The Determination of Vehicle Speeds from Delta-V in Two Vehicle Planar Collisions

Abstract

The change of a vehicle's velocity, delta-V ( v), due to an impact is often calculated and used in the scientific investigation of road traffic collisions. In isolation however, this figure does not yield any information concerning the actual velocities of the vehicles and such information is often of prime concern to those investigating collisions. In this paper a method is developed which uses the change in velocity sustained by a vehicle in a planar collision to estimate the velocities of the vehicle before and after a collision. The key equations are derived from conservation of momentum, conservation of energy and restitution. As with the calculation of delta-V, the method requires an initial estimate of the principal directions of force. The pre and post impact angles of the vehicles' velocities can be used to obtain better estimates of the principal directions of force and of the coefficient of restitution. In collisions where it is difficult to analyse the vehicles' post-impact motion, this method provides a way to estimate the actual speeds of vehicles. To demonstrate the method, it is used to analyse one of the RICSAC collisions. The results of an analysis of other staged collisions illustrate the accuracy of the method.

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

“…The momentum-only models cannot provide a solution as to the pre-impact speed of either vehicle. However the CRASH model could be used to establish the change in velocity of each vehicle and thereby the pre-impact velocities using the model developed by Neades and Smith [8].…”

“…When 0   , the numerator in Equation (12) It is clear that a non-zero tangential coefficient of restitution may be required to maintain consistent results between the models. This necessitates some modification to the closing speed algorithm developed by Neades and Smith [8] through the incorporation of a tangential coefficient of restitution. Neades [14] describes how this may be achieved by substituting the definition of tangential restitution given by equation (6) into equation (23) …”

Vector model of vehicle collisions for inferring velocity from loss of kinetic energy with restitution on residual crush surface

“…In fact equation 3.5, which provides the key to solving for change of velocity, may be transformed through the substitution of I = mk 2 where k is the radius of gyration, J = mΔv, R ×Ĵ = h and δ = 1 + This corresponds in form to the main result in [22], a generalisation of 'the formula commonly used to calculate velocity change' to include restitution and confirms the correction subsequently noted regarding the index numbers of the delta terms [19]. The closeness of equation 4.1 to equation 3.5 is by no means apparent at first sight and the same is true of their derivations.…”

“…The standard model [19] considers the case where there is no relative movement between the vehicles at point P perpendicular to the direction of impulse after impact (figure 2, right) which may be expressed as…”

“…This is implemented by imagining that the impulse point P lies in the central region of the crush zone and that the restitution point Q lies on the residual crush surface. The mathematical model employed for this purpose is based on [22] for the assessment of velocity change as extended by [19] to include relative velocity and absolute velocity. This is one of the commonly used, 'essentially equivalent' models surveyed by [18], referred to here as the Smith-Neades or standard model.…”