To find out: How your motion can be described in mathematical language
How to translate between different mathematical representations: graphs and equations.
Materials: data logging software, motion detector
Introduction: In this investigation, you will use a motion detector to plot a distance
(position)-time graph of your motion. As you walk (or jump, or run) the graph on the computer screen displays how far away from the detector you are.
• "Distance" is short for "distance from the motion detector."
• The motion detector is the origin from which distances are measured.
• It detects the closest object directly in front of it (including your arms if you swing them as you walk).
• Our motion detectors will not correctly measure anything closer than 0.20 meter. When making your graphs don't get closer to the detector than 0.20 meter.
Activity 1-1: Making Distance-Time Graphs
1.
Be sure that the motion detector is connected to the computer. Open the experiment file called L01IN1A1 to display distance vs. time axes (in folder Additional Physics, Tools for Scientific Thinking,
Mechanics).
2.
Stand about 0.2-m away from the detector, begin graphing, and have someone move the detector until the reading is 2 m.
3.
Begin graphing and make distance-time graphs for different walking speeds and directions, and sketch your graphs on the axes. Label important features of the graph, such as the values of the x- and y- intercepts.
4.
After creating each graph, use the curve fit tool, to create a mathematical model of your data.
5.
Record the mathematical model and its coefficients.

a. Start 0.20 meter away and make a distance-time graph walking away from the detector (origin) slowly and steadily . Sketch the graph on the right. b. Make a distance-time graph, walking away from the detector (origin) medium fast and steadily. Sketch the graph. c. Make a distance-time graph, walking toward the detector (origin) slowly and steadily. Sketch the graph. a. Make a distance-time graph, walking toward the detector (origin) medium fast and steadily. Sketch the graph.

Question 1-1: Describe the difference between the graph you made by walking away slowly and the one made by walking away more quickly. How are these differences shown in the mathematical models? What do the coefficients represent?
Question 1-2: Describe the difference between the graph made by walking toward and the one made walking away from the motion detector. How are these differences shown in the mathematical models? What do the coefficients represent?
Activity 1-2: Curved Position-Time Graphs
1.
Can you make a curved position-time graph? Try to make the graph shown below. Before you attempt to make each graph, discuss with your group what you think you will have to do to create it. Then collect data. Keep trying until you get as close as you can to the shape. After you are satisfied with your graph, use the curve fit tool,
to find a mathematical model. Record the equation and its coefficients of the best model. Also write a description of the motion that created your graph.
2.
Describe how you must move to produce the position-time graph.

Question 1-5: What is the general difference between motions that result in a straight-line position-time graph and those that result in a curved-line position-time graph?
What is different about the mathematical models?
Is there more than one model that will fit your data? How would you choose the best model? Justify your decision.

Many graphs of position vs. time look like parabolas. Mathematicians usually give the equation for the parabola as follows: y = ax
2
+ bx + c where a, b , and c are constants. Although all parabolas have the same basic shape, some are skinny, some are fat, some start one place on a coordinate axis and some start another place.
Some parabolas are even upside down! The magnitudes of the constants and whether they are positive or negative determine how a given parabola looks. In the next few activities you are going to use spreadsheet calculations and graphing to demonstrate to yourself how the values of each of the constants affect the shape of a parabola.
Some skinny, fat, and upside down parabolas in different locations on a graph (A) that uses typical mathematics notation of y vs. x .
Open the “Playing with Parabolas” spreadsheet

Activity: Parabolas for different a, b, and c values a.
First let’s explore the effect of the value of c on the location of the parabola. Examine the equation for the parabola and predict what you think the effect of changing the value of c will be.
Suppose it is changed to +100? To –100? What will happen in each case? Explain the reasons for your prediction. b.
Test your prediction by changing the values of c graphs for these three different values of c from +1 to +100 and then to –100. Sketch in the space below using solid, dotted, and dashed lines and marking the appropriate values of c on each sketch. c.
Describe in words what c has to do with the y –intercept of the graph. Does the basic shape
(curvature) of the parabola change in any way? d. Does b actually determine the slope of the parabola when x = 0? To check this, we suggest that you return the values of the constants to +1, +1, and +2 and then try changing b from +1 to +30 and then to –30. Sketch graphs for these three different values of b in the space below using solid, dotted, and dashed lines and marking the appropriate values of b on each sketch.

e.
Describe in words what b has to do with the slope of the graph at the y -intercept. Does the basic shape (curvature) of the parabola change in any way? f. Explore the effect of the value of a on the shape of the parabola. To do this, it is suggested that you return the values of the constants to +1, +1, and +2 and then try changing a from +2 to –2 and then to +4. Sketch graphs for these three different values of a in the space below using solid, dotted, and dashed lines and marking the appropriate values of a on each sketch. g.
Describe in words what a has to do with the shape of the graph. Does the basic shape (curvature) of the parabola change in any way?
How do your findings about the coefficients for parabolic graphs compare to what you have learned about linear graphs?

Objective: To create a mathematical model for a real data set.
Open the file balltoss.cmbl.
Play the movie clip. As you watch the motion of the ball, think about what you have learned about mathematical models for motion data. Sketch your prediction of the graph of the height of the ball vs. time. What model do you think will best suit this data? Justify your choice.
Use the Video Analysis tools to track the ball during the motion. Use the meterstick in the movie to scale the data.
What type of function will best model this data? Why do you think so?
Select the “Analyze” menu, and then “model”. Create a model for this data. Tweak the values of the constants until you get a good fit. Record the values of the coefficients and the form of the function.

Objective: To create a mathematical model for a real data set.
Open the file shuttlelaunch.cmbl
Play the movie clip. As you watch the motion of the shuttle, think about what you have learned about mathematical models for motion data. Sketch your prediction of the graph of the height of the shuttle vs. time.
Use the Video Analysis tools to track the shuttle during the motion. Use the meterstick in the movie to scale the data.
What type of function will best model this data? Why do you think so?
Select the “Analyze” menu, and then “model”. Create a model for this data. Tweak the values of the constants until you get a good fit. Record the values of the coefficients and the form of the function.