Okay.
When you want to go somewhere you program your flying car with a start and an end point of the journey. The onboard computer of every flying car is linked to a database of routes, so when you program a new route it'll be calculated such that it doesn't cross any other car's routes at any point. And that's the collision problem done for. (Only works with flying cars, it wouldn't work if you were confined to roads.)
Possible disadvantage: you have to get in early on the flying car craze to get good routes. After 20 years or so the only routes from Point Your House to Point Grocery Store left will have to go via Argentina and the North Pole, and you'll have to pack a huge jerry can.
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