Stop on a dime
I had a vague memory about the phrase "stop on a dime" being connected with trains. A search online leads to some references about the area of contact between a train wheel and the track being about the size of a dime. So trains really do "stop on a dime". However, in this case the fact that trains are stopping on a dime means that it takes them a great distance to stop. I haven't come across anything more definitive than that.