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11.5 Power by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

vCalc Companion Formulas | |

vCalc Formulary | 11.5 Power |

`P=(DeltaE)/(Deltat)` | Power |

A car may have plenty of energy in its gas tank, but still may not be able to increase its kinetic energy rapidly. A Porsche doesn't necessarily have more energy in its gas tank than a Hyundai, it is just able to transfer it more quickly. The rate of transferring energy from one form to another is called **power**. The definition can be written as an equation,

where the use of the delta notation in the symbol `DeltaE` has the usual interpretation: the final amount of energy in a certain form minus the initial amount that was present in that form. Power has units of J/s, which are abbreviated as watts, W (rhymes with “lots”).

If the rate of energy transfer is not constant, the power at any instant can be defined as the slope of the tangent line on a graph of `E` versus `t`. Likewise `DeltaE` can be extracted from the area under the `P`-versus- `t` curve.

? The electric company bills you for energy in units of kilowatt-hours (kilowatts multiplied by hours) rather than in SI units of joules. How many joules is a kilowatt-hour?

? 1 kilowatt-hour = (1 kW)(1 hour) = (1000 J/s)(3600 s) = 3.6 MJ.

? A typical person consumes 2000 kcal of food in a day, and converts nearly all of that directly to heat. Compare the person's heat output to the rate of energy consumption of a 100-watt lightbulb.

? Looking up the conversion factor from calories to joules, we find

`DeltaE=2000 kcal×(1000 cal)/(1 kcal)×(4.18 J)/(1 cal)=8×10^6 J`

for our daily energy consumption. Converting the time interval likewise into mks,

`Deltat=1 day×(24 hours)/(1 day)×(60 min)/(1 hour)×(60 s)/(1 min)=9×10^4 s`.

Dividing, we find that our power dissipated as heat is 90 J/s = 90 W, about the same as a lightbulb.

It is easy to confuse the concepts of force, energy, and power, especially since they are synonyms in ordinary speech. The table on the following page may help to clear this up:

force | energy | power | |

conceptual definition | A force is an interaction between two objects that causes a push or a pull. A force can be defined as anything that is capable of changing an object's state of motion. | Heating an object, making it move faster, or increasing its distance from another object that is attracting it are all examples of things that would require fuel or physical effort. All these things can be quantified using a single scale of measurement, and we describe them all as forms of energy. | Power is the rate at which energy is transformed from one form to another or transferred from one object to another. |

operational definition | A spring scale can be used to measure force. | If we define a unit of energy as the amount required to heat a certain amount of water by a `1°C`, then we can measure any other quantity of energy by transferring it into heat in water and measuring the temperature increase. | Measure the change in the amount of some form of energy possessed by an object, and divide by the amount of time required for the change to occur. |

scalar or [4] vector? | vector --- has a direction in space which is the direction in which it pulls or pushes | scalar --- has no direction in space | scalar --- has no direction in space |

unit | newtons (N) | joules (J) | watts (W) = joules/s |

Can it run out? Does it cost money? | No. I don't have to pay a monthly bill for the meganewtons of force required to hold up my house. | Yes. We pay money for gasoline, electrical energy, batteries, etc., because they contain energy. | More power means you are paying money at a higher rate. A 100-W lightbulb costs a certain number of cents per hour. |

Can it be a property of an object? | No. A force is a relationship between two interacting objects. A home-run baseball doesn't “have” force. | Yes. What a home-run baseball has is kinetic energy, not force. | Not really. A 100-W lightbulb doesn't “have” 100 W. 100 J/s is the rate at which it converts electrical energy into light. |

11.5 Power by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.