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?

We have that t * a = V, where t is time taken, a is acceleration. However, we also have that D = (a t©÷)/2, right? Therefore, we have that t = ¡î(2 D / a). Since the limit as a goes to 0 of that is infinity, the particle can take as long as it wants.