The greatest possible acceleration and deceleration that the train may have is a and its maximum speed is v. Find the minimum time in which the train can get from one station to the next if the total distance between the stations is s. The train starts at rest and comes to a stop at it's destination.

this is a problem of kinematics, we will divide it in part to solve it more easily, one part is when it is accelerated and another when it is at constant speed

Let's start when it has acceleration, tell us what part of the rest

Vf = Vo + at

Vf = a t

S

ince the maximum speed is "v", let's calculate the time it takes to reach this speed

t = v / a

Let's call this time t1

Let's look for the time it takes to slow down when it reaches the other station

0 = v + (-a) t

.t = v / a

Let's call this time t2

Now we have all the times of the accelerated movement we can calculate how far they travel in these times.

Starting X = Vo t + ½ a t² = 0 + ½ a t²

X1 = ½ a (v / a)²

X1 = v² / 2 a

Braking x = Vot + ½ a t² =

X2 = v (v / a) - ½ a (v / a)²

X2 = v² / a - ½ v² / a

X2 = v² / 2 a

The same distance

Let's calculate the distance you travel at constant speed

X3 = Xt -x1-x2

X3 = s -v² / 2 a -v² / 2 a

X3 = s -v² / a

We can calculate the time it takes to travel this distance

V = x / t

t = x / v

t3 = (s- v² / a) / v

t3 = s/v - v/a

Total travel time is

t all = t1 + t2 + t3

t all= v / a + v / a + (s / v -v / a)

t all= s/v + v/a

This is the minimum time between the two stations