Newton's Law of Cooling Calculus, Example Problems, Differential Equations

This calculus video explains how to solve newton's law of cooling problems. It provides the formula and explains how to derive the equation using differential equations.
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I love that during this topic in both precalculus and calculus1, we were only given "Ae^kt" and never told about ANY of this temperature stuff, whatsoever. Nobody thought it would be beneficial to let us know.

[@davem45]

Awesome video and great instruction. I actually did this over the weekend with a bottle of soda and worked through this formula first, got my solution and then took another bottle and took the temp every 15mins for 2hrs 45mins and I came within .7 degrees of the math answer which I chalked up to opening and closing the fridge door to get the bottle's temp.

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Thank you for this. This video is the best explanation for Newton's Law of Cooling that I have found. Excellent job.

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MR. Organic Chemistry Tutor, thank you for a massive analysis of the classical Newton Law of Cooling and its relationship to initial and final temperature.

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I legit still have no idea how this works but the example did help me on one problem

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[@SeanBenson23]

The derivation had me laughing out loud. That’s incredible. So cool

[@MegumiJA]

hi, may i ask, why do we need to derivative the formula

[@SeanBenson23]

@@MegumiJA Derivatives help us find instantaneous rate of change. We would like to know the instantaneous rate of change of the temperature with respect to time. The rate of change of the temperature with respect to time is proportional to some constant (-k, the decay constant, because we are cooling) and the difference between the starting temperature of the object and the surrounding temperature. Hope this helps!

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This was really helpful, much appreciated.

[@emergile7584]

Can you discuss this A temperature with an initial temperature of 70° is placed in a room with a controlled temperature of 22 degrees after 3 minutes the thermometer with 60 degreesdetermine how long will the thermometer reads 30° determine also on the reading after 8 minutes

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[@robyrogo1843]

Can anyone explain me why when he integrated 1/(T-T_s) to be ln(T-T_s) he didn't add the constant C , but when he integrated -k he added the constant?

[@kushagrashrivastava9452]

It doesn't create a difference. Suppose, after integration, he had a constant on both LHS and RHS, LHS + m = RHS + n We can now subtract m on both sides, LHS = RHS +n - m Now as n and m are both arbitrary constants, we can let n - m = c Thus, LHS = RHS + c So he just skipped this one step of adding constants on both sides. Hope this helps :)

[@robyrogo1843]

@@kushagrashrivastava9452 yeah i got it some time ago but thanks so much for your time anyway!

[@robyrogo1843]

@@kushagrashrivastava9452 the friend who helped me used the same words ,, he just skipped one step " :)

[@SeanBenson23]

Fantastic question

[@jogennoble4931]

Very helpful for students like me.

[@Oscar-jg9gg]

If what you said is true, then why does the object reach room temperature. Wouldn't it take an infinite amount of time?

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[@salamatuabubakar2839]

what if the object is placed in water?? will Ts still be a constant??

[@Idekato]

what is dt? I knows its delta time but how do I find delta time.

[@alexkemos3176]

Isnt the first equation e^kt (not e^-kt)?

[@takiawashington2253]

80 X 12?

[@kevinperkins7123]

This is an awesome video!

[@emmanuelking5933]

forever grateful

[@maegayagi2639]

6:05 I though that its 100-50 not 80-50. Could u please explain in that part... Thanks. In advance

[@amirhanna5990]

He corrects himself shortly after.

[@Unconqueredxd]

I don't know if you figured it out, but he explains the reason he did that. We are trying to figure out how much LONGER it is going to take going from 80 degrees to 70 degrees.

[@tomatrix7525]

Amir Hanna, no he did not as he is correct. T0 in this case is 80 because we are rinding the time from 80 to 70 degrees, here we are beginning at 80. That is why. Imagine we are starting at 80 not 100, as we already know 100 to 80 takes 12 mins. If we find 80 to 70, which he does, we can add it to 12 to find 100 to 70, thus we let 80 be T0

[@johnwhite7253]

What is the formula for law of heating?

[@illangageperera1605]

Explaining good👍

[@abhijitgogoi8348]

Thanks a lot sir

[@manueltouyaa6800]

what if you don't have the time for an object at x degrées to cool to y degrées?

[@scritch9549]

How do I find K of real materials?

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Perfect👌

[@sanopann1896]

watching this while don't have any idea about derivative. me stop school 😨😨😨😣😵‍💫😵 whats going here

[@34_ophuthuan90]

why T0 is =80 i dont get it

[@deleted-user-82u3ht92]

6:15

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thank you

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it is good content

[@alexhagdahl3321]

6:04 why is the initial temperature equal to 80 there and not 100?

[@alexhagdahl3321]

nvm lol 21.52 - 12 minutes = 9.52 minutes LONGER

[@alexhagdahl3321]

I should probably finish the video before asking for help

[@blackeli]

@@alexhagdahl3321 thanks for your comment I was turning my brain upside down tryin to figure out why i got 21,52 instead of 9,52

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Why did you use e

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[@trustme835]

can u help me to solve this using same methods u describe here?..A substance takes 3 min in cooling from 50C to 45 C and takes 5 min in cooling from 45C to 40C. What is the temperature of surrounding?

[@lanceleesunga6635]

34.6 C

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Austin Lee how’d you do it

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[@vaiebhavpatil2340]

2 min silence for those who can't find this vid

[@davidearlsantos5628]

Your first example is wrong. Try checking it. At t=9.5mins the temperature will be 50 not 70.

[@davidearlsantos5628]

It should be 2.16 minutes for the temperature to be 70.

[@ninoacierto8670]

at t=0 the T must be 100C and t=12 T must be 80C. Am I right??

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[@34_ophuthuan90]

nah turning the T0 into =80 is a bad idea

[@itsbetty5070]

where did you get the 80 from?in the first beginning, I don't get it.

[@lf198]

i think it should be 100

[@zachp2389]

it is because at 12 minutes the object cools to 80 degrees as stated in the problem

[@stanza2768]

at T(12) the object has already cooled to 80 degrees

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[@orozcodiego7905]

18:04 why does the dT dissapear from the equation?

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