Comparing arrows and bullets is ridiculous. They will cause death by different methods. Small high velocity projectiles (bullets) kill by vital organ interruption, but part of the equation is kinetic energy (ft/lbs). The impartation of that energy to the target (animal, etc.) has an effect, mostly shocking the central nervous system. But the shock effect alone does not kill. I've seen deer shot in the ass with a 338WM run away, not to be recovered.
Arrows, on the other hand, are much different. They kill by vital organ interruption as well, but purely by hemorrhage. The amount of kinetic energy transferred to the target is negligible. Getting hit, even by a 1/4# war arrow @ 170 fps, would get your attention in armour, but the amount of kinetic energy would not kill you. And armour penetration, as has been shown over and over, is only possible at very close range... 20-30 yards. War arrows are no different, they killed by hemorrhage by hitting vulnerable areas, and were devastating to unarmoured horses.
I have shot a vast number of WT deer through the chest with my 300WM, and not ONE of them was knocked off its feet. Some didn't even show any sign of being hit. Most bullets were a complete pass through. The only devastating, knocked off their feet results, were head shots, but for a whole different reason... massive trauma to the CNS.