Part 2- Stating the relationship or calculation

Part 3- Specifying the output

Notice that each calculation or operation can be explained in a text box beside it. Adding text descriptions are not necessary but add greatly to the understanding of the template. Notice also that Mathcad performed the calculation symbolically taking the givens in Step 1 and performing the calculation in step 2. The result is output in Step 3. This simple output statement in step 3 could have just as easily been a table of displacements or a graph. Now on to some examples.

I have prepared the solution of the motion formula of a vehicle as an example of Mathcad's basic possibilities that are useful in a field of accident reconstruction. The one-dimensional model of Newton's second law for the slowing of a vehicle (without engine braking) would appear like:

where: m - mass of a vehicle, x - distance, t - time, g - gravity acceleration, a  - road slope, f  - rolling friction coefficient, k - f change coefficient (f dependence on velocity), g  - air density, A - effective vehicle front surface, c_x  - aerodynamic drag coefficient, m  - friction coefficient. The analytic solution of this differential equation is rather difficult.

The simple way to find the solution of the common differential equation:

is numerical integration with the Euler method. The basis for this method is to approximate the solution for all variables between x_n and x_n+1 with the solution for x_n like:

The solution to equation (1) consists of presenting this equation as a recurrent formula, like (3). In Mathcad it appears as follows (it is part of the template named vehicle.mcd included to this article):

Solutions for distance, speed, acceleration, and kinetic energy are presented in diagrams on the template. As an example Figure 1, included below, is the diagram for acceleration (deceleration in fact - note the change after the first second - this is the decreasing influence of aerodynamic and rolling drag):

The general approach is to prepare templates for the calculations of many types of accidents (sometimes with several variants). For example, pedestrian impacts by a skidding vehicle, impacts with a tree or post after skidding, car to car collisions, and so on. Each of these types of accidents is calculated in similar way. So if I have to analyze an impact with a tree once, the next impact with a tree will be similar. If I have a method for one pedestrian impact I can calculate the next one in a similar way. Of course each event must be considered individually but the basic calculation method, and template, will be similar.

Example 2

For example, an accident with an impact into a tree might occur like this: a vehicle is braked (skidding), it impacts the tree and rebounds to rest. The vehicle loses its kinetic energy during each phase of the event. Each loss can be evaluated and the sum will represent the initial velocity of the vehicle as illustrated below (this is a portion of the tree.mcd template):

Figure 2

Figure 2 above is a portion of Mathcad's template for this problem. Here vu is initial speed, Eh - energy loss during braking, Ed - energy loss on impact, Ep - energy loss during post-impact motion. The second row is an error calculation with total differential method. The particular derrivatives (dEh, dEd, dEp) are successively calculated in the template. It is worth noting how the differential as seen in the second row of illustration is calculated. Mathcad can calculate the derivative (in place and symbolically), so I can instruct Mathcad to: "calculate the derivative of defined function F(x) on variable x (with specific value)" and I do not care how Mathcad calculates it (This is not quite true, but it is close enough). As a solution to such calculations I get the value of the initial velocity and calculated deviation, so the true solution is a range of velocities which accounts for the uncertainty.

The next example is collision with pedestrian. Let us imagine that a person walks across a street in front of approaching car. The driver brakes but he doesn't avoid impact. I can determine the impact point, the pedestrian's pre collision path, and the length of locked wheel tire marks before and after impact. I assume pedestrian's pre collision velocity and friction coefficient. The questions are: What was the vehicle's initial speed? What was the distance between the car and the pedestrian when the driver 'should have' recognized the hazard? How long did the emergency last? What was the collision speed? This speed and time/distance analysis will be prepared as a Mathcad template and will answer all these questions.

As with other problems of this type we will work from rest back toward the approach. The first step is to calculate the impact velocity from the length of tire marks after to collision with the energy conservation law. Next I have to account for the increasing (still working backwards) vehicle velocity as it was braked before impact. Assuming the friction coefficient I can calculate the time of vehicle deceleration before and after impact. The time before impact must be increased by the brake actuation time- when brake forces increase prior to lockup. Brake actuation times range from 0.2-0.3 sec. Now I have total time of braking before collision. If I compare this time with the time of pedestrian motion I will know where the driver's actions began. I can also calculate the distance between pedestrian trajectory and vehicle at start of the perceived dangerous situation. Finally I can draw the diagram illustrating vehicle's and pedestrian's motion. This diagram is presented below:

trr = Time of first contact with Brake Pedal

tzh = full brake application

srr = pedestrians location at perception

szh = pedestrians location at full brake application

This diagram is the solution to the problem presented above, the template named ped.mcd.

The goal of this article was to present computer calculators in general and Mathcad specifically as useful tools for the traffic accident Reconstructionist. It is difficult to exhibit all of the features of this powerful and complex application without writing a book. I hope that this article and the three examples linked below give a basic idea of the numerous ways these types of programs may be used.

I have found Mathcad particularly helpful when repeated calculations are necessary- for example changing estimated values that occur in a particular problem. Mathcad templates also make excelent supplements for reports as Exemplified by the illustrations in this article. All were created in Mathcad. Seeing the calculations is preferred and welcomed by juries and police.

About the Author

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