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Shuttle uses is not because of its cost (cheap or expensive) but because of its specific impulse.
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Exactly. The cost isn't important for the shuttle.
Factors like weight and a specific impulse are more important for shuttles.
For cars cost is more important than the specific impulse.
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You may find problems to store lots of electric energy because of the limitation of design of batteries, while if you separate H2 and O2 you have the energy being stored.
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There different ways to store energy. I don't know which one will be used for cars in 20 years, since I don't know what kind of new technology will be invented till then.
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Kerosene is a hydrocarbon which requires oil. Electrolysis requires water.
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Hydrocarbons can also made from ethonal or the way Craig Venter wants them to make via a specially designed bacteria.
In addition the difference between transforming hydrocarbons into kerosene and doing electrolysis is that electrolysis needs external energy.
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Electrolysis is used to produce oxygen inside the shuttle, and you get a remnant of H2.
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Where did you get that information from?
I don't think it makes sense to not do the electrolysis process at earth, since you don't get new water anyway while you are up there.
The space station might recycle waste from the astronauds that way, but it doesn't make sense for other water to be electrolysed up there.
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Of course it's free energy.
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Solar cells provide (once they are built) free energy. That hasn't much to do with electrolysis.