Will a thinner object rust faster than a thicker object?

Depends how much rust are you looking for, the quality of your iron and the properties of your sample water types. Most often, nails are galvanized or treated in some way to resist rusting and may takes weeks or longer to show any real signs of rusting. Why does water rust less than coca cola, or apple juice?

And why does the iron in the apple juice turn black? Coca-Cola and apple juice have a lower pH than water. Low pH solutions will rust iron faster, and the reason it turns black in apple juice is much like a blue and yellow marker make green; the compounds in apple juice, when mixed with iron, form a material black in color.

I have an indoor water fountain with metal cups are now rusting...” To remove the rust, drain the fountain let it dry, Go buy CLR and a stiff brush rinse and scrub. As for preventing the rust from coming back, the short answer is there's nothing you can do.