"mature," late 14c., past-participle adjective from grow up. The noun meaning "adult person" is from 1813, short for grown-up person, etc.
"a working to overtake a leading rival," by 1971, probably a figurative use from U.S. football in reference to being behind in the score. The verbal phrase catch up was used from early 14c. in the sense of "raise aloft," it is attested from c. 1400 as "to take up suddenly," and by 1846 in the sense of "get to the same point, overtake;" see catch (v.) + up (adv.).
also lockup, "detention cell for offenders," 1838, perhaps short for earlier lock-up house; from the verbal phrase. Meaning "action of locking up" is from 1845. The verbal phrase lock (someone) up in a dwelling, prison, etc., is from early 15c. Of things, "to hold in safekeeping or concealment," also early 15c. See lock (v.) + up (adv.). To lock up (intransitive) "lock all the doors" (of a house, shop, etc.) is from 1901.