Particle Chasing Kinematics

People seemed to enjoy the previous problem (equilateral triangle kinematics), so here's the next one (again, from Irodov's book, but much tougher in my opinion):

Point A moves with constant speed (but not constant velocity!) |v|=5 m/s so that the vector v is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity u=4 m/s. At the initial moment of time, v is perpendicular to u and the points are separated by a distance of 27 m. How soon will the points converge (in seconds)?

This problem is quite difficult, but the solution is very interesting; it's one of the most interesting (and hardest) kinematics problems I know, so enjoy! If nobody gets the solution within a week (unlikely, but you never know), I'll post it so you're not kept hanging :).