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Activity Walk the Line: Straight Line Distance Graphs 1 When one quantity changes at a constant rate with respect to another, we say they are linearly related. Mathematically, we describe this relationship by defining a linear equation. In realworld applications, some quantities are linearly related and can be represented by using a straight-line graph. In this activity, you will create straight-line, or constant-speed, position versus time plots using a Motion Detector, and then develop linear equations to describe these plots mathematically. OBJECTIVES • Record position vs. time data for a person walking at a uniform rate. • Analyze the data to extract slope and intercept information. • Interpret the slope and intercept information for physical meaning. MATERIALS computer Vernier computer interface Logger Pro Motion Detector PROCEDURE 1. Position the Motion Detector on a table or chair so that the gold disc is pointing horizontally out into an open area where you can walk. There should be no chairs or tables nearby. 2. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. 3. Launch Logger Pro and open the file “01 Walk” in the Real-World Math with Computers folder. 4. Stand about a meter from the Motion Detector. When you are ready to collect data, click . Walk away from the Motion Detector at a slow and steady pace. You will have five seconds to collect data. Real-World Math with Computers 1-1 Activity 1 5. Examine your graph. It should show a nearly linearly increasing function with no spikes or flat regions. If you need to repeat data collection, return to Step 4. DATA TABLE y-intercept b optimized slope m optimized line equation x1, y1 x2, y2 regression line equation ANALYSIS 1. Click the Examine button, , to display a cursor to read values from your graph. 2. The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept value. The independent variable is x, which represents time, and y is the dependent variable, which represents position in this activity. Use your mouse to trace across your graph to the left edge to read the y-intercept. Record this value as b in your data table. 3. One way to determine the slope of your position vs. time graph is to guess a value and then check it by viewing a graph of the line with your data. You can do this using the Manual Curve Fit feature of Logger Pro. a. b. c. d. Select Curve Fit from the Analyze menu. Choose the Manual Fit Type option. Select mt + b Linear from the General Equation field. Now you can adjust the parameters. Enter the y-intercept you determined in the b field. You can now adjust the value stored in the slope field m by entering values or by using the up and down arrows to the right of the value field. Adjust m so that the model line is as close as possible to the data. e. Record the optimized value for the slope in your data table. f. Click to return to the main graph window. 4. Use the values of the slope and intercept to record in your data table the equation of the line that best fits your position vs. time data. 5. Another way to determine the slope of a line to fit your data is to use two well-separated data points. The analyze mode should still be enabled. Choose two points (x1, y1) and (x2, y2) that are not close to each other and record them in the data table. 1-2 Real-World Math with Computers Walk the Line 6. Use the points in the table to compute the slope, m, of the position vs. time graph and record it here. y − y1 = m= 2 x 2 − x1 How does this value compare with the slope you found by trial and error? 7. You can also use Logger Pro to automatically determine an optimized slope and intercept. a. Click the Linear Fit button, , to perform a linear curve fit. b. Use the parameters m and b to write down the equation of Logger Pro’s best-fit regression line, and record it in your data table. c. (optional) Print your graph by selecting Print Graph from the File menu. 8. How do the values of the slope and intercept as determined by Logger Pro compare to your earlier values? Would you expect them to be exactly the same? 9. Slope is defined as change in y-values divided by change in x-values. Complete the following statement about slope for the linear data set you collected. In this activity, slope represents a change in ______________________________ divided by a change in ______________________________. 10. Based on this statement, what are the units of measurement for slope in this activity? 11. The y-intercept can be interpreted as the starting position or the starting distance from the Motion Detector. What does the slope represent physically? Hint: Consider the units of measurement for the slope you described in the previous question. Real-World Math with Computers 1-3 Activity 1 TEACHER INFORMATION Walk the Line: Straight Line Distance Graphs 1. Place the Motion Detector at waist-high level for the walker. The walker should not be closer than 0.5 meter to the detector when data collection begins. Clear the area of other materials such as desks or chairs. 2. The walker must maintain a constant rate while walking directly away from the Motion Detector. 3. Either a Vernier Motion Detector or a TI CBR can be used. SAMPLE RESULTS Data with manual curve fit Real-World Math with Computers 1-1T Activity 1 Logger Pro linear fit DATA TABLE y-intercept b xxxx optimized slope m xxxx optimized line equation xxxx x1, y1 xxxx x2, y2 xxxx regression line equation xxxx ANSWERS TO ANALYSIS QUESTIONS Answers have been removed from the online versions of Vernier curriculum material in order to prevent inappropriate student use. Graphs and data tables have also been obscured. Full answers and sample data are available in the print versions of these labs. 1- 2 T Real-World Math with Computers