Power in physics

Power P is the rate of doing work or transferring energy: P = W/t, or instantaneously P = dW/dt. SI unit: watt (W = J/s). Widespread alternatives: kilowatt (kW = 10³ W), megawatt (MW), imperial horsepower (1 HP = 745.7 W), metric horsepower (1 CV/PS = 735.5 W). Power measures how fast energy is delivered, distinct from total energy — a lightning bolt delivers GW for microseconds but little total energy; a bulb delivers watts for hours, accumulating more.

Mechanical power

For a force F moving its application point at velocity v (angle θ between F and v): P = F · v · cos(θ). Parallel force (cos θ = 1): P = F·v. Example: a car at 100 km/h (27.8 m/s) against 400 N of drag needs P ≈ 11.1 kW ≈ 14.9 HP. Fundamental for sizing motors, hoists, pumps — any machine applying force in motion.

Electrical power (DC and AC)

In DC: P = V · I. A 5 V, 2 A USB charger delivers 10 W. In AC with phase shift φ: P_active = V_RMS · I_RMS · cos(φ) (watts actually dissipated), P_apparent = V_RMS · I_RMS (VA), P_reactive = V_RMS · I_RMS · sin(φ) (VAR). Power factor cos(φ) rates efficiency of transfer: 1 = purely resistive, 0 = purely reactive. Motors and transformers typically have 0.7-0.9; capacitor banks compensate.

Rotational power

For a body rotating at ω (rad/s) under torque τ (N·m): P = τ · ω. Heart of motor sizing: torque and power curves relate through ω. From RPM: P = τ · rpm · 2π/60. A 200 N·m motor at 3000 rpm: P ≈ 62.8 kW ≈ 84.3 HP. Electric motors peak torque at low speed; combustion engines peak power at higher RPM.

Efficiency

Efficiency η = P_out / P_in (dimensionless, often in %). Always ≤ 1 by the second law of thermodynamics; losses become heat, friction, radiation, noise. Typical values: modern electric motor 90-96%; petrol engine 25-35%; diesel 35-45%; PV panel 20-22%; LED 50-60%; coal plant 40%; combined-cycle gas 60%. Maximum theoretical for a heat engine between T_hot and T_cold (K): Carnot efficiency η_C = 1 − T_cold/T_hot.

Unit conversions

• 1 HP (imperial) = 745.700 W (SAE convention).
• 1 CV/PS (metric) = 735.499 W (Europe, Japan).
• 1 BTU/h = 0.293 W.
• 1 ton of refrigeration = 3517 W.
• 1 ft·lbf/s = 1.356 W.
• 1 kcal/h = 1.163 W.
• 1 erg/s = 10⁻⁷ W.

Power and energy

Power is instantaneous; total energy is the time integral E = ∫P dt. For constant P: E = P·t. Common energy unit: kilowatt-hour (kWh = 3.6 MJ = 1 kW × 1 h). Washing machine at 2 kW for 1.5 h → 3 kWh. 100 W LED TV on 4 h/day → 0.4 kWh/day = 146 kWh/year.

How to use

Four modes: mechanical (W, t or F, v → P); electrical (V, I, optional cos(φ) → P active/apparent/reactive); rotational (τ, ω or rpm → P); efficiency (P_in, P_out → η%, losses). Auto-conversion to kW, HP, CV. SI units.

Notable values

LED bulb 10 W; kettle 2000 W; hair dryer 1800 W; small car 55 kW (75 HP); high-speed train 8-10 MW; nuclear reactor 1 GW; Sun 3.83·10²⁶ W; human brain 20 W; walking ~200 W; elite cyclist sprinting 1500 W; Usain Bolt at peak ~2500 W. Global average per person ≈ 3 kW continuous (≈ 26 MWh/year).

The many faces of power and its units

Power is the rate at which energy is transferred or transformed, i.e. energy divided by time. The same quantity takes different forms by context: in mechanics P = W/t = F·v (force times velocity), in rotation P = τ·ω (torque times angular velocity), in electrical engineering P = V·I·cos φ. The SI unit is the watt (W = J/s), named after James Watt, the Scottish engineer who improved the steam engine in the 18th century. It was Watt who introduced horsepower as a commercial comparison with real horses: today 1 imperial HP equals 745.7 W and 1 metric CV 735.5 W. In practice multiples such as the kilowatt (kW) and megawatt (MW) are used: a nuclear plant delivers about 1000 MW.

Efficiency, apparent power and power factor

No real machine converts all input energy into useful output: efficiency η = P_out/P_in is always below 100% due to friction and heat losses. An electric motor reaches 90-95%, a petrol engine barely 25-35%. In alternating current, power splits: active power P = V·I·cos φ (in watts) does real work, reactive power Q = V·I·sin φ (in var) oscillates back and forth without doing work, and apparent power S = V·I (in volt-amperes) is their vector combination. The power factor cos φ measures the transfer efficiency: low values imply high currents and line losses, which is why industries correct their power factor with capacitor banks. These concepts are central to power engineering.

Updated July 2026