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# Newton's Law of Cooling : Application to Falling Body Temperature

In a murder investigation, the crime scene investigators arrive 5am to examine the body. They found the body temperature to be 30 degrees celcius in a constant room temperature of 20 degrees celcius. (Normal body temperature is taken to be 37 degrees celcius). At 5.30 am the coroner arrived and measured the body temperature to be 28 degrees celcius and estimated the time of death to be (t) hundred hours. Find (t) using Newtons law of cooling, which states that the rate of cooling at any instant is directly proportional to the difference in temperature between the object and its surroundings, ie.

dT/dt=k(To -Ts )

#### Solution Preview

In the given problem, surrounding temperature,
Ts = constant = 30 degree celcius = 20+273 = 293 K

At 5.00 am,
T1 = 30 degree celcius = 30+273 = 303 K

At 5.30 am,
T2 = 28 degree celcius = 28+273 = 301 K

Because,
dT/dt = k(To-Ts)
=> dT/(To-Ts) = k*dt
Integrate both sides, we get,
=> ln(To-Ts) + ...

#### Solution Summary

Newton's Law of Cooling is applied to finding time of death using falling body temperature.

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