Motion Graphs: What Do They Tell Us?


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1 Motion Graphs: What Do They Tell Us? Physics HS/Science Unit: 02 Lesson: 01 Introduction: Motion graphs allow us to visualize motion and to specify quantitative aspects of that motion. We will be primarily concerned with displacementtime graphs and velocitytime graphs. From the displacementtime graph we will be able to trace the location of an object at any time directly from the graph. We will also be able to determine the sense (positive or negative) of the velocity and can, through slope analysis, create the velocitytime graph. We will learn that the slope of the displacementtime graph at any instant is the velocity of the moving object at that time. We will develop some terminology that is used throughout the course. Our emphasis will be on constant accelerated motion, but the ideas are valid for other types of motion as well. Terminology of motion physics style: In the course of talking about motion, certain words are closely related and can be confused in casual conversation. In this context vector quantities have both a magnitude and a direction. Examples are displacement, velocity, and acceleration. Scalars do not have an associated direction. In addition, we make graphs of these same quantities and are sometimes concerned about the coordinates on the graph of these quantities. Velocity is a vector, and a proper description would give both the speed (how fast) and the direction of the motion. Speed is the magnitude of the velocity vector. In one dimensional motion this would generally be designating the direction as positive or negative. Displacement is a vector, which is the total distance from the beginning of the motion to the end of the motion and includes the direction of this motion. The displacement magnitude is often different from the total distance traveled, as might be given by the odometer on an automobile. For example, moving 4 meters north followed by 3 meters south has a displacement of 1 meter north and not 7 meters. Acceleration is a vector, but unlike velocity, does not have a common term for the magnitude or value of the vector. The same word, acceleration, is normally used for both; thus, it is important to be aware of the context. Line Graphs Equation of linear graphs: One of the simpler kinds of graphs, where the values fall on a straight line, is called a line graph or a linear graph. In Cartesian coordinate systems (x and yaxes perpendicular), all linear graphs are described by a rather simple equation, which you have undoubtedly studied in a math class. The form of this equation is y = m x + b. Each value of x has a single value of y given by this equation. The symbol m is the slope of the line and b is the yintercept (the place where the line crosses the y axis). For example, the equation y = m x + b y = 3 x + 2 is the general form of linear graphs and describes the graph at the right. The slope is 3; meaning that for every change in x of value1 that the value of y changes by +3, and the graph crosses the yaxis (x = 0) at 2. In physics, the variables for the motion graphs are not y and x but displacement and time or velocity and time. Thus, if the equation describing the displacement of an object were d = 3 t + 2, the graph would look identical with the vertical axis 2012, TESCCC 06/08/12
2 Motion Graphs: What Do They Tell Us? representing the displacement and the horizontal graph representing the time of travel. The slope would be 3 and the value of displacement at t = 0 would be 2. Finding the Slope of the curve from the DistanceTime Graph (velocity): If the equation of the line is known, the slope is automatically available. However, if the equation is not known, which is the normal situation when taking data in the laboratory, the graph will yield the slope through a relatively simple process. Recognize that the term slope refers to the rise or fall of the curve as the values of abscissa (time axis) increases. A large positive value indicates a steep upward slope; a slope of zero is a horizontal line and so on. In fact, the slope is the ratio of increase in displacement (Δd) to the increase in time (Δt). The symbol delta (Δ) is used to indicate a difference between two values. Thus Slope = Δd/Δt. Since velocity is (displacement)/(time), when we find the slope in the motion graph, we are finding the velocity of the moving object. The units of velocity would be (meters/sec) as is appropriate. Velocity = Δd/Δt Using the graph above as an example, we can choose the time interval from t = 1 to t = 1 for Δt = 2. The corresponding value for Δd is given by 5 minus 1 or 6. Thus, Δd = 6 and the slope (velocity) is 3 m/sec. This value, of course, agrees with the value given by the equation of the line. We would have obtained the same velocity value for any time interval because the slope has the same value throughout the graph. However, if the curve was not a straight line, we would have found different values over different time intervals. In each case we would have found the average velocity over that time interval. If the time interval is chosen sufficiently short, we would effectively find the instantaneous velocity at the time of the brief interval. Important Facts About the Slope of DisplacementTime Graphs: The slope at any point on the graph represents the velocity at that point (time). If the slope is a straight line, then the velocity is constant and the equation for the motion can be written as position = (velocity) (time interval) + starting position. Technically each slope determination is the average velocity over the time interval chosen to find the slope. If the time interval is very short, the value becomes the instantaneous velocity. The motion detector software, which takes position points very rapidly, not only plots the distance data but also takes the displacement between each pair of points and computes the slope or average velocity for that time interval. Since the time intervals between points are short, the velocitytime curve is nearly the instantaneous velocity. If the velocity is not constant, but is changing, then the object is accelerating. More about that below.
