Conventional, "fixed-block" train control systems might best be assumed for this puzzle.  Such a system is designed to detect trains in “track circuits” by the "shunting" of their wheels.  Each track circuit defines a "block" of track, which can be hundreds or thousands of feet in length.  Changes in speed commands must be delivered from the "wayside" to a  train based on the location of its "lead car" ("cab signaling," it's called).  New speed commands are also received as the train moves from block to block (see Confusion Free Codes).

At any given moment, the location of the train ahead is determined by its "tail car," which is assumed to be situated at the rear-most boundary of the track block in which it is detected.  For safety, it is also appropriate to assume that an impenetrable barrier is located at that boundary -- and that the barrier will stay there until the train ahead no longer occupies the block.  Accordingly, as the preceding train moves forward, the so-called "brick wall" gets torn down and immediately re-built at the next block boundary, leaping logically from block to block.

Sketches like the one below apply what is known in mathematics as a "phase-plane plot" -- a graph of velocity against displacement (in this case, train speed versus track location).

xC0 = vC² / 2 dS

...where dS is a constant deceleration as customarily modeled for "service braking."  The train comes to a stop at Point 3, which is deliberately targeted at a distance xB in approach to the brick wall located at Point 4.  The size of xB is mostly attributable to the fixed-block design and represents a "buffer" distance that allows for uncertainty in the stopped location of the train-in-trail.

Keep in mind that the train ahead is probably not stopped at all but traveling down the track at, say, vC, with its tail car at Point 5 or beyond.  If so, the brick wall at Point 4 may actually be torn down prior to the train-in-trail reaching Point 1, and no command for Trail Braking will be given.  Trains operating on their respective "time-table" schedules will generally stay much farther apart, but if the train ahead is delayed, a minimum headway distance xH must nevertheless be assured for "following move" separations.  To estimate the minimum headway time tH for trains operating at vC, then, we calculate...

tH = tM + [(vC² / 2 dS) + xB] / vC.

Solutions for the Trail Braking puzzle are provided by the graph on the right using "base-case" values as follows: train length xL = 700 feet, mode-change delay tM = 4 seconds, and service brake rate dS = 3 miles per hour per second (mphps).
 

Higher cruise speeds are desirable, obviously, for they will improve trip-times between stations, but higher speeds will also increase the minimum headways between trains.  That will seem counter-intuitive to some readers.  It may help to imagine driving a race-car in a video game always approaching a brick wall that randomly jumps ahead the length of a football field along the race-track.

Meanwhile, the abbreviation "CBTC" in the graph legend stands for Communications Based Train Control, an advanced system which achieves the effect of extremely short block lengths -- as if the brick wall gets torn down permanently.  The result is called "moving-block" train control.  The graph shows that CBTC will support the same headway (tH = 34 seconds) at cruise speeds over 90 miles per hour -- a potential savings of more than 30 seconds per mile in trip-time.  By the way, even a fixed-block system can handle more than a train a minute (tH = 60 seconds) at cruise speeds exceeding 100 miles per hour.

Oh right, there is just this one other matter.  Passengers.  And Station Stops.