Graphical Analysis of Data The scientist is 'equently presented with huge amounts of data in need of analysis. Typically this data is purely numerical and is in the form of a data table. To extract meaningful information from such a table is of extreme importance. However, any trends or relationships that might exist in the data are typically obscured when in the form of a table of raw numbers. To analyze the data, a graph is generally constructed. Suppose we have done a study comparing quantity X to quantity Y. The data from our experiment is recorded in the data table below. A plot of this data would appear as follows: Y versus X 307 207 107 0T . t ' t ' t ' i U D 2 D d D 5 D 3 0 Notice that the points all fall along X what appears to be a straight line. We then connect the points with a solid line referred to as the "line of best t". In this case we have discovered a very simple relationship between X and Y: as X increases, so does Y in direct proportion. We say X and Y are directly proportional because as one quantity changes, another changes in exactly the same manner. When two quantities are directly proportional, a straight line results. Next consider the data and graph below: Here we do not obtain a straight line but a curved line referred to as a arabola. Note that a small change in X corresponds to a large change in Y. Y versus X 30 I 20* C 10* I U C 0\" I I . I . I [H] DU 40 EU X Y versus X i . ,_ I 7 . . Here we obtaln a curve referred to as a : ' . hyperbola. A hyperbola is Indicatlve of 00 I . . . 1'0 I . . I 2'0 I I I I 3'0 an Inverse proportlon. Inverse x proportions occur when the product of two quantities is constant. Many other cases exist, including the possibility of no signicant relationship between the two quantities being investigated. Such data would not produce a single curve but would tend to scatter all over the graph. As evidenced by the rst graphical example, when two quantities are directly proportional to one another, a straight-line graph results. This is signicant due to the simplicity of the relationship between the quantities and the fact that whenever a straight line is obtained, the slope of that line can be computed. In the example that follows, the data is shown with the corresponding graph: Y versus X 30 5'\" (x2, "1) = (5.0, 25.1) N7 HJJJ 10* l l . I DD 20 40 50 80 There appears to be a linear relationship between Y and X. A straight line of best t has been drawn through the data points. As previously indicated, this is interpreted to mean that as X increases so does Y increase proportionally. Although the fact that Y and X are directly proportional is of considerable importance, this information does not allow one to write a relationship between the variables. To do this a proportionality constant (call it "k" ) is needed. Then it can be written as Y = k X. Fortunately "k" is numerically equivalent to the slope of the straight line. The slope of the tted line is the \"rise\" divided by the \"run\": kzslopezAY :M 25.25 AX 6.0 2.0 Study the example graph (and the slope calculation) carefully and note that it conforms to the following general rules: 1. The scale for each axis is selected so as to make the most efcient use of the entire sheet of graph paper while making certain that all data "fits" onto the graph. 2. The graph is titled, the axes are indented and labeled with proper quantities, and the scale size is marked at convenient intervals. 3. Each data point is plotted as a single point. 4. A smooth curve (line of best t ) is drawn so as to lie as near as possible to all data points. 5. Any set of intervals may be used to determine the slope but larger intervals are more desirable since they reduce error. In this example, the proportionality constant ( k) was found to be 5.25. In the case of the parabola or hyperbola it is not desirable to calculate the slope of the line because the slope continually changes along the length of the curve. Methods exist to overcome this problem, but are beyond the scope of this course. Procedure 1. Refer to the data table below, Exercise #1. An object is subjected to a constant acceleration along a frictionless track. A student measures its velocity (v) after specic durations (At). The student uses a graph to analyze the truck's motion. Duration, At, (s) Velocity, v, (m/s) 2.0 6.0 4.0 7.0 6.0 8.0 8.0 9.0 10.0 10.0 12.0 11.0 A. Plot the velocity (in meters/sec) vs. time (seconds). The velocity is the y-axis and time is the x-axis. Use any graphing software you like or graph this data in pencil on graph paper. Excel has a nice graphing package. Calculate the slope of this graph. You will B. What is the physical signicance of the slope value computed in part A? C. Having determined the slope of the line, you can now write (1 = kt. Use this equation to predict a value of "(1" when t = 1.5 s