3 Motion Graphs: What Do They Tell Us? Physics HS/Science Unit: 02 Lesson: 01 Finding the Slope of the Curve from the VelocityTime Graph (acceleration): The velocitytime graph can be analyzed in the same way that the displacementtime graph is analyzed and the resulting slope at any instant is the acceleration. In other words, acceleration = Δv/Δt. Finding the slope for the velocitytime graph yields the acceleration of the object being studied. The acceleration may be a linear graph, as it is for falling objects, or an oscillating graph, as it is for a pendulum. Important Facts About the Slope of VelocityTime Graphs: The slope at any point on the graph represents the acceleration at that point (time). If the slope is a straight line, then the acceleration is constant and the equation for the velocity can be written as velocity = (acceleration) (time interval) + starting velocity. Technically, each slope determination is the average acceleration over the time interval chosen to find the slope. If the time interval is very short, the value becomes the instantaneous acceleration. The motion detector software, which calculates velocity points very rapidly, not only plots the distance and velocity data but also takes the change in velocity between each pair of points and computes the slope or average acceleration for that time interval. That software will provide graphs of acceleration vs. time. Most of the motion studied in mechanics (this semester) will have constant or zero acceleration. Using Excel: Graphing and Curve Fitting Function (constant velocity motion): While the motion detector software creates quality graphs and also provides some data analysis such as velocity and acceleration graphs, it is useful to perform this analysis outside of the program and to be able to generate custom graphs. The use of a computer to provide attractive graphs of raw data and to analyze these graphs has been made relatively easy by the use of Excel. Excel is part of the Microsoft Office Suite of programs and has excellent graphing and curve fitting capabilities. Since you have already imported data from your motion detector into Excel, we will assume that you have that skill. If you do not have suitable data for analysis, tables of data are provided below. The data has random measurement errors to simulate real data. You will perform a curve fit to the distancetime data to provide the best value of the velocity. The curve fitting technique used by Excel for linear graphs is called regression analysis, and the line fit through the data points is called the regression line. Constant Velocity Motion data is available (see your instructor) Graphing Procedure and Slope Determination Step by Step: 1. Open the Excel program. 2. Enter the data into Excel with the Time in column A and Distance in Column B. 3. Once all the data has been entered, click in the top left corner and drag to highlight the entire data set. 2012, TESCCC 06/08/12
4 Time (sec) Distance (m)
5 Physics HS/Science Unit: 02 Lesson: 01 Motion Graphs: What Do They Tell Us? 4. On the tool bar, choose the chart wizard icon. 5. From the Standard types tab, choose scatter; choose next. 6. Chart wizard step 2 of 4, click on the next button. 7. Chart wizard step 3, fill in the blanks. a. chart title: distance vs. time b. value (x) axis: time (sec) c. value ( y) axis: distance (m) 8. Chart wizard step 4, click on the finish button. Your graph should be next to your data table. 9. Click on Chart on the menu bar. 10. Choose add a trendline. 11. On the Type tab choose linear. 12.On the Options tab change the forecast forward and backward a few units. Place a check next to: a. display equation on chart b. display R 2 value on chart, choose okay at the bottom. 2012, TESCCC 06/08/12
6 Visually inspect the scatter