Hello friends……
On this occasion, I will share how the role of mathematics in forensics, maybe, my friends have watched detective movies, both cartoon-type movies and movies that are lifted from real stories like CSI Files and Forensic Files.

When friends pay attention, the most frequent cases featured in films and real stories are cases of bodily discovery. In the films, a detective and forensic expert can determine when the body died.

For detection systems that record discrete events, the dead time is the time after each event during which the system is not able to record another event. source

Determining the estimate of when a victim dies is critical in determining who the victim is killed. On this occasion, I will discuss how detectives and forensic experts determine the time of death, on the victim by using Newton's cooling law.
This law was discovered by Sir Isaac Newton, the same scientist as the inventor of the law of motion.

Humans are warm-blooded creatures. That is, the human body temperature is not much affected by environmental temperature. Normal human body temperature is 37.5 oC. At the time a person dies, his body temperature will not be 37.5 oC.

It will be assumed that the events are occurring randomly with an average frequency of f. That is, they constitute a Poisson process. The probability that an event will occur in an infinitesimal time interval dt is then f dt. It follows that the probability P(t) that an event will occur at time t to t+dt with no events occurring between t=0 and time t is given by the exponential distribution (Lucke 1974, Meeks 2008). source

But it goes down slowly so that within a certain time the temperature will be equal to the ambient temperature. In this case, the ambient temperature is considered lower than the normal human body temperature. In general, the ambient temperature is 27 ° C.

The process of decreasing the temperature in the human body when it dies turns out to follow Newton's cooling law.
in this law, the temperature drop of an object having a higher temperature than its environment is directly proportional to the difference in temperature of the object to its environment. Mathematically, Newton's cooling law can be written to be :

In the above formula, dT / dt is the temperature change with time, k is a constant with units of 1 / time, T is the temperature of the body as a function of time and Ta is the ambient temperature.

The value of k constant can be determined by measuring the temperature at different times, but for the human body, in general, is 0.696 / hr. We can solve the above equation by doing a simple integral process:

While C is a constant which can be determined by entering the initial temperature T0. The final solution of the above equation becomes:

Having successfully derived the above equation, let us review the real case example. In the case of homicide was found corpses inside the house with a body temperature of 28 oC and the temperature inside the house at that time 27 oC. Bodies were found on Saturday at 15:00. When was the murder committed?

From this case, we can specify some quantities: T (t) = 28 oC, To = 37.5 oC, and Ta = 27 oC. Substituting these quantities into Newton's law of cooling equation, the following results will be obtained.

If we reduce the time of discovery of the corpse with the results of the calculation, the estimated time of the death of the victim is Tuesday at 11:38. On the results can be seen the calculation of human body temperature after death for 8 hours.

The curved line indicates an ambient temperature of 27 ° C. How? Friends interested in being a detective? The science of mathematics and physics is quite useful for detectives!

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REFERENCE

For detection systems that record discrete events wikipedia source
It will be assumed that the events wikipedia source

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