Equivalence of impact-phase models in two-vehicle planar collisions

Determining the pre-impact velocities of vehicles is of prime importance when investigating road traffic collisions. Two types of impact-phase model are in common use to achieve this purpose: those based on the conservation of linear and angular momentum as exemplified by the models presented by Brach and Brach and by Ishikawa, and those based on the computer reconstruction of automobile speeds on the highway (CRASH) program which explicitly includes the conservation of energy. A summary of the various models is provided to show how the models are related to each other, together with a brief discussion of their strengths and weaknesses. Of particular significance is the fact that, although there are differences between these models, it is shown that they are equivalent provided that certain conditions are satisfied, namely that the crush or impact plane is oriented perpendicular to the impulse. In addition, it is shown that they produce identical results from consistent input data. Explicit conversion factors between the models are provided, together with a novel method to transform the coefficients of restitution between various orientations of the crush plane. This facilitates comparison and movement between the models and it is shown that the choice of model utilised for an individual collision depends largely on the availability of particular data.