crack (n.)youdaoicibaDictYouDict[crack 词源字典]
"a split, an opening," mid-15c., earlier "a splitting sound; a fart; the sound of a trumpet" (late 14c.), probably from crack (v.). Meaning "rock cocaine" is first attested 1985. The superstition that it is bad luck to step on sidewalk cracks has been traced to c. 1890. Meaning "try, attempt" first attested 1830, nautical, probably a hunting metaphor, from slang sense of "fire a gun."
At their head, apart from the rest, was a black bull, who appeared to be their leader; he came roaring along, his tail straight an end, and at times tossing up the earth with his horns. I never felt such a desire to have a crack at any thing in all my life. He drew nigh the place where I was standing; I raised my beautiful Betsey to my shoulder, took deliberate aim, blazed away, and he roared, and suddenly stopped. ["A Narrative of the Life of David Crockett, Written by Himself," Philadelphia, 1834]
Adjectival meaning "top-notch, superior" (as in a crack shot) is slang from 1793, perhaps from earlier verbal sense of "do any thing with quickness or smartness" (Johnson). Grose (1796) has "THE CRACK, or ALL THE CRACK. The fashionable theme, the go." To fall through the cracks figuratively, "escape notice," is by 1975. [crack etymology, crack origin, 英语词源]
fall (v.)youdaoicibaDictYouDict
Old English feallan (class VII strong verb; past tense feoll, past participle feallen) "to drop from a height; fail, decay, die," from Proto-Germanic *fallan (cognates: Old Frisian falla, Old Saxon fallan, Dutch vallen, Old Norse falla, Old High German fallan, German fallen, absent in Gothic).

These are from PIE root *pol- "to fall" (cognates: Armenian p'ul "downfall," Lithuanian puola "to fall," Old Prussian aupallai "finds," literally "falls upon").

Meaning "come suddenly to the ground" is from late Old English. Of darkness, night, from c. 1600; of land sloping from 1570s; of prices from 1570s. Of empires, governments, etc., from c. 1200. Of the face or countenance from late 14c. Meaning "to be reduced" (as temperature) is from 1650s. Meaning "die in battle" is from 1570s. Meaning "to pass casually (into some condition)" is from early 13c.

To fall in "take place or position" is from 1751. To fall in love is attested from 1520s; to fall asleep is late 14c. To fall down is early 13c. (a-dun follon); to fall behind is from 1856. Fall through "fail, come to nothing" is from 1781. To fall for something is from 1903.

To fall out is by mid-13c. in a literal sense; military use is from 1832. Meaning "have a disagreement, begin to quarrel" is attested from 1560s (to fall out with "quarrel with" is from late 15c.).