I have a stupid question:

How fast is a bath tub supposed to drain?

That's hard to say -- A bathtub drain with a short trap arm (the portion of the drain between the p-trap and the vent) will drain much more quick than a bathtub with a longer trap arm.

According to code, the maximum developed length of an 1-1/2 trap arm cannot exceed 42 inches and cannot exceed more than a 90 degree change in direction.