If i hook up a 12v car battery to a car alternator that is driven by a 12v (1/2 horsepower) electic motor...?

yes the battery will go out it'll probably take less than 24hrs with a fully charged new deep cycle battery

Thats called perpetual motion, and it will never work. when it generates 12v and uses 12v to turn the motor that generates it, FRICTION will always play as a negative effect and will slow down and quit eventually.

With the losses inherent in the motor and the alternator, the battery will gradually go flat. Actually a motor itself puts some power back into the circuit, it uses a lot to start but them the amount of current drops considerable as it runs. I'm not sure what your question has to do with cars and transportation though.

No, and here's why. It takes more energy for the motor to turn the alternator than the alternator produces. You also have energy losses due to vibration, heat and friction.

sure is the way my grand dad kelp his boat battery charged.. but really a battery charger is cheaper.. think magnets and spinning poles north one side and south the other seen that on johnny Carson was a non stop motor that produce electric current til the bearing when out thnk he said he had one running 10 yrs before it stop

ya it works, but the battery will always have a 12v drain on it so you wont be doing anything, you need a motor that draw less an runs around 1500 rpm, the voltage regulator on the alternator will keep the battery form over charging, an if you add a voltage converter you can get AC power to run things off of. i have a patent on the idea.

why would you want to!

Good luck. There is a scientific law that you can never get out more energy than you put in because of gravity and friction. What you are talking about has been tried and failed so many times. Try it for yourself and fail for yourself.