After analyzing existing vehicle motion data for low speed rear-end impacts, this paper develops an efficient mathematical model to simulate such testing. The proposed idealization is believed to be as simple a model as is possible for delivering reasonable time history responses of the two vehicles involved, i.e. a linear two-degree of freedom mass/spring/dashpot system. A closed-form solution of the equations is then developed and numerical results based upon it are compared to the time histories of the actual vehicle motions and peak loads. Experimental data is used to determine the effective stiffness and damping for the two degree of freedom model. Comparisons of results obtained from the analysis with scaled test data agree favorably, thus identifying the basic design parameters which could be used to minimize vehicle damage and passenger loading. It is shown that the availibility of certain gross vehicle data would obviate the need for running future elaborate, costly and destructive low speed crash tests. Recommendations are made regarding the basic vehicle bumper stiffness and damping data needed to employ the present model, and to design vehicles with improved responses to low speed impacts.