Power is related to other quantities, for example the power involved in moving a ground vehicle is the product of the traction force on the wheels and the velocity of the vehicle. The output power of a motor is the product of the torque that the motor generates and the angular velocity of its output shaft. Likewise, the power dissipated in an electrical element of a circuit is the product of the current flowing through the element and of the voltage across the element.
Power is the rate with respect to time at which work is done; it is the time derivative of work:
where P is power, W is work, and t is time.
If a constant force F is applied throughout a distancex, the work done is defined as . In this case, power can be written as:
If instead the force is variable over a three-dimensional curve C, then the work is expressed in terms of the line integral:
As a simple example, burning one kilogram of coal releases much more energy than detonating a kilogram of TNT, but because the TNT reaction releases energy much more quickly, it delivers far more power than the coal.
If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula:
It is the average amount of work done or energy converted per unit of time. The average power is often simply called "power" when the context makes it clear.
The instantaneous power is then the limiting value of the average power as the time interval Δt approaches zero.
In the case of constant power P, the amount of work performed during a period of duration t is given by:
In the context of energy conversion, it is more customary to use the symbol E rather than W.
Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is also described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral:
where x defines the path C and v is the velocity along this path.
If the force F is derivable from a potential (conservative), then applying the gradient theorem (and remembering that force is the negative of the gradient of the potential energy) yields:
where A and B are the beginning and end of the path along which the work was done.
The power at any point along the curve C is the time derivative:
If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the mechanical advantage of the system.
Let the input power to a device be a force FA acting on a point that moves with velocity vA and the output power be a force FB acts on a point that moves with velocity vB. If there are no losses in the system, then
The similar relationship is obtained for rotating systems, where TA and ωA are the torque and angular velocity of the input and TB and ωB are the torque and angular velocity of the output. If there are no losses in the system, then
In a train of identical pulses, the instantaneous power is a periodic function of time. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. It is also called the duty cycle (see text for definitions).
In the case of a periodic signal of period , like a train of identical pulses, the instantaneous power is also a periodic function of period . The peak power is simply defined by:
The peak power is not always readily measurable, however, and the measurement of the average power is more commonly performed by an instrument. If one defines the energy per pulse as:
then the average power is:
One may define the pulse length such that so that the ratios
are equal. These ratios are called the duty cycle of the pulse train.
Power is related to intensity at a radius ; the power emitted by a source can be written as:
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^Fowle, Frederick E., ed. (1921). Smithsonian Physical Tables (7th revised ed.). Washington, D.C.: Smithsonian Institution. OCLC 1142734534. Archived from the original on 23 April 2020. Power or Activity is the time rate of doing work, or if W represents work and P power, P = dw/dt. (p. xxviii) ... ACTIVITY. Power or rate of doing work; unit, the watt. (p. 435)
^Heron, C. A. (1906). "Electrical Calculations for Rallway Motors". Purdue Eng. Rev. (2): 77–93. Archived from the original on 23 April 2020. Retrieved 23 April 2020. The activity of a motor is the work done per second, ... Where the joule is employed as the unit of work, the international unit of activity is the joule-per-second, or, as it is commonly called, the watt. (p. 78)
^"Societies and Academies". Nature. 66 (1700): 118–120. 1902. doi:10.1038/066118b0. If the watt is assumed as unit of activity...
^Halliday and Resnick (1974). "6. Power". Fundamentals of Physics.CS1 maint: uses authors parameter (link)
^Burning coal produces around 15-30 megajoules per kilogram, while detonating TNT produces about 4.7 megajoules per kilogram. For the coal value, see Fisher, Juliya (2003). "Energy Density of Coal". The Physics Factbook. Retrieved 30 May 2011. For the TNT value, see the article TNT equivalent. Neither value includes the weight of oxygen from the air used during combustion.