Two identical metal balls connected at the ends of a light spring of force constant k form a dumbbell like structure. The dumbbell rests on a frictionless horizontal floor and third identical ball is placed at distance $l$  from the right ball of the dumbbell. All the three balls are in a line. A fourth identical ball moving with velocity u collides with left ball of the dumbbell

If all the collisions between the balls are elastic and head-on, the minimum distance $l$  so that the rightmost ball acquires speed u is $\pi u\sqrt{\frac{m}{xk}}$ where $x$ is------