh only after 100 hours of light, and a 1 GW power plant produces 1 TWh in 1,000 hours, that is six weeks.

For very long times it is easier to think in hours and days, which is why the result panel shows clock format and picks the unit automatically. For very short ones — milliseconds and microseconds — the instantaneous power can be enormous despite a small energy; that is why pulsed lasers with energies of single joules reach powers of terawatts. The calculator computes in seconds and displays full precision, so both ends of the scale are available.

Sources

The definition, derivation, and historical note come from general physics textbooks and metrology sources: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapter on work, energy, and power; H. D. Young, R. A. Freedman, University Physics, chapter “Work and Kinetic Energy,” section “Power”; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapters 4 (Conservation of Energy) and 13 (Work); BIPM, The International System of Units (SI), 9th edition (joule, watt, non-SI units: kWh); NIST SP 811 (conversion factors for kcal, ft·lb, hp); the specific heat of water per the NIST Chemistry WebBook; H. W. Dickinson, James Watt: Craftsman and Engineer, 1936. This page is for educational purposes.