Regression (I think) of Newton's Law of Cooling

In summary: So, for his equation, k is the slope of the line, and A0 is the y-intercept. However, you are correct that for the equation lny = kt + A0, c = ln(A0). In summary, the conversation discussed two methods for modeling data collected from two cooling cups, with the first method using Excel and the second method involving finding the constants A0 and k by constructing a linear function relating lny and t. The OP chose to use Excel for method 1 and attached the resulting graphs, but was stuck on how to approach method 2. The conversation then clarified that for the equation lny = kt + A0, k is the slope of the line and A0 is the y-intercept
  • #1
HalcyonStorm
7
0

Homework Statement


Using a data logger, I have collected data for two cooling cups: the temperature (c) at 1 second intervals. My task was to model this data using two methods.

"METHOD 1: Use EXCEL or the regression analysis capability of your graphic calculator"
"METHOD 2: Find the constants in the model (A0 and k) by constructing a linear function relating lny and t."

I chose to use Excel for method 1. I have attached the spreadsheet.

Homework Equations


Newton's Law of Cooling: y=A0*e^kt


The Attempt at a Solution


I have attached the graphs that I have been able to produce (method 1) in a word document. I am unable to do method 2, as I'm really stuck for ideas.

Thank-you! Any help is much appreciated :)
 

Attachments

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  • u1.xls
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Last edited:
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  • #2
HalcyonStorm said:

Homework Statement


Using a data logger, I have collected data for two cooling cups: the temperature (c) at 1 second intervals. My task was to model this data using two methods.

"METHOD 1: Use EXCEL or the regression analysis capability of your graphic calculator"
"METHOD 2: Find the constants in the model (A0 and k) by constructing a linear function relating lny and t."

I chose to use Excel for method 1. I have attached the spreadsheet.

Homework Equations


Newton's Law of Cooling: y=A0*e^kt


The Attempt at a Solution


I have attached the graphs that I have been able to produce (method 1) in a word document. I am unable to do method 2, as I'm really stuck for ideas.

Thank-you! Any help is much appreciated :)

For method 2, the idea is to find constants A0 and k for which lny = kt + A0. If you plot lny versus t, this equation represents a straight line.
 
  • #3
Thanks for your response!

Would it be safe for me to assume that (from y= mx + c), k is equal to m and A0 is equal to c?
 
  • #4
HalcyonStorm said:
Thanks for your response!

Would it be safe for me to assume that (from y= mx + c), k is equal to m and A0 is equal to c?
Yes.
 
  • #5
Be careful! - Its really "c = ln(A0)"
 
  • #6
TheoMcCloskey said:
Be careful! - Its really "c = ln(A0)"
No, not if the OP is working with the equation lny = kt + A0.
 
  • #7
No, not if the OP is working with the equation lny = kt + A0.

Yes, that is is correct, and I stand corrected. I was assumming the OP's original equation was, as he stated in "Relevant equations", of the form [itex]y = A_0 \cdot e^{kt}[/itex].
 

FAQ: Regression (I think) of Newton's Law of Cooling

What is the concept of regression in Newton's Law of Cooling?

Regression in Newton's Law of Cooling refers to the mathematical process of finding the best-fit line or curve that describes the relationship between the temperature of an object and time when it is cooling down. It helps to determine the rate at which an object loses heat and the initial temperature of the object.

How is regression used in Newton's Law of Cooling?

Regression is used in Newton's Law of Cooling to analyze and predict the cooling behavior of an object. By using regression analysis, we can plot the temperature data over time and determine the slope of the line or curve, which represents the rate of cooling. This helps us to understand and quantify the cooling process and make predictions about future temperatures.

What are the types of regression used in Newton's Law of Cooling?

The two most commonly used types of regression in Newton's Law of Cooling are linear regression and exponential regression. Linear regression is used when the cooling process follows a straight line, while exponential regression is used when the cooling process follows a curved or exponential pattern.

How do we determine the regression equation in Newton's Law of Cooling?

The regression equation in Newton's Law of Cooling can be determined by using statistical software or by hand calculations. The equation can be derived by finding the slope and intercept of the best-fit line or curve using the temperature data. The equation can then be used to make predictions about the temperature at any given time during the cooling process.

What are the limitations of regression in Newton's Law of Cooling?

Regression in Newton's Law of Cooling is based on the assumption that the cooling process is continuous and that the rate of cooling remains constant throughout the process. However, in real-life situations, the cooling process may be affected by external factors such as air flow or changes in temperature. This can lead to inaccuracies in the regression analysis and affect the accuracy of predictions made using the regression equation.

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