Wired article discussing the physics of the black arrow.
Bad news: it shouldn't have enough speed to kill the dragon.
But my question is: What if the arrow is lighter, i.e. mithril arrow? In fan discussions, mithril is assumed to be aluminum or titanium....because it is hard, I always backed titanium.
The main argument against the black arrow being of mithril is that is that mitril is silver, and the arrow is black..similar to Turin's black blade Gurthang This sword was reforged from Anglachel, which was meteorite meta; in origin but stronger than earthen steel.
Turin used Gurthang to kill the dragon Glaurung...so if the black arrow was made from a similar metal, it would indeed be able to kill Smaug.
a discussion of meteorite weapons can be found at strangehistory.
and at wikipedia.