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I know that normally we say travel by car or in a car, but would it be acceptable to say 'on a car'?
If you mean on top of a car's roof then "on a car" would be correct. Otherwise it would be "in a car"
Honestly I don't see why "on a train" is correct english. "on a boat" makes sense, because originally people would stand on top of a boat. Same for "on a bike", since you're definitely not in it.