Introduction to the Activity Maybe you`ve left a hot cup of coffee on Introduction to the Activity
Maybe you’ve left a hot cup of coffee on your desk and then got involved in an email and forgot
it was there. Minutes later, you think “coffee” and take a sip. It is too cool and needs warming.
You might not consider it, but what you’ve witnessed is an application of physics. Isaac Newton
himself thought about it, he just didn’t have a microwave to remedy the problem of a cold cup of
coffee.
Consider a hot cup of coffee, or water, placed on a desk. List up to four factors you believe will
influence the rate at which the coffee cools.
1.
2.
3.
4.
After making your list, discuss your list with another student. Make changes if you want, but
note why you made the change. That is, what your partner said to cause you to make the change.
Finally, with your partner, select the one graph below that you believe represents a good model
for the temperature of the cooling liquid over time.
a
b
c
d
e
f
Introduction to Module 5
The Right Stuff: Appropriate Mathematics for All Students
Introduction to Module 5
The Right Stuff: Appropriate Mathematics for All Students
Module 5
Newton’s Law of Cooling – An Application of Exponentials and Logarithms
Find an appropriate model for the temperature of a cooling liquid over time.
Given Information:
Given a “hot” liquid, Newton's Law of Cooling states that the rate at which the liquid cools is proportional
to the difference in the temperature between the liquid and the room temperature (or ambient temperature,
Tα). This relationship is modeled by the formula:
T t = a + be kt
The values of a, b and k are affected by variables in the experiment.
What variables? What will change the rate at which the liquid cools?
These are questions that go beyond a cup of coffee left on a desk! Consider a winter day. When the
temperature drops to zero, you want to know what you can do to keep the inside of your house warm.
The class will perform three experiments that will help determine at least one of the factors that can change
the rate of cooling.
Group Work
Step 1
Read the directions in Step 2. Discuss the experiment within the group.
Q1. What factors will influence the cooling process and affect the values of a, b and k?
Step 2
The class will be divided into groups. Each group has a temperature probe, a TI-Nspire ® handheld, and a
cup of hot liquid. Each group will also have either a porcelain coffee cup, a Styrofoam coffee cup, or a
Styrofoam coffee cup with a lid.
Record the room temperature (also called ambient temperature, Tα, pronounced “T sub alpha”). Heat the
liquid to a boil and pour it carefully into the appropriate container. Connect the temperature probe to the
handheld and turn the handheld on. The temperature probe will show the current temperature of the liquid.
To begin automatic data collection (for three minutes – measurements every second), press “MENU”, then
“1: Actions”, and then “9: Data Collection”. Once the temperature display has hit a maximum, move the
cursor to the right arrow in the temperature display and press the round white button. Data collection will
then begin in the spreadsheet. Record the actual time that this begins. <You may want to practice this
procedure at least once before using the hot liquid.>
The graph and data are automatically displayed on the handheld.
Record:
Room Temperature (Tα):
________
Time:
________
Initial Temperature (T0):
________
Ending Temperature (T180 ): ________
Newton’s Law of Cooling – An Application of Exponentials and Logarithms
The Right Stuff: Appropriate Mathematics for All Students
One person in the group will remain with the liquid and take temperature measurements at the ten, twenty,
and thirty minute marks.
Record:
Temperature (T600 ):
________
Temperature (T1200 ):
________
Temperature (T1800 ):
________
Step 3
We know the model should be Tt = a + be k t .
Q2. Should k be greater than or less than zero? Why?
Q3.
What value should a have?
Hint: Over a long period of time, maximum cooling will have taken place. The value of be k t
would be negligible.
Q4.
What value should b have?
Hint: Let t = 0.
Q5.
Compute the value of k .
Hint: Use the temperature at t = 180.
Write the model for the data here
using the values you found above:
Step 4
Use the statistical package in TI-Nspire to find a model for the data.
Write it here:
Step 5
Copy the data from the handheld into an Excel spreadsheet using the computer link cable.
Use the tools in Excel to find an exponential model for the data.
Hint: After copying the data into Excel, edit the data by removing the constant a. Find the model
from the revised data.
Write it here:
Q6.
Compare the models. Are they equivalent? What reasons would cause these two models to be
different?
Newton’s Law of Cooling – An Application of Exponentials and Logarithms
The Right Stuff: Appropriate Mathematics for All Students
Step 7
Use a model to predict the temperature of the liquid at the ten, twenty, and thirty minute marks. Compare
those predictions with the actual temperature. Discuss the results compared to the observed values.
Step 8
Other groups in the class used different containers (porcelain cup, Styrofoam cup, Styrofoam cup with a
lid…).
Q7. How do the models compare? What component of the model changes as the container changes and
why?
Q8.
What other factors could be introduced to change the model?
Conclusion
Fill in the blanks with appropriate words or phrases.
The temperature of a an object placed in an atmosphere of a different temperature will _______ at a rate
proportional to the difference between the temperature of the object and the temperature of the atmosphere. The
rate at which the temperature changes can be controlled by the type of _______ in which the object is placed.
The more the object is exposed, the ______ the rate will be.
The function that best models this phenomenon is _________. The exponent is ________ if the temperature of
the object is decreasing and _________ if the temperature of the object is increasing. As t is large, the model
will always level out at the _______ temperature, if conditions remain the same.
Newton’s Law of Cooling – An Application of Exponentials and Logarithms
The Right Stuff: Appropriate Mathematics for All Students