flying (adj.)youdaoicibaDictYouDict
early 15c., replacing forms from Old English fleogende "flying, winged;" present participle adjective from fly (v.1). The meaning "attached so as to have freedom of movement" (1670s) is the source of the nautical use (flying jib, etc.). Meaning "designed for rapid movement" (especially in military terms) is from 1660s; meaning "passing, hasty, temporary, rapidly constructed" is from 1763.

Flying fish is from 1510s; flying buttress is from 1660s. Flying Dutchman, ghost ship off the Cape of Good Hope, is attested since 1803 [John Leyden, "Scenes of Infancy," who describes it as "a common superstition of mariners"]. Flying colors (1706) probably is from the image of a naval vessel with the national flag bravely displayed. Flying machine is from 1736 as a theoretical device. Flying saucer first attested 1947, though the image of saucers for unidentified flying objects is from at least 1880s.
way (n.)youdaoicibaDictYouDict
Old English weg "road, path; course of travel; room, space, freedom of movement;" also, figuratively, "course of life" especially, in plural, "habits of life" as regards moral, ethical, or spiritual choices, from Proto-Germanic *wegaz (cognates: Old Saxon, Dutch weg, Old Norse vegr, Old Frisian wei, Old High German weg, German Weg, Gothic wigs "way"), from PIE *wegh- "to move" (see weigh).

From c. 1300 as "manner in which something occurs." Adverbial constructions attested since Middle English include this way "in this direction," that way "in that direction," both from late 15c.; out of the way "remote" (c. 1300). In the way "so placed as to impede" is from 1560s.

From the "course of life" sense comes way of life (c. 1600), get (or have) one's way (1590s), have it (one's) way (1709). From the "course of travel" sense comes the figurative go separate ways (1837); one way or (the) other (1550s); have it both ways (1847); and the figurative sense of come a long way (1922).

Adverbial phrase all the way "completely, to conclusion" is by 1915; sexual sense implied by 1924. Make way is from c. 1200. Ways and means "resources at a person's disposal" is attested from early 15c. Way out "means of exit" is from 1926. Encouragement phrase way to go is short for that's the way to go.