Reasoning behind the mathematics in 'planner' module (in the key of 'Mathematica')
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s == speed, a == acceleration, t == time, d == distance

Basic definitions:

  Speed[s_, a_, t_] := s + (a*t) 
  Travel[s_, a_, t_] := Integrate[Speed[s, a, t], t]

Distance to reach a specific speed with a constant acceleration:

  Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, d, t]
    d -> (m^2 - s^2)/(2 a) --> estimate_acceleration_distance()

Speed after a given distance of travel with constant acceleration:

  Solve[{Speed[s, a, t] == m, Travel[s, a, t] == d}, m, t]
    m -> Sqrt[2 a d + s^2]    

  DestinationSpeed[s_, a_, d_] := Sqrt[2 a d + s^2]

When to start braking (di) to reach a specified destionation speed (s2) after accelerating
from initial speed s1 without ever stopping at a plateau:

  Solve[{DestinationSpeed[s1, a, di] == DestinationSpeed[s2, a, d - di]}, di]
    di -> (2 a d - s1^2 + s2^2)/(4 a) --> intersection_distance()

  IntersectionDistance[s1_, s2_, a_, d_] := (2 a d - s1^2 + s2^2)/(4 a)