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#### Proceed through the steps one by one.

Step 1: Driving Motivation and Setup - Show

Driving Motivation: Now that you've learned new concepts, it's time to apply them in a real context! The motivation behind much of research in math is to come up with models of how real-life objects operate. Once we've developed a model, we can extrapolate to answer other questions. For example, if we have a model for the path of a bullet fired in the air, we can use it to find the precise time the bullet hits the ground. If we had a model for the positions of two cars on a one-lane road, we could use it to see if they would ever crash. In the experiment below, we can use the model to extrapolate who might reach a position first!

Setup: For our experiment's setup, we'll simply be using a stopwatch and a measuring mechanism (in this case, a ruler). We'll use a pause button to stop the video so that we may consider the position of the object in relation to the time elapsed. This setup is quite similar to that of current experiments in UNC's Fluids Lab, where researchers use a camera and then track time and position! As you go through the animation, remember to record your data in the table on the notesheet you have.

Step 2: Perform the Experiment - Show

Step 3: Modeling - Show

Now that you've recorded your data, you can make approximate plots of time versus displacement from the origin. Though we don't have specific functions for the data, we can look at the shape of the graphs to interpolate what's happening. Using these plots, you'll be able to consider questions about velocity, concavity, and acceleration.

### Consider which car belongs to each plot of time versus position, and mouse over the plot to check your answer.

Step 4: Apply Your Knowledge - Show

Questions:

• Are any of the cars ever at a negative position? Answer
No. Within the time of our experiment we only see the cars at a positive displacement from the origin.
• Which of the cars moves at a negative velocity? Answer
Watching the experiment, we see that Car 2 is moving in the left direction. This is confirmed by the position diagram, which has a negative slope.
• Do Cars 1 and 2 have the same value for a(t)? Answer
Yes. Though the cars are moving in opposite directions, they both have negative values for acceleration. Notice that the graphs for each car are concave down, meaning that the second derivative of the position function is negative.
• What’s the acceleration value for Car 3? Why so? Answer
Car 3 is moving at a constant velocity, so its acceleration value is zero. Because the position scatter plot is linear, this implies that its first derivative is a constant, and its second derivative is zero.
• If a finish-line were 25 inches from the origin, which car would win the race? Answer
Looking at the models, Car 3 would likely win. It has a constant, positive velocity, whereas Car 2 is coming to a rest and Car 3 is moving in the opposite direction.

Step 5: Reflection - Show

Prompt: Now that you've performed the experiment, write a few sentences reflecting on how you used math in a 'scientific' context. Consider the tools you used and how you were able to model with the cars. Can you think of examples of how modeling can be used in other subjects? Which ones, and how?

Show a sample reponse.

Sample: I believe it's fascinating that we're able to observe an object - in this case a car - and describe its motion with precise language using what we know about kinematics. We used rather rudimentary tools, a stopwatch and a ruler, and yet we were able to perform the experiment, collect data, and then analyze it. Using the concepts of velocity, acceleration, and concavity, we were able to examine the position graph, even though there were only six data points! We were able to describe what we saw with logical arguments, and even extrapolate to describe what might happen next (i.e. the third car would likely reach a certain position first). Modeling is a powerful tool, and it could be used in a variety of other contexts. Some examples of this might include modeling drug reactions in the human body, developing financial forecasts in economics, or even predicting animal populations in biology!