Traffic injury prevention · 2026
New modelling of equations for Delta-V calculation in low-speed rear-end collisions between vehicles.
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Garrote-Márquez Álvaro, Balboa-Vega Santiago, Fernández-Cabrera David, de-la-Prada-Rus Rafael, Pro-Martín José Luis, Zabala-Ibáñez Isabel
Аннотация
OBJECTIVE: Rear-end vehicle collisions at low speeds are among the most common types of traffic accidents in Europe, often leading to whiplash injuries. Precise estimation of the change in velocity (Delta-V, DV) is essential for accident reconstruction, particularly when evaluating injury thresholds. Traditional momentum-based methods assume conservation of linear momentum and neglect energy losses, which may produce significant inaccuracies when the striking vehicle brakes before impact. This study aimed to evaluate the discrepancies between theoretical and empirical DV under braking conditions and to develop a practical correction methodology to enhance the accuracy of accident reconstructions.
METHODS: Crash test data were extracted from the AGU Zürich database in collaboration with the Dynamic Test Center (DTC) and insurance partners. Ninety-seven low-speed rear-end collisions (≤21 km/h) involving contemporary vehicles (manufactured post-1998) were selected. Two groups were defined: 21 tests without braking and 76 tests with braking applied by the striking vehicle. Empirical and theoretical DVs were calculated using classical restitution-momentum equations. Errors were analyzed for both groups. Linear correlation analyses were conducted to identify predictors of error, and multiple regression models were developed using vehicle mass and differences in momentum (DP). Cross-validation techniques were employed to minimize bias given the relatively small dataset.
RESULTS: Predictive regression lines were then generated to correct DV estimates in braking scenarios because for the unbraked group, empirical and theoretical DVs were closely aligned, with mean errors below 4% for nearly all cases. In contrast, collisions with braking exhibited substantially greater discrepancies, with mean errors exceeding 11%. Strong correlations were observed between DP and DV errors (R2≈0.82). Multiple linear regression using the mass of the striking vehicle and DP achieved higher predictive accuracy (R2≈0.93 for errors in DV of the striking vehicle and R2≈0.91 for DV of struck vehicle). Application of the corrective regression models reduced DV errors to approximately 3%, representing an improvement in accuracy.
CONCLUSIONS: This study confirms that braking in low-speed rear-end impacts disrupts momentum conservation, leading to systematic under- or over-estimation of DV using traditional methods. By incorporating vehicle mass and momentum differences into regression-based corrections, DV estimates can be substantially improved using minimal and easily obtainable parameters. The proposed workflow offers accident reconstructionists a practical, empirically validated method to enhance the reliability of DV calculations in braking scenarios. While results are robust across diverse makes and models, future research should expand to other impact types, such as lateral and oblique collisions.
METHODS: Crash test data were extracted from the AGU Zürich database in collaboration with the Dynamic Test Center (DTC) and insurance partners. Ninety-seven low-speed rear-end collisions (≤21 km/h) involving contemporary vehicles (manufactured post-1998) were selected. Two groups were defined: 21 tests without braking and 76 tests with braking applied by the striking vehicle. Empirical and theoretical DVs were calculated using classical restitution-momentum equations. Errors were analyzed for both groups. Linear correlation analyses were conducted to identify predictors of error, and multiple regression models were developed using vehicle mass and differences in momentum (DP). Cross-validation techniques were employed to minimize bias given the relatively small dataset.
RESULTS: Predictive regression lines were then generated to correct DV estimates in braking scenarios because for the unbraked group, empirical and theoretical DVs were closely aligned, with mean errors below 4% for nearly all cases. In contrast, collisions with braking exhibited substantially greater discrepancies, with mean errors exceeding 11%. Strong correlations were observed between DP and DV errors (R2≈0.82). Multiple linear regression using the mass of the striking vehicle and DP achieved higher predictive accuracy (R2≈0.93 for errors in DV of the striking vehicle and R2≈0.91 for DV of struck vehicle). Application of the corrective regression models reduced DV errors to approximately 3%, representing an improvement in accuracy.
CONCLUSIONS: This study confirms that braking in low-speed rear-end impacts disrupts momentum conservation, leading to systematic under- or over-estimation of DV using traditional methods. By incorporating vehicle mass and momentum differences into regression-based corrections, DV estimates can be substantially improved using minimal and easily obtainable parameters. The proposed workflow offers accident reconstructionists a practical, empirically validated method to enhance the reliability of DV calculations in braking scenarios. While results are robust across diverse makes and models, future research should expand to other impact types, such as lateral and oblique collisions.