Remember that work done *W* by a force *F* acting on an object through a distance *s* is given by

If we differentiate both sides of the equation with respect to time, we get

This means that if a force *F* is acting on an object, and that object is moving at speed *v*, then *F* is doing work (on the object) at the rate of *Fv*. In other words, the power *P* delivered by a force *F* is given by *Fv*.

Consider a car fitted with an engine that provides a constant power *P*_{engine}. As the car moves, it experiences a drag force *D* that increases with speed of the car. Let’s assume that *D* is proportional to *v*^{2}

This means that the drag force is causing the car to losing energy at the rate of

At low speed, the car can gain KE since . On the other hand, at high speed, the car must lose KE since . The maximum speed of the car is thus given by

The power-speed graph illustrates the power dynamics pretty neatly.

The force-speed graph is also interesting.

Note that an engine that provides constant power cannot provide the same engine force *F*_{engine} at different speeds.

In fact, the math shows that *F*_{engine} is inversely proportional to *v*. (That’s why the *F-v* graph has a shape). So as the car accelerates, not only does the retarding drag force increases, the propelling engine force also decreases. Maximum speed occurs at the point where the two graphs meet.

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**Video Explanation **

Power Delivered by a Force

**Concept Test **

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