OED states that both "a bow" as in the weapon and "to bow" or "a bow" as in to incline at the knee share a common etymology:
Etymology: Common Germanic: Old English boga, corresponding to Old Frisian boga, Old Saxon bogo (Middle Dutch booghe, Dutch boog), Old High German bogo (Middle High German boge, modern German bogen), Old Norse bogi (Swedish båge, Danish bue) < Germanic *bugon-, < stem bug- of beugan, to bend.
However the two words are pronounced quite differently. I was wondering when and why the two meanings diverged in pronunciation.