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) Distance (m) 0 0.0 1 4.9 2 19.6 3 44.1 4 78.5 5 122.6 6 176.6 7 240.3 8 313.9 9 397.3 10 490.5

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). 2. Describe the relationship between the two quantities. Be as complete and specific as you possibly can.

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?

Explanation - The equation is a simplified version of 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 and object would have fallen after 0.5 seconds. Show your work in the space below.

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.

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.

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?

9. Are there any modifications could be made in order to make it useful on Mars? How could you accomplish this?