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Crime fighting, etc. with Inverse Math

July27

Inverse problems are mathematical detective problems. An example of an inverse problem is trying to find the shape of an object only knowing its shadows. Is it possible to do this at all? What sort of errors are we likely to make, and how much extra information is needed?

Other examples of inverse problems are remote sensing of the land or sea from satellite images, using medical images for diagnosing tumours, and interpreting seismographs to prospect for oil. Another example is perhaps not the first thing that springs to mind when you think of maths: fighting crime.

When a crime has been committed police must look at all the evidence left at the crime scene and work backwards to deduce what happened and who did it. Often, the evidence is a result of a physical process that is well understood — like a speeding car causing skid marks. So to find out the exact cause of the evidence — the speed of the car — the maths that describes the physics needs to be run backwards. This means solving an inverse problem.

Let’s step into an ordinary day in the life of a police unit and see how mathematics can help fight crime. We are investigating a car accident and need to answer the question: was the car speeding?

The evidence available is the collision damage on the vehicles involved, witness reports, and tyre skid marks. Examining the skid marks can help reconstruct the accident. The marks are caused by the speed of the car as well as other factors such as braking force, friction with the road and impacts with other vehicles.

Mathematically we can use mechanics to model this event in terms of $s$, the length of the skid, $u$, the speed of the vehicle, $g$, the acceleration due to gravity and $\mu $, the coefficient of friction times braking efficiency.

The model links the cause (the speed of the car) to the effect (the distance of the skid). As long as we have an accurate estimate of $\mu $, the value describing friction and braking efficiency, we can solve the problem and determine the speed of the car from its skid marks. Read more on this fascinating application of Mathematics in the real world here at plus math.

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