**
***A large amount of energy is lost each time a car is brought to a stop by applying the brakes. The kinetic energy is converted into heat energy, which is useless in getting the car going again. For this reason, some engineers have experimented with the idea of storing energy in a flywheel when a car comes to a stop.*

*A flywheel is a massive ring which is free to spin about its center, like a bicycle wheel. The kinetic energy of a flywheel is given by*

*E _{K}*

*= I*

*ω*

^{2}*/2*

*where I is the moment of inertia and** **ω **is the angular frequency in radians per unit time. Thus the frequency f (in cycles per unit time) is f =** **ω**/2**π**. The moment of inertia is given by*

*I** **= MR ^{2}*

*where M is the mass of the flywheel and R is the radius.*

*Ideally the kinetic energy of the car would be transferred to the flywheel as the car comes to a stop. When the driver wants to go again, the energy would be transferred back to forward kinetic motion. Unfortunately, the efficiency of the two transfers will be less than 100%, so energy will be lost to heat. This energy can, of course, be made up by conventional means, such as burning gasoline.*

*For the following questions, use the notation:*

*M _{car}*

*is mass of the car*

*M is mass of the flywheel*

*R is radius of the flywheel*

*ω** **= 2**π**f** **= angular frequency of the flywheel*

The efficiency for converting forward kinetic energy to rotational energy is a, and the efficiency for converting rotational energy to kinetic energy is b. The car is initially going velocity v and the flywheel is still. The car slows to a stop by converting energy to the flywheel, and then it gains velocity again until the flywheel is still. What is the resulting speed of the car?