Ed M. answered 04/26/16
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Help with grammar, French, SAT Writing, the TOEFL and ESL.
I presume you're referring to the novel The Watsons Go to Birmingham--1963. It would also have been helpful if you had provided a specific passage from this novel that includes the phrase a mile a minute so that the figurative use of this phrase in the context could be illustrated and explained.
I don't have direct access to the entire novel, but according to an Internet search the following is found in the fiftieth paragraph of the second chapter:
I knew they weren't going to waste any time with this new guy, it was going to be real easy and real quick with him. He was like nobody we'd seen before. He was raggedy, he was country, he was skinny and he was smiling at everybody a mile a minute.
In the last sentence of this passage, a mile a minute is an adverbial phrase modifying the verb smiling, and thus this use is figurative since no human mouth--in spite of how big some people's seem to be--can extend for one mile in a smile or otherwise. Moreover, the phrase a mile a minute in its most literal sense refers to velocity, i.e., the distance (one mile) covered in a specific amount of time (one minute), but of course a smile cannot "move" in the literal sense. So because one mile per minute, i.e., 60 mph, is relatively fast speed, a mile a minute is often used figuratively, as in this example from the novel, to indicate rapid and/or consistent action not necessarily involving the covering of any kind of distance, as is the case with "this new guy" who is "smiling at everybody," i.e., continuously and presumably eagerly.