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Linear Graphs Graphing data shows if a relationship exists between two quantities also called variables. If two variables show a linear relationship they are directly proportional to each other. 2 Examine the following graph:

Linear Graphs 3 Independent Variable Dependent Variable

Linear Graphs – Slope of a Line The slope of a line is a ratio between the change in the y-value and the change in the x- value. This ratio tells whether the two quantities are related mathematically. Calculating the slope of a line is easy! 4

Linear Graphs – Slope of a Line 5 y x Rise = Δy = y 2 – y 1 Run = Δx = x 2 – x 1 Slope = Rise Run Slope = y 2 – y 1 x 2 – x 1 y2y2 y1y1 x2x2 x1x1

Linear Graphs – Equation of a Line Once you know the slope then the equation of a line is very easily determined. 6 Slope Intercept form for any line: y = mx + b slope y-intercept (the value of y when x =0) Of course in Physics we don’t use “x” & “y”. We could use F and m, or d and t, or F and x etc.)

Linear Graphs: Area Under the Curve Sometimes it’s what’s under the line that is important! 7 Work = Force x distance W = F x d How much work was done in the first 4 m? How much work was done moving the object over the last 6 m?

Non Linear Relationships  Not all relationships between variables are linear.  Some are curves which show a squared or square root relationship 8 In this course we use simple techniques to “straighten the curve” into linear relationship.

Non Linear Relationships 9 This is not linear. Try squaring the x-axis values to produce a straight line graph Equation of the straight line would then be: y = x 2

Non Linear Relationships 10 This is not linear. It is an inverse relationship. Try plotting: y vs 1/x. Equation of the straight line would then be: y = 1/x

Meaning of Slope from Equations Often in Physics graphs are plotted and the calculation of and the meaning of the slope becomes an important factor. 11 We will use the slope intercept form of the linear equation described earlier. y = mx + b

12 Unfortunately physicists do not use the same variables as mathematicians! For example: s = ½ x a x t 2 is a very common kinematic equation. where s = distance, a = acceleration and t = time Meaning of Slope from Equations

13 Physicists may plot a graph of s vs t, but this would yield a non-linear graph: s t s t2t2 Meaning of Slope from Equations To straighten the curve Square the time

14 But what would the slope of a d vs t 2 graph represent? Let’s look at the equation again: s = ½ at 2 {s is plotted vs t 2 } What is left over must be equal to the slope of the line! slope = ½ x a {and do not forget about units: ms- 2 } y = mx + b Meaning of Slope from Equations

15 Now try These. A physics equation will be given, as well as what is initially plotted. Tell me what should be plotted to straighten the graph and then state what the slope of this graph would be equal to. Example #1:a = v 2 /r a v Plot a vs v 2 to straighten graph Slope = 1/r Meaning of Slope from Equations a = (1/r)v 2 Let’s re-write the equation a little: Therefore plotting a vs. v 2 would let the slope be:

16 Example #2:F = 2md/t 2 F t Slope = 2md Go on to the worksheet on this topic Meaning of Slope from Equations F 1/t 2 Plot F vs 1/t 2 to straighten the graph

Error Bars on Graphs 17 You already know about including errors with all measured values. These errors must be included in any graph that is created using these measured value. The errors are shown as bars both in the horizontal and vertical direction. For example: (horizontal ) (vertical) This would be shown like this on the graph. Error Bars!

Error Bars on Graphs Plot the following data and add in the error bars: 18 time (s) (+0.2) Distance (m) (+0.5)

Error Bars on Graphs 19 Best fit line Max. slope Minimum slope