Putnam 1972

A particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse.

Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$ and $B^{2n-1} = 1$ for some positive integer $n$. Find the value of $B$.