Hi I have been interested in this subject for a few years now and have came up with loads of little ideas. My most recent one being free energy through using a water wheel.
Could you run a small water wheel which generates enough electric to pump the water back up so it flows back down and turns the wheel leaving spare electric???
No. The energy to pump up the water in optimal circumstances (with a 100% efficiency water wheel, generator, wire, and pump) would be exactly equal to the energy sending it back down. In reality, however, some water misses the wheel and doesn't help to generate (but can only go back up if the pump pushes it), the generator loses energy to heat as do the wires and pump, and the pump itself generates shaking motion (if we assume it's like a bicycle pump) that wastes energy uselessly jiggling the pipes as a byproduct.
One kilogram dropped one meter will have a velocity of 4.429 m/sec and therefore a momentum of 4.429 units. Two kilogram dropped one meter will have a velocity of 4.429 m/sec and therefore a momentum of 8.859 units. Place all of that 8.859 units of momentum into one kilogram and it will rise to 4 meters. At four meters height a one kilogram mass has an energy of 39.24 joules. Two kilograms at one meter height has 19.62 joules. The input energy is 19.62 joules and the output is 39.24 joules. So: yes you can produce such a pump.