A: Linear Momentum has many assumptions which leads to sensitivities of the solution procedure in many impact configurations. Evaluating sensitivities in any solution procedure can be accomplished with a Monte Carlo Analysis. One of the key factors underlying a Monte Carlo analysis is a probability distribution of individual factors within the analysis.
There is no single Monte Carlo method; instead, the term describes a large and widely-used class of approaches.
A recent SAE paper by Wade Bartlett includes a presentation of how to do it with your spreadsheet program:
Conducting Monte Carlo Analyses with Spreadsheet Programs, Society of Automotive Engineers, SAE paper 2003-01-0487
The Monte-Carlo method or some other sensitivity analysis procedure is essential when applying a linear momentum solution procedure to accident reconstruction. Testing the ranges of input approximations is essential for a proper linear momentum solution.
A classic example of problems with the sensitivity of a linear momentum solution is when it is applied to a t-bone type collision.
- When a lighter car crosses the path of a heavier car/truck and the lighter vehicle is struck in the side by the heavier vehicle. If the heavier striking car/truck happens to swerve before the impact, either to the left or the right, the degree or two of change of impact angle can result in dramatic changes in the results of a linear momentum solution. Why? The swerve by the heavier vehicle will produce a change in the separation angles. The change in separation angles, if all attributed to the smaller vehicle speed (which it will be if the impact angle of the striking heavier vehicle is assumed to be 0 (zero) degrees) will dramatically change the linear momentum solution approximated speed of the smaller lighter vehicle. The result is that depending on the direction of the swerve, the small vehicle will be 'reconstructed' as either going very fast in the forward or reverse direction. Depending on the difference in the weights of the vehicles the assumption for impact angle of the heavier vehicle can result in very large errors in the analysis.
For an example of a program which tests ranges of inputs, see VSTAR. The program creates a graphics/map of the effects of the ranges of uncertainties in inputs to the solution procedure.
Another way to test and refine a linear momentum solution procedure is to run a sophisticated simulation program like SMAC. However it is not essential that you run SMAC.
You can alternatively (or in combination with a SMAC analysis) run a sensitivity analysis of your linear momentum solution to determine which input variables produce the most dramatic changes in the linear momentum results.
The following is a illustration of a sensitivity analysis: