lagyoudaoicibaDictYouDict[lag 词源字典]
lag: English has three distinct words lag. The verb ‘fall behind’ [16] is perhaps of Scandinavian origin (Norwegian has lagga ‘go slowly’), although a link has been suggested with the lag of fog, seg, lag, a dialect expression used in children’s games which represents an alteration of first, second, last. Lag ‘insulate’ [19] comes from an earlier noun lag ‘barrel stave’, which was also probably borrowed from a Scandinavian language (Swedish has lagg ‘stave’); the original material used for ‘lagging’ was wooden laths.

And finally the noun lag ‘prisoner’ [19] seems to have come from an earlier verb lag, which originally meant ‘steal’, and then ‘catch, imprison’; but no one knows where this came from.
[lag etymology, lag 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.).