Graphing Module IA. Table of Contents. UCCS Physics Labs. Best-Fit Line 2 Equation of Straight line 4 FYI. Graphing Module IA - 1

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1 Graphing Module IA UCCS Physics Labs Table of Contents Best-Fit Line 2 Equation of Straight line 4 FYI FYI On average, 100 people choke to death on ball-point pens every year. Graphing Module IA - 1

2 The Best-Fit Line At this point all you know is how to plot data points. For a graph to become truly useful we need to find the function that interrelates the data points. The simplest functional relationship is the linear relationship. A linear relationship will produce data that lie along a straight line. The data points will most likely not lie perfectly along a straight line because of error in the measurements. We will need to make a line that best-fits the data, which is called a best-fit line, surprise! Let s try an example now. Examine the following graph: y These two lines are other possible best-fit lines for this data. They don t appear to do as good of a job representing all the data as the solid black line. This line divides the data points evenly. The line also gives the general trend the data is following. data points 0 x Drawing a best-fit line is an art form that you will quickly learn. Just remember that the best-fit line should be as close as possible to all the data points. Graphing Module IA - 2

3 Getting information from a graph distance distance Graph A Graph B You should have learned in your lecture class that the velocity of an object is the measure of how much distance it can cover over a length of. Example: 60 miles per hour is a velocity measurement, at this velocity the car will travel 60 miles in a period of an hour. So velocity is the ratio of the distance traveled divided by the it took to travel that distance. velocity = distance Says that s great, but how do we get the velocity from the distance verse graphs? Well, remember that a graph is nothing more than a series of measurements. To find the velocity all we need to do is select to data points and find how much distance was traveled and divide that number by the it took to travel that distance. Look closely at the two graphs above, in Graph A not a lot of distance is traveled in a length of, this means the velocity is small. However, compare this to Graph B where a greater distance is traveled in the same amount of, the velocity will be larger. The steeper the graph of distance vs. the larger the velocity. Graphing Module IA - 3

4 You should also have learned that the acceleration of an object is the change in it s velocity over. Therefore, acceleration = velocity. Since the units of velocity are distance per ( m/s ), the units of acceleration become distance per per ( m/s/s = m/s 2 ). Equation of the best-fit line Once we have a best-fit line, we need to know how to represent it mathematically. A line is a functional relationship between the two variables plotted on the vertical and horizontal axis. The equation of a line is as follows: y = m x + b y the variable located on the vertical axis. m the slope of the line. (Explained below, wait for it!) x the variable located on the horizontal axis. b y-intercept. (Also explained below, try to be more patient!) To find the equation of any line is to find the value of the slope and the y-intercept. The slope (m) represents how rapidly the line will rise or fall, the larger the value for the slope the steeper the line. If the value of the slope is positive, then the line will be increasing in y as x is increasing. If the slope is negative, the line will be decreasing in y as x is increasing. If the slope is zero the line will not increase or decrease but will stay at a constant value of y no matter the value of x. The y-intercept (b) is the value of the vertical axis where the line intersects that axis. If the line does not reach the vertical axis, then simply extend the line until it does and record the value. The slope is defined by the ratio rise over the run, which is how many vertical units the line will rise or fall divided by the number of units the line will run in the horizontal direction. A more mathematical way of looking at the slope is: The change in the y coordinate ( y) divided by the change in the x coordinate ( x) as you travel along the line. Graphing Module IA - 4

5 Let s try calculating the slope of a straight line: To find the rise or the y, just take the difference between the two y coordinates: rise = 3.8 m m = 2.3 m To find the run or the x, just do the same difference calculation for the x coordinates: run = 4.3 s - 1 s = 3.3 s Now to find the slope divide these two numbers: "rise" 2.3 m slope = = = 0.7 m "run" 3.3s s If you remember, the ratio of the distance divided by was called something else, velocity. Therefore, the slope of a distance vs. graph is equal to the velocity. Similarly, if the velocity were plotted verses the slope of the graph would be equal to the acceleration of the object. Graphing Module IA - 5

6 Some tips and tricks for calculating the slope Try to spread out the two points you are using to calculate the slope over the length of the line. This will give a more accurate slope calculation. Remember you are measuring the slope of the line, so use the x and y coordinates of the line. If it is possible use the coordinates of data points that lie on the line. This will reduce any extra error gained in reading coordinates off the line because you already have the exact coordinates from the data. We now have a value for the slope of our line. All we need to complete the equation of this line is to find the value of the y-intercept (b). This should be a simple task. The line crosses the y-axis at a value of about 0.8 m, so b = 0.8 m. The equation of a line is: y = m x + b Filling in the values for the slope and the y-intercept from our example gives: y = (0.7 m ) x m That s it! s This brings up an important point. As you have probably already guessed, not all graphs will be in a straight line. Some graphs are very curvy and can wiggle all over the place. In this experiment we have a constant acceleration, g. Because of this, the velocity must be constantly increasing (definition of acceleration). This means that a plot of distance vs. will not be a straight line (straight line = constant velocity). Since the velocity is constantly increasing the slope of the line must also be constantly increasing. distance If you look at this graph you will notice that the slope at any point will increase as goes by. Therefore, the object must be accelerating. Graphing Module IA - 6