# Eratosthenes and Auto Collisions: Science At Work

## Investigators Can Learn from Greek Mathematicians

(Editor's Note: The following article was contributed by Tom Morris, the owner of St. Louis Traffic Accident Reconstruction.)

Eratosthenes was a Greek mathematician who lived in Egypt in the 3rd Century BC. He observed that in Syene, on the longest day of the year, columns cast no shadows when the sun was directly overhead. However, at the same time, the columns in Alexandria cast a significant shadow. This simple observation led Eratosthenes to conclude that the only way for both columns to cast shadows of differing lengths at the same time was if the surface of the earth was curved. A few simple geometric calculations later, he had calculated the circumference of the planet.

The story of Eratosthenes illustrates that a lack of information isn’t what prevents one from discovering the truth. Everyone could see the same shadows. However, nobody saw their significance because intuition and daily experience led them to erroneously conclude that the earth was flat. If we look at auto collisions a different way than we normally do, then we might also see something that previously went undetected.

Control of the Intersection

This is a common argument used when a vehicle covers a significant distance within the intersection prior to impact. Photos show the damages to the right quarter panel and the police report has the vehicle crossing five lanes of traffic, almost clearing the intersection. The driver crossing the intersection claims to have been traveling at the posted limit, entering on a yellow light that turned to red. The other driver claims to have accelerated from a stop after his light turned green. Did anyone see the vehicle run a red light? No.

So the adjuster deftly argues that the law requires a stopped vehicle presented with a green signal to yield the right of way to any vehicle lawfully within the intersection before proceeding.  Since the vehicle had crossed five lanes, the driver of that stopped vehicle must have been inattentive prior to starting into the intersection, right?

It at least seems reasonable. After all, look at that distance the vehicle traveled within the intersection before being struck.

Let’s take a look at this another way. The one thing that is missing from this analysis is the consideration of time. A collision occurs because two vehicles attempt to occupy the same space at the same time. The impact point gives us a common time and location for each vehicle.

Because everyone is willing to concede that the vehicle accelerated from a stop on a green light, we have the means to calculate our second time. Take a look at the stop bar and the collision point. How far did the vehicle accelerating from a stop travel prior to the collision? The stop bar is about 12 feet back from the curb and maybe a half a lane beyond that, or 6 feet. Let's figure a total of 18 feet. Before we can calculate the time, we must calculate how fast the vehicle was going at 18 feet. We use the formula…

The typical acceleration rate for a passenger vehicle from a stop is around 0.2g or 6.4 ft/s/s.  The initial velocity is zero.

Now that we know the end velocity, we can calculate the time using

So now we know that the collision occurred approximately 2.4 seconds after the vehicle accelerated. However, we want to know when things happened in relation to the changing of the light, as it is part of everyone’s testimony. Did the vehicle accelerate at the instant the light turned green? Likely not. An average driver, according to AASHTO requires approximately 1.3 seconds to respond to a green light. Add that to the acceleration time of 2.4 seconds, and you can now say that the collision occurred 3.7 seconds after the stopped vehicle’s light turned green.

The next question to answer is “where was the crossing vehicle when the light turned green for the stopped vehicle?” How fast does this driver claim to be traveling? Was he or she driving at the posted limit perhaps? At 35 miles per hour or 51 ft/s, we can now calculate its position relative to the impact point.

If the vehicle crossed five lanes at 12 ft per lane, that accounts for 60 ft. Add the length of the vehicle of 15 ft and that accounts for 75ft traveled within the intersection. Then subtract that 75 ft from the total distance to impact of 188.7 ft and it places the crossing vehicle 113.7 ft outside the intersection when the stopped vehicle received the green light.  Since most electric traffic signals have a conflict monitor, you can effectively state that under the testimony provided, the crossing vehicle was 113.7 ft from the intersection when the light turned red.

Not only did the crossing vehicle enter the intersection unlawfully through a red light, but it was also a significant distance from the intersection when the driver decided to proceed. When applying a Monte Carlo analysis to this scenario, we discover that there was a 0.02 percent chance that the driver entered on a yellow, and a 99.98 percent that the driver entered through a red.

Until now, this would just be viewed as a “swearing match,” with no visible means of resolution.  A new approach provides answers that were there the whole time. It could be the right time to start looking at shadows a little differently.

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