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The purpose of this study was to develop a numerical analytical model of collinear low-speed bumper-to-bumper crashes and use the model to perform parametric studies of low-speed crashes and to estimate the severity of low-speed crashes that have already occurred. The model treats the car body as a rigid structure and the bumper as a deformable structure attached to the vehicle. The theory used in the model is based on Newton's Laws. The model uses an Impact Force-Deformation (IF-D) function to determine the impact force for a given amount of crush. The IF-D function used in the simulation of a crash that has already occurred can be theoretical or based on the measured force-deflection characteristics of the bumpers of the vehicles that were involved in the actual crash. The restitution of the bumpers is accounted for in a simulated crash through the rebound characteristics of the bumper system in the IF-D function.