Differentialligning

Ja, det er korrekt. Hvis du antager, at k er positiv, kan du skrive:

y` = -k(y-y0)

Hva hvis jeg skal beregne proportionalitetsfaktoren for et metalstykke hvis temperatur er 100grader, der anbringes i omgivelser, hvis temperatur y0 er 20grader. Efter 30 sek er metalstykkets temperatur faldet til 95grader.

hvordan gør jeg dette?

#0,

Ja, dit forslag er korrekt, med den bemærkning, at y0 er omgivelsernes temperatur. Dette kaldes iøvrigt for Newtons afkølingslov, og er den behandlet vidt og bredt i litteraturen. Fx læste jeg engang en artikel, hvori man forsøger at opstille en bedre model for afkøling, end Newtons afkølingslov, da Newtons afkølingslov et eller andet sted er en for simpel model. Den tager fx ikke højde for, at vandet fordamper under afkøling.

Artiklen, jeg talte om før, er ikke frit tilgængelig på nettet, så jeg kan ikke linke til den. Her er dog et abstract:

Newton's Law of Cooling:
How Quickly Does Water Cool?
Rustem Onkal
Mathematics
Advisor: Stan Wagon

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the surrounding ambient temperature. Using T ambient for the ambient temperature, the law is dT/dt = -K (T-Tambient), where T is temperature, t is time, and K is a constant related to efficiency of conductive heat transfer. Most mathematicians assume that the Newton’s Law of Cooling is the rule that governs the cooling of hot water to ambient room temperatures, and so the decline is a simple exponential decay. However, recently collected data does not follow the exponential decay suggested by this law. The problem with Newton's Law of Cooling is that it only takes conductive/convective component of heat transfer into account, ignoring the roles played by radiation and evaporation. The aim of this study is to come up with an alternative equation that can be used to accurately model the rate of cooling of water. The appropriate experiments were run to collect the data in order to (1) test the accuracy of the equations used to model evaporation, radiation and conduction/ convection individually, (2) estimate the best values for the constants of the equations (K1 for conduction and K2 for evaporation) of this new model which combines all three forms of heat transfer, and (3) test the accuracy of the final model. We worked with simple differential equations and made use of Mathematica’s remarkable fitting facilities throughout our analysis. Our final novel model, which involves all three forms of heat transfer (conduction, evaporation and radiation), now, is able to easily and accurately model and predict the rate of heat transfer from hot water to cooler ambient temperatures.

#2,

Først skal du løse ligningen y` = -k(y-20). Du får to konstanter: k, som er proportionalitetskonstanten, og c, som er en integrationskonstant. Disse konstanter kan findes ud fra de to oplysninger om, at y(0)=100 og at y(30)=95.

ok.. fik k til 0,0116.....

Sidste opgave lyder:

Hvor længe varer det før metalstykkets temperatur er faldet til 40 grader??