## Power, Force and Motion

Motion without friction is in most practical situations an unachievable ideal. There is always some friction and some work must always be done to overcome it.

If the friction force is constant for a distance moved then the work done to overcome that force is We can obtain another very useful equation from this by differentiation.  is constant so Consider the meaning of this equation. When a car is moviing at a velocity v on a level road, it is doing work at a rate Fv to overcome friction. The car engine can only supply power up to a certain maximum rate, When the speed is such that the car cannot accelerate any more because to do so would require that some of the energy supplied by the car engine is used to increase the cars kinetic energy, which is impossible because all the power supplied by the engine is being used to overcome friction.

In general though, the power supplied by a car is used up in several ways. It can be used to drive the car uphill (if the car is driving downhill, this energy is changes sign), increase the kinetic energy and speed, or overcome friction. We write  Assuming to be constant (in fact the car is losing mass because fuel is being burnt), we can write We can use the chain rule to simplify obtaining hence Finally, there is a relationship between and Assume the slope is at an angle to the horizontal then from the diagram below  Hence we can write  