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### Course: Differential equations>Unit 1

Lesson 5: Exponential models

# Newton's Law of Cooling

Newton's law of cooling can be modeled with the general equation dT/dt=-k(T-Tₐ), whose solutions are T=Ce⁻ᵏᵗ+Tₐ (for cooling) and T=Tₐ-Ce⁻ᵏᵗ (for heating).

## Want to join the conversation?

• Does that mean that ice cream pulled out from a refrigerator at -4 C' will get hotter more quickly than that pulled out from a refrigerator at 0 C'?
• Yes, since the temperature difference will be greater with the cooler ice cream, that one will be subjected to a faster increase in temperature.

Still, by the time it gets to 0℃, the rate of temperature increase will be the same as the ice cream that was originally at 0℃, so the colder one will always take more time than the not so cold to reach the same temperature.
• dT/dt=-k(T-Ta) i don not understand the negetive k,can't it just be positive?
because later we need to take the absolute value and write two functions according to the object is hotter or cooler?thanks in advance!
• For Newton's law of cooling you do not need to have the negative sign on the k, but you do need to know/understand that k will be a negative number if an object is cooling and a positive number if the object is being heated. This makes intuitive sense as you would need a positive exponent to increase temperature and a negative exponent to decrease temperature. The main reason I can see for putting the negative k in is to keep you from forgetting it later.
As far as the two equations go, I can tell you that I was able to solve a few problems using either equation. It requires a little bit of manipulation and you really have to think about what you are doing in order to achieve this, but it can be done. Know that if you perform it with the wrong equation, then you will end up with a negative t, which just means that you were going back in time to warm or cool your object. It is probably best to know that there are two equations, and when to use them in order to save yourself the mental anguish of having to perform these manipulations.
To summarize, the negative sign is put in front of the k as a means to prevent you from accidentally omitting it later, and the 2 equations are to keep you from having to wrestle with even more awkward equations and ending up with a negative time.
• At Sal starts to integrate, why do the dT and dt terms vanish in the process?
• The dT and dt tell you what you are supposed to integrate with respect to, or simply what variable is to be integrated. After you have performed the integration, the dt (or dT) becomes useless and disappears.
• Doesn't the cooling depend on the other factors as well like the nature of matter?
• Absolutely, The k is a ratio that will vary for each problem based on the material, the initial temperature, and the ambient temperature. Most of the problems that I have seen for this involve solving for C, then solving for k, and finally finding the amount of time this specific object would take to cool from one temperature to the next.
• What are the factors that influence the speed of the temperature to get cool?
• the environment of the subject can affect the speed.
(1 vote)
• If the cooling of the coffee is affected by external factors, the calculation is still accurate
• yes, because the original calculation does not change, we just need to consider how the new factors work.
(1 vote)
• Please, can you use actual NUMBERS in reference to the LETTERS. If I could see NUMBERS I might actually understand. Just letters is so confusing. If you have a link to another video that shows numbers, please post here. Thank you so much.
• Head on over to the next video, entitled "Worked example: Newton's law of cooling," and you'll see Sal work a problem like this with numbers.

Just on a side note, though, I'd be remiss not to point out that the way Sal solves this, using arbitrary constants, is probably the way that makes things easiest in the long run. Sure, we could "remove" two of the constants here (k and T_a) by replacing them with numbers. Then you have a number to look at instead of a letter (although we can't get around adding the constant C to the mix). But ultimately, writing a letter is really no different conceptually than writing a number -- they're just different symbols for a constant.

Also, defining the constants first is not particularly helpful if you're trying to solve an initial value problem or otherwise trying to fit your equation to real-world situations. Typically you'll have no idea what the constants are, but you'll know what values the function should have at different points along the t axis. You'll run into constants extremely frequently that are similar to the ones in this video. C is an integration constant, and k is a proportionality constant. Both show up in almost every exponential model you'll see in a differential equations course, and I'm not sure you can get by without knowing how to solve them this way.

Hopefully all that doesn't sound rude -- I don't intend it to be. But being uncomfortable using letters/symbols instead of numbers will definitely hold you back in pretty much every branch of mathematics.
• how and why would the equation be if the heat from the hot cup changed the temperature in the room?
(1 vote)
• It would be a completely different, and much more complicated equation. As you already noticed, one of the simplification that Newton's Law of Cooling assumes is that the ambient temperature is constant, but it's not the only simplification. Newton's Law of Cooling also assumes that the temperature of whatever is being heated/cooled is constant regardless of volume or geometry.

If you wanted to create a more realistic (and therefore more complicated) model of temperature exchange, the Diffusion Equation is probably a good starting point, since it does considers geometry.