The speed of a train increases at a constant rate from zero to v, and then remains constant for an interval, and finally decreases to zero at a constant rate . If L be the total distance travelled , then the total time taken is 

Time taken to accelerate to v:

 

t₁ = v/α

 

Time taken to decellerate:

 

t₂ = v/β

 

Distance travelled in these time intervals:

 

In t₁:

s₁ = v²/2α

 

In t₂:

s₂ = v²/2β

 

Thus, distance for which speed is constant:

s₃ = L - v²/2(1/α + 1/β)

Time taken for this:

t₃ = s₃/v = L/v - v/2(1/α+1/β)

 

Total time taken:

 

t₁+ t₂+ t₃ = L/v + v/2(1/α+1/β)