How could you find out whether copper is harder or softer than the plastic used in a plastic cup?
Scratch them with something you know is harder than both of them. Like a nail. Nails are usually made of iron or stainless/galvanized steel, so they're harder than both plastic and copper.
There is a certain kit that you use to find the hardness of things, I think it was a hardness test. Unfortunately I don't know where you get them...
test drill both of them and compare (?)