What are fast clocks? How do I use them?
Fast clocks, as their name implies, are clocks that run faster than the standard 12-hour clock, Typically, they run 6 or 12 times faster which translates into a "fast day" every 2 (6:1) or 1 (12:1)
hours.
The purpose of a fast clock is to permit timetable operations within the constrained environment of a model railroad, No more, no less, Three principal factors are accounted for in this manner.
The first, of course, is distance, If a real train were to travel an hour between stops, at, say, 50 mph, we'd need 50 scale miles of track -- over 3000 feet in HO -- to absorb that hour, Most of us
don't have quite that much room for our layouts.., By scaling time at, say, 10:1, we reduce that to 300 feet -- still a lot, for most people, but it's starting to approach something manageable, Note,
that an N-scaler would need ``only'' 1650 feet of track for the same situation, and hence could get away with a slower clock -- but your equations would have the clock run faster.
The second major reason for scale time is to absorb the differences in how long certain operations take, For example, when we couple or uncouple cars, we don't have to worry about air hoses, pumping
up the brake line, having someone physically there to pull the pin, etc, We also don't have to have our brakier walk the full length of the train to throw switches, put down fuses, etc, All of these
things are largely invariant with respect to scale, too.
Finally, most of us don't have ``real time'' to devote to operations, (If nothing else, we need to do things like build cars, engines, factories, etc.) If I run my clock at 10:1, I can reasonably
expect to get a full day's operation into one operating session -- and thus I can have trains going out and back, meeting lots of other trains, etc.
Fast time is not meant to try to scale down time, but to compensate for lack of distance on a layout and to fit operating sessions between a two and four hour time frame (some do go longer, depending
on the ratio and if they're run on a weekend).
Anyway, I've operated on layouts which have used a 12:1 ratio (one real hour equals 12 fast ones) and 6:1 (one real hour equals six fast ones), I've found the 6:1 sessions to more enjoyable as you
have time to do things like switch at reasonable speeds and yes, even, think, At 12:1, you sometimes feel that you are moving 87 times faster,
Should also note that fast time is used to schedule trains to different parts of a layout and that having a train depart a town and arrive at another one an hour later is more realistic than having
it arrive 10 minutes later real time.