mid-14c., also simply coat (mid-14c.); originally a tunic embroidered or painted with heraldic armorial bearings (worn over armor, etc); see coat (n.) + arm (n.2) and compare Old French cote a armer. Sense transferred in Middle English to the heraldic arms themselves. Hence turncoat, one who put his coat on inside-out to hide the badge of his loyalty (1550s).
1917, from air (n.1) + force (n.); first attested with creation of the Royal Air Force. There was no United States Air Force until after World War II. The Air Corps was an arm of the U.S. Army. In 1942, the War Department reorganized it and renamed it Army Air Forces. The National Security Act of 1947 created the Department of the Air Force, headed by a Secretary of the Air Force, and the U.S.A.F.