ABSTRACT
This chapter discusses the theory and practice behind a method for using energy and momentum conservation to reconstruct the crash phase of a coplanar accident. It starts with known exit conditions (as obtained from reverse trajectory analysis) and ends with the unknown impact conditions. It is backward-looking and is a reconstruction, as opposed to a simulation. Using energy conservation, it utilizes three equations instead of the usual two. The method assumes that the velocity vector direction of one vehicle is known. The three unknowns are the magnitude of that velocity vector, and the X- and Y-components of the other vehicle’s velocity vector. Calculations are performed on a rotated coordinate system aligned with the resultant linear momentum vector, which avoids the problem of the solution blowing up for the degenerate case of uniaxial collisions. Angular momentum at impact and separation are calculated, but not conserved. The method calculates as an output the Principal Direction of Force (PDOF) from the vehicle headings and velocities at impact, as it should be, rather than requiring it as an input. It is not necessary to define a center of impact or impulse. Crush energies are based on vehicle structural characterizations and crush profiles, and are inputs.
Since kinetic energy involves the squares of velocities, one of the equations is nonlinear. The three simultaneous equations are solved in closed form using the Quadratic Formula. This produces two roots, which have physical meanings. Linear and angular positions of the two vehicles at impact are related to those at separation by the assumed crash duration and the average velocities during the crash. The positions are adjusted by moving one or both of the vehicles, or changing the assumed impact duration, so that physical evidence at the scene and on the vehicles, including vehicle matchup, is recognized and accounted for.
Calculations are accomplished using a single spreadsheet, an example of which is provided. Other topics discussed in Chapter 26 include the development of the governing equations, the physical meaning of two roots, relative speeds and restitution coefficient, vehicle center of mass positions, vehicle heading angles at impact, calculation of the PDOF, and showing proof that the reconstruction satisfies the conservation laws upon which it is based.
