## Conservation of Energy for a Car Moving on a Slope

The power supplied by a car engine is in general used in several different ways.

1. It is used to overcome air resistance.

2. Some is used to overcome friction between the road and the wheels, in the internal machinery of the car.

3. The air in the wheels is set in motion and experiences heating.

4. Some is used to increase the kinetic energy of the car, or changed to heat in the brakes when the car slows down.

5. Some is used to overcome the force of gravity if the car moves up the slope, or, if the car is moving down the slope, the gravitational potential energy lost is changed into other forms of energy.

In general we can say that the power provided by the engine is equal to the sum of the powers produced in each of the processes described above. We consider only friction, kinetic energy and gravitational potential energy here. A car is shown below moving up a hill at an angleto the horizontal.

The driving force supplied by the cars engine isInseconds the car moves a distanceso the engine supplies energy equal to

The resistance to motion isso the energy needed to overcome it is, by the same argument as above, equal toAs shown by the diagram below, the height of the car increases byin 1 second, so the increase in gravitational potential energy inseconds is

Finally, suppose that the velocity of the car changes fromtoinseconds, The increase in kinetic energy is

Putting all these together gives

Dividing bygives

Now letthenis the instantaneous acceleration andso and

Finally, in this expression,is a common factor so