I said "tungsten terminal velocity (for a man-shaped piece)". I.e. take some tungsten and make a statue out of it - I wanted to use existing numbers for humans to make a quick calculation, not recalculate for a tungsten arrow. Since it has a density 20 times that of an actual human (and thus 20 times the weight), the statue tops out at 4,47 times the terminal velocity. Given that a human diving head-first tops at about 300 mph, we get a bit over 1300 mph for the tungsten statue. Now streamline the same material into an arrow that has about 1/6 the projected area for its mass that a human would have. That's another 2,5x increase in terminal velocity. We are now looking at roughly 3250 mph terminal velocity, or close to a mile a second. Of course, said object is not going to actually reach that speed unless you drop it from about 100 miles up.
Now, to impart the same kinetic energy on an arrow weighing 5 pounds, you need 2.600.000 joules. For comparison purposes, the muzzle energy of a .50 BMG round fired from typical HMGs is 15.000 joules, the muzzle energy of an AK-47 round is 2.000 joules and a typical medieval arrow has about 1500 joules kinetic energy when fired.
Giving such kinetic energy however is difficult. You'd need a bow with a draw weight of 75 tons to manage 2/3 of it. If you are a giant and get a big metal bow made out of a huge laminated spring, you can reach those draw weights. For human-sized nonmagical bows the numbers are problematic. The typical laminated spring used in car suspensions could provide only about 2,5 tons of draw weight in a bow that is still sized for humans. You'd need considerably more advanced materials than steel and a good bit of engineering to get to the needed 75 tons. This doesn't mean it's impossible. It just means the bow would be a really, really expensive custom work and it would only work for someone of your mythic strength. You could not hand it off to someone else and hope they can use it - even guys with supernatural strength could not hope to draw it.
OTOH, it should be fairly simple to give vampires or other strong non-people powerful bows. 2,5 tons of draw weight is right in the alley of inhuman strength so go make some bows out of laminated metal springs and give them thin steel arrows.