Lab 1 Uniform Motion - Graphing and Analyzing Motion Objectives: < To observe the distance-time relation for motion at constant velocity. < To make a straight line fit to the distance-time data. < To interpret the slope as the velocity of the motion. < To observe that the average mean square error is smallest for the closest fit. Equipment: < Motion sensor < Pasco 1.2 m track and dynamics cart < Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis software Physical Principles: The position of an object moving along a line is indicated by its displacement. The displacement is ±1 times the distance of the object from a reference point called the origin, the numbers being positive on one side of the origin and negative on the other side. Denoting the displacement as x and the time as t Figure 1: Object with displacement +2 from origin. x ' vt%x o (1) In a graph of x (on vertical axis) versus time (on horizontal axis) the velocity of the motion v is equal to the slope of the line. The initial position, the location at the beginning when time is zero, is x o. This value is where the line crosses the vertical axis and is called the intercept. The best fit of a straight line to a data set is the one with the smallest value of the average square deviation. Experiment 1 Page 1
The slope is given by v ' slope ' rise run ' )x )t ' x 2 &x 1 t 2 &t 1 (2) It is often possible and convenient to take x 1 and t 1 to be zero. Predictions: Draw a rough graph in your journal of what you think the motion will be. Plot the displacement x versus the time t. Do this when the cart starts at an initial position of 50 cm and travels for a time of 2 seconds at a speed of 50 cm/s. Will the curve be a straight line or a curved Figure 2 Slope, v, and intercept, x o. line? If it is straight will it slope up or down. If it is a curved line will it curve up or down? Explain why you think it will behave this way. Do this for two cases and label the graph for each. The two cases are: 1. Motion toward the origin, 2. Motion away from the origin. Procedure: Setup: Plug the motion sensor s phone plugs into digital channels 1 and 2 with the yellow banded plug into channel 1. Place the motion sensor about 40 cm from the end of the track opposite the bumper with the center of the sensor about 12 cm above the track. Align the sensor so that the sound waves will travel directly along the track. Place the cart on the track at the end near the sensor. Data Collection: Double click the left mouse button on the physics labs folder to open it if necessary (it is usually open). Double click on the scwkshp icon in the folder to open Science Workshop. See Figure 3 below. Click and drag the phone plug icon to digital channel 1, choose Motion Sensor. Click on the REC button and at the same time push the cart away. Wait until data collection stops. Drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on the rescale icon (fourth from the left in the lower left of the graph window). Drag the Table icon onto the Motion Sensor icon. Click on the clock to the right of the E at the upper left of the Table window to display the times. Click just above the time-distance data columns to select all of the data or click and drag to select Experiment 1 Page 2
Figure 4 Science Workshop window. Figure 3 Graphical Analysis window. Experiment 1 Page 3
the portion of the data that is valid. Under the Edit menu, choose copy to store the data temporarily in the Window s clipboard. Graphing Data: Double click on the VernierGA icon in the physics folder to open the graphing analysis program, click on OK and click on the restore (upper right center icon). Click on the row 1, x data position. Under the Edit menu option choose paste data to copy your data from temporary storage in the clipboard. Analyzing Data: Note that the displacement is plotted vertically (y-axis) and the time data is plotted horizontally (x-axis). Click on the graph of your data on the right to select the graph. Choose Analyze from the main menu and click on Manual Curve Fit. Select the Stock function M*x + B to select a linear (straight-line) model. (According to Eq. (1), the x here corresponds to your time values, the M corresponds to your velocity values, and B corresponds to your beginning location x o ). Change the values in the intercept box B = at the lower left and the slope box M = to vary the intercept and slope of the model line. Note values of the Mean Square Error at the lower right of the graph for each value of slope M and intercept B. Do this until the model line visually fits most closely to the data and then make further adjustments until the Mean Square Error is as small as possible. In table 1 in the data recording section record the values of the slope M, intercept B and the Mean Square Error. Click on OK-Keep Fit. In the Main menu click on Analyze and choose Automatic Curve Fit, click on the Stock function M*x + B and click on OK. Click on OK-Keep Fit. In table 2 in the data recording section record the values of the slope M, intercept B and the Mean Square Error. Click on the Linear Regression icon (the rightmost icon under the Data menu) to obtain again a linear fit to the data. Record the regression coefficient. A value close to one indicates a close fit to the line. Compare these values with those obtained in your manual fit Click on the graph title and change the title to Displacement versus Time. Click in the text window and enter your name, experiment name, date and experiment details, ie motion away from detector. Choose File in the main menu, then Print, click on Selected Display and click on OK. Include this graph in your journal. How does your observed curve compare with your predicted curve? What is the speed of the cart? How far from the detector is the cart when the detector begins measuring its motion. What does the value of the Mean Square Error indicate? Motion in opposite direction: Experiment 1 Page 4
If you have time return to the Science Workshop window and repeat the experiment placing the cart on the end opposite the motion sensor and pushing it toward the sensor. At the same time click on the REC button. Repeat the analysis above. A suggested format of your entries into your journal is shown below. Experiment Title and Number Date of Experiment Experimenter s name Names of Partners List of objectives including the questions about nature that you are asking in the experiment. Description of the apparatus that you are using. Description of current ideas about the quantities involved in the experiment, that is the relevant understood physical principles. Your prediction of what the answers will be to your questions. Description of the measurement process. Data collected. Graphical representation of the data usually done with linear models. Discussion of the experimental results, values measured, percentage errors from other published values, sources of error in the experiment. Answer the question Is the data consistent with the proposed model? Experiment 1 Page 5
Recording Data: Table 1 Manual Curve Fit Motion relative to detector v slope M x o intercept B Mean square error Table 2 Automatic Curve Fit Motion relative to detector v slope M x o intercept B Mean square error Regression coefficient R = away from the detector R = toward the detector Experiment 1 Page 6