Mathematical Methods for Accident Reconstruction

Summary: Accidents can be mathematically reconstructed to model accident risk and to improve safety equipment designs.

Accident reconstruction is important for understanding how accidents happen and for preventing accidents in the future. Principles and techniques from physics, mathematics, engineering, and other sciences are used to quantify critical variables and calculate others. For example, the initial speed of a suddenly braking vehicle can be determined by mathematically analyzing tire skid and yaw marks. The length of skid marks is a function of vehicle velocity and the amount of friction between the wheels and the road surface. In the case of yaw or circular motion, the radius of the yaw mark is also a factor in the calculation, as well as the elevation of the road. Speed can also be calculated from the trajectories, angles, and other characteristics of objects struck by a speeding vehicle, or between two or more colliding vehicles. Investigators may use distances and angles to determine the original positions of passengers who have been ejected from a vehicle. For more complex modeling, mathematicians, engineers, and other accident reconstructors rely on principles and equations from physics, such as those governing energy and momentum, as well as vehicle specifications, mechanical failure analyses, geometric characteristics of highways, and quantification of visibility, perception, and reaction. Data from both real accidents and staged collisions, along with statistically designed safety analyses and other methods such as stochastic modeling, are often used to construct accident simulations and visualizations for use in a wide variety of contexts, including legal proceedings. Actuaries use accident data to model accident risk, which in turn influences insurance rates and public policy, such as seat belt and helmet laws.

Modeling Accident Reconstructions

Accidents related to travel and transportation can have a variety of negative consequences including personal injury and death. The analysis of accidents can lead to improved designs of vehicles and reduced fatalities as well as warning travelers about potential risks of travel. In reconstructing accidents, evidence from photographs, videos, eyewitnesses, or police reports is collected. Decision trees are used to ask questions at each stage of reconstruction and help decide the closest accident scenario dictated by the available evidence. In such reconstructions, probability must be assigned for the likely cause of the accident and for the particular accident type among the possible accident scenarios based on the available evidence. Stochastic modeling is used to help solve such problems in accident reconstructions.

Uses of Accident Reconstructions

Another important aspect of accident reconstructions is to estimate the probability of occurrence of various types of injuries one may suffer in accidents. Such probability estimates are used to help calculate travel insurance. By nature, accidents happen randomly and—since the types of injuries suffered in accidents also vary randomly—it is important to model accident types and predict the kinds of injuries one may suffer in different accident types. Such models can help prepare communities with the optimal number of emergency services and also help doctors prepare for any unique types of injuries they are likely to deal with.

A typical problem is determining the types of special medical facilities that should be established to deal with travel-related accidents in a city. Such problems require stochastic modeling based on past data, which will help in simulating different types of accidents. Simulations help in planning emergency services to deal with accidents. Accident reconstructions may also help in forecasting the number of accidents of different types likely to happen in the near future, which may lead to better planning of the health, emergency, and disaster management facilities in the city.

Safety and Design Using Accident Reconstructions

Accident reconstructions also may help in improving vehicle design. Incorporating safety devices in vehicles is also a very important aspect of design. Safety devices, which help in avoiding severe injuries to passengers because of accidents, are designed with the help of accident reconstruction and are always a matter of high priority. Simulations can be used to develop sensors that can give an early warning about impending accidents or reduce the speeds of vehicles—thereby reducing the severity of an accident. In creating such designs, mathematical optimization methods are used to determine the optimal cost and space to be allotted. Another crucial application of accident reconstruction and accident modeling is driver training. Sophisticated simulators can be used to simulate different accident scenarios and train drivers to react appropriately to each situation in real time. These simulators are based on algorithms and use random number generators to simulate accident situations. Well-developed algorithms that closely simulate real accidents are needed to reduce—or even eliminate—major accidents.

Bibliography

Brach, Raymond, and R. Matthew Branch. Vehicle Accident Analysis and Reconstruction Methods. Warrendale, PA: SAE International, 2005.

Franck, Harold, and Darren Franck. Mathematical Methods for Accident Reconstruction: A Forensic Engineering Perspective. Boca Raton, FL: CRC Press, 2009.