Project 1

Comparing the Newton and the Stefan-Boltzmann Cooling Laws

Determination of time of death. The chief of police hands you the following report prepared by detectives handling a recent murder case.

Police Report. Police arrived at the scene of a murder at 12 am. They immediately took and recorded the temperature of the corpse, which was 33oC, and thoroughly inspected the area. By the time they finished the inspection, it was 1:30 am. They again took the temperature of the corpse, which had dropped to 30oC, and had the corpse sent to the morgue. The temperature at the crime scene had remained steady at 23oC.
This kind of report is, unfortunately, routine. The temperature information is used to determine approximately when the person was murdered. The idea is to set up a differential equation via an appropriate cooling law, determining parameters from the data given, and integrating backwards to normal body temperature (37oC) to find the time of death.

What the police chief wants to know is how accurately the time of death can be determined using Newton's law of cooling rather than the more accurate, but also more complicated, Stefan-Boltzmann law that Newton's law approximates. Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the ambient temperature and the body temperature. The Stefan-Boltzmann law states that the temperature changes at a rate proportional to the difference of the fourth power of the ambient temperature and the fourth power of the body temperature, provided these temperatures are given on the absolute (Kelvin) scale. Write a brief report to the chief addressing his concerns. He hasn't got a lot of time, so keep the report to five pages, including any plots. He wants the report by Tuesday, February 13.