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. The output of the model for a numerical simulation is the acceleration vs. time information for each vehicle in the simulated crash. Three low-speed crash tests were performed and the dynamic IF-D curve was measured in each crash. The analytical model was used to simulate the three low-speed crash tests in order to demonstrate the model's ability to describe the vehicle dynamics in a crash that has already occurred. The model is also used to perform parametric studies that show how the structural characteristics of the vehicles' bumpers and the closing speed affect the crash pulse and to demonstrate a technique to estimate the maximum severity of a low-speed crash that has already occurred.