为什么警察会用cop一词？

这个问题很有意思，查了一下GOOGLE，发现以上两位的都不怎么准确。

其实cop和police或“原指警察制服上的铜扣子，慢慢人们就以扣子代警察了”都没什么关系，但是cop可以说是从copper而来的。

根据下面的英文，我们可以看到cop是1700左右进入英语词汇（原本属于法语中的caper及后来的拉丁语capere，其实是今天capture的意思），是“抓捕、抓获”的意思。1846年，有报刊在cop后面加上了“er”的后缀，表示“抓捕或逮捕犯罪的人”。在1859年，cop作为copper的缩写开始流行起来。

我把英文原文引下来，大家可以做个参考：
Theories for the origin of cop abound, including the copper badge explanation you mention. We'd always been under the impression that the term was an acronym for "constable on patrol." Well, it seems we're both mistaken, as we learned from our roundabout quest for the answer.

Around the year 1700, the slang verb cop entered English usage, meaning "to get ahold of, catch, capture." By 1844, cop showed up in print, and soon thereafter the -er suffix was added, and a policeman became a copper, one who cops or catches and arrests criminals. Copper first appeared in print in 1846, the use of cop as a short form copper occured in 1859.