Creating a Mathematical Model
Objective – to create a mathematical model using real data, verify the model with additional observations, and use the
model to make predictions
Description - This is meant to be a graphing activity where students gain understanding of how an equation can be used
to accurately predict the behavior of a physical property. The data in the table below show how far and object has fallen
each second for 10 seconds.
Distance Covered by a Falling Object
Time (sec)
0
1
2
3
4
5
6
7
8
9
10
Distance (m)
0.0
4.9
19.6
44.1
78.5
122.6
176.6
240.3
313.9
397.3
490.5
Directions
1. Using Excel or similar graphing software, create a scatter plot of the data in the table shown above. Be sure that your
graph has a descriptive title and labeled axes (including units).
Distance Covered by a Falling Object
600.0
y = 4.905x2 + 2E-13x - 5E-13
R² = 1
500.0
Distance (m)
400.0
300.0
200.0
100.0
0.0
-100.0
0
2
4
6
8
10
12
Time (sec)
2. Describe the relationship between the two quantities. Be as complete and specific as you possibly can.
Students should be able to say that there is a positive relationship between the two variables and some may even realize
that it is not linear and is some type of exponential/power type of function.
Explanation - Scientists like to use relationships like this to interpolate (determine values between measured values) or
extrapolate (to make predictions about future scenarios that have not been measured yet). In order to do this, they need
to come up with an equation that represents what is happening.
3. Right click on any of the points on the graph and select Add Trendline. Select Polynomial and down at the bottom
check the boxes that say Display Equation on Chart and Display R-squared value on Chart.
4. The equation presented can be used to determine the distance covered by a freely falling object after any amount of
time. What fundamental physical process is this equation modeling?
The physical process is gravity since that what determines the force of attraction of an object and therefore how it will
behave as it falls.
1
Explanation - The equation is a simplified version of 𝑑 = 2 𝑎𝑡 2 where d is the distance the object has fallen after time t
and a is the acceleration due to Earth’s gravity or 9.81 m/s2. Be sure you understand how the equation off the graph is the
same as the one described above.
Explanation – The R-squared value tells how closely the two values are correlated. An R-squared value of one shows a
perfect mathematical relationship and means that if you know the value of one variable you can predict the other value
correctly 100% of the time.
5.
Once scientists have an equation, they must verify it against observational data to make sure their model is correct.
Use the equation to predict how far an object would have fallen after 0.5 seconds. Show your work in the space
below.
1
1
𝑚
𝑑 = 𝑎𝑡 2 = (9.81 𝑠 ) (0.5 𝑠)2 = 1.22 𝑚
2
2
6. Compare this to the observational/experimental number that your teacher provides you with. How do the two values
compare? If there are any differences, discuss why those differences exist and how they affect the accuracy of the
model. Do differences between the model and observational data mean that the model is not useful? Explain.
To demonstrate this either set up a demonstration with photogate or video capture software and perform an experiment or
find a video analysis clip showing a freely falling object. Depending on the results of your individual experiment the
measured value may differ from the calculated value. The main reason for any difference would be that the equation does
not account for air resistance which affects objects differently depending on their mass to surface area ratios. The main
point to make here is that models are not perfect by any means but represent a reasonable degree of accuracy.
7. The real value of a mathematical model comes from being able to extrapolate or predict values beyond the range of
those you have data for. For instance, pretend that you work for the Kramerica corporation and are developing a
bladder system for an oil tanker to prevent oil spills in the ocean. As part of your research you are dropping giant
balls filled with oil from a tall office building. If someone is standing beneath the window that is 600 meters tall, how
many seconds will they have to get out of the way? Use the model you have created, show all your work in the space
below, and describe the level of certainty you have about your prediction.
(2)(600𝑚)
2𝑑
𝑡=√ =√
= 11.1𝑠
𝑎
9.81𝑚
𝑠
The level of certainty will vary based on the difference between the calculated and measured values from the previous
question. Students should realize that the person may have slightly more or less time to move but it will be somewhere in
the neighborhood of 11 seconds.
8. Scientist are also interested in the possibility of applying the model to new and different situations. Would this model
be useful for describing freely falling objects on Mars? Why or why not?
The equation would not be useful as is since it includes the acceleration due to gravity on Earth and that value will be
different for Mars.
9. Are there any modifications could be made in order to make it useful on Mars? How could you accomplish this?
By using either a calculated or experimental value for the acceleration due to gravity on Mars the model could be adapted
for use on Mars or any other planet.