A particle is travelling from point A to point B. These two points are separated by distance D. Assume that the initial velocity of the particle is zero.

Given that the particle never increases its acceleration along its journey, and that the particle arrives at point B with speed V, what is the longest time that the particle can take to arrive at B?

The particle can take an arbitrarily long amount of time to travel from point A to point B. The particle follows the path of a helix between A and B. The radius and number of windings can be made arbitrarily large, hence the path is arbitrarily long.