Newton's Law of Cooling: Finding the Time of Death of a John Doe

Zrcohfc Kcr tlc pchrv @ lhc bccg cgmkp`gi tlc dcn`iltv ko FV@. GF@V hgd v`a`nhr “okrcgv`f‐ T_ vlkyv' Hzhrt orka tlc`r bnhthgt tcgdcgfp tk i`c tlc “ahvv!vzcf‐ ohr tkk asfl h`r!t`ac. hgd trcht `t n`ec tlc ik tk vknst`kg cvzcf`hnnp `g FV@. @ lhc cvzcf`hnnp bccg `gtr`iscd lky tlc acd`fhn c|ha`gcrv hrc hnyhpv tlc zk`gt ko okfsv ylcg vkac kgc bciv tlc xscvt`kg ko  t`ac ko dchtl' Ccg tlksil @ lhc akdcnncd ahgp nsazcd fhzhf`thgfc vpvtcav. hgd ccg v`asnhtcd tlca `g V@ASN@GE. @ d`d gkt rchn`vc tlc vknst`kg tk vsfl h zrkbnca yhv h d`rcft hzzn`fht`kg ko Gcytkg‐v Nhy ko Fkkn`gi ! hv @ v`aznp gccr ihc tl`v xscvt`kg asfl tlksilt bcokrc' @g zhrt`fsnhr. @ ﬈gd tlc hghnpt`f vknst`kg ko tlc ﬈rvt!krdcr KDC xs`tc hg cncihgt vknst`kg tk tl`v zrkbnca. cvzcf`hnnp v`gfc `t `gvthgtnp vlkyv tlc rcvsnt `v tlht ko h dcfhp`gi c|zkgcgt`hn osgft`kg ! hv kgc yksnd c|zcft `t tk bc' H fkazstht`kghn hzzrkhfl tk tlc vknst`kg `v chv`np kbth`gcd bp vkn`gi tlc KDC bp sv`gi Csncr‐v Actlkd' Tl`v thve `v v`aznp hfl`ccd bp hzzrk|`aht`gi tlc dcr`ht`c y`tl tlc ﬈g`tc!okryhrd d`oocrcgfc. ylcrc o ,| % l) o ,|) . tlcrcbp kbth`g`gi h rcfsrv`c fkazstht`kghn hnikr`tla' o  ,|) 1 n`al : l −



→

— A`flhcn dc V`nh. Kftkbcr 2::0

Fkaacgtv. Fr`t`f`va. hgd Fkgtr`bst`kg Fkgtr`bst`kg @o pks lhc hgp vsiicvt`kgv kr d`vfkcr crrkrv `g tl`v dkfsacgt. yr`tc tk a`flhcnJaydcv`nh'fka tk lcnz ahec tl`v dkfsacgt bcttcr' Tlhgev `g hdhgfc okr pksr fkgtr`bst`kg' Fkgtr`bstkrv y`nn bc hfegkyncdicd `g tl`v dkfsacgt'

A`flhcn A' Y`mctsgic dc V`nh AVf ,D`vt). BCgi ,Lkgv). A@CCC. HA@AcflC,SE) cah`n= a`flhcnJaydcv`nh'fka a`flhcnJaydcv`nh'fka • bnki= bvkda`ec'fka • rcvsa´ c´ vsac= c´ = aydcv`nh'fka

8

O`gd` O`gd`gi gi tlc tlc T`a T`acc ko Dcht Dchtl l ko h Mklg Mklg Dkc Dkc

O`rvt!krdcr n`gchr d`oocrcgt`hn cxsht`kgv hrc y`dcnp svcd bp vf`cgt`vtv hgd cgi`gccrv tk vknc h hr`ctp ko  zrkbncav `gkn`gi tcazcrhtsrc'

Gcytkg‐v Nhy ko Fkkn`gi= Tlc rhtc ht yl`fl tlc tcazcrhtsrc T ,t) flhgicv `g h fkkn`gi bkdp `v d`rcftnp zrkzkrt`kghn tk tlc tcazcrhtsrc d`oocrcgfc bctyccg tlc bkdp hgd tlc fkgvthgt tcazcrhtsrc vsrrksgd`gi acd`sa' T v . ko tlc vsrrksgd`gi

Gcytkg‐v Nhy ko Fkkn`gi `v akdcnncd hv h ﬈rvt!krdcr `g`t`hn!hnsc zrkbnca.

 1 ,  ,:) 1 dT  dt T 

e T 

−

T v )

,8)

T :

ylcrc T : `v tlc `g`t`hn `g`t`hn tcazcrhtsrc tcazcrhtsrc ko tlc bkdp hgd e `v tlc fkgvthgt ko zrkzkrt`kghn`tp' @o  T v `v fkgvthgt. bp vczhrht`gi hr`hbncv.



8 T 

−

T v



dT  1 e dt

,2)

@gtcirht`gi bktl v`dcv.

  

8 T 

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T v

ng ,T 

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−

T 

dT  1

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e dt

T v ) 1 et % F 

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,6)

T v 1 cet%F 

T ,t) 1 T v % F cet

Hzznp`gi tlc `g`t`hn fkgd`t`kg.

T ,:) 1 T v % F c:

→

F  1 T :

−

T v

,4)

8

A`flhcn A' Y`mctsgic dc V`nh AVf ,D`vt). BCgi ,Lkgv). A@CCC. HA@AcflC,SE) cah`n= a`flhcnJaydcv`nh'fka a`flhcnJaydcv`nh'fka • bnki= bvkda`ec'fka • rcvsa´ c´ vsac= c´ = aydcv`nh'fka

Lcgfc.

T ,t) 1 T v % ,T :

−

T v ) cet

,3)

cet 1 :' Tlcrcokrc. n`at @o  e > :. n`at ko `tv vsrrksgd`giv' →∗

→∗

T ,t) 1 T v . hgd tlc tcazcrhtsrc ko tlc bkdp hzzrkhflcv tlht

C|haznc= Ht 0 ha kg Kftkbcr 80. 2::0 h bkdp yhv oksgd `g rkka 629 ht tlc Sg`crv`tp Fcgtcr' Tlc rkka `v eczt ht h fkgvthgt tcazcrhtsrc ko 92 O' Tlc acd`fhn c|ha`gcr yhv fhnncd hgd lc hrr`cd `g c`ilt a`gstcv' Tlc ﬈rvt tl`gi lc d`d yhv tk thec tlc tcazcrhtsrc ko tlc bkdp' @t yhv