Graphs in Physics

Why do we graph data? Graphing data shows if a relationship exists between two quantities, also called variables. There are many types of possible relationships between the variables.

Linear Graphs If two variables show a linear relationship they are directly proportional to each other. Examine the following graph:

Linear Graphs Dependent Variable Independent 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!

Linear Graphs – Slope of a Line y x y2 Rise = Δy = y2 – y1 Slope = Rise Run y1 Run = Δx = x2 – x1 Slope = y2 – y1 x2 – x1 x1 x2

Linear Graphs – Equation of a Line Once you know the slope then the equation of a line is very easily determined. Slope Intercept form for any line: y = mx + b y-intercept (the value of y when x =0) slope 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 is under the line that is important! For example: 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 differnet type of relationship, maybe a square or square root relationship, for example. In this course we use simple techniques to “straighten the curve” into linear relationship. Into to extract information from our graph in a easy way. This is called linearizing.

Ex 1. Non Linear Relationships This is not linear. It is an exponential relationship. Try squaring the x-axis values to produce a straight line graph Equation of the straight line would then be: y = x2

Ex 2. Non Linear Relationships 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 and the meaning of the slope becomes an important factor. We will use the slope intercept form of the linear equation described earlier. y = mx + b

Meaning of Slope from Equations Unfortunately physicists do not use the same variables as mathematicians! d = ½ x a x t2 For example: is a very common kinematic equation. where d = displacement, a = acceleration and t = time

Meaning of Slope from Equations d t Physicists may plot a graph of d vs t, but this would yield a non-linear graph in this case: d t2 To straighten the curve Square the time

d = ½at2 Meaning of Slope from Equations But what would the slope of a d vs t2 graph represent? Let’s look at the equation again: d = ½at2 {d is plotted vs t2} y = mx + b d is y and t2 is x… so whatever is before t2 must be equal to the slope of the line! slope = ½ a {and don’t forget about the units: m/s2}

Meaning of Slope from Equations 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. Plot a vs v2 to linearize the graph Example #1: a = v2/r a v Let’s re-write the equation a little: a = (1/r)v2 Therefore plotting a vs. v2 would let the slope be: Slope = 1/r

Meaning of Slope from Equations Example #2: F = 2md/t2 F t Plot F vs 1/t2 to straighten the graph F 1/t2 Slope = 2md Go on to the worksheet on this topic