The kinematical properties of the fragment ejection processes in the events that produced the presently known asteroid families have long been poorly known due to the presence of a couple of unknown angles (the true anomaly and the argument of the perihelion of the parent body at the epoch of the family formation) in Gauss' equations. In the present paper a general procedure for obtaining a reliable estimate of the unknown angles starting from the known values of the proper semi--major axis, eccentricity and inclination of the family members is described and extensively tested. The results of numerical simulations are highly encouraging when we consider some plausible structures of the ejection velocity fields, in agreement with those observed in laboratory experiments. In these cases, we show that the unknown angles can be reliably determined, and the ejection velocities of the fragments can be fairly well reconstructed. An application of this technique to a sample of real asteroid families shows structures of the resulting ejection velocity fields fairly plausible according to the evidence coming from laboratory experiments. An interesting by--product of this analysis is also an evaluation in each case of the direction of impact of the original projectile in the reference frame of the orbiting parent body. The relevance of these results for the physical studies of the asteroids and their collisional evolution is discussed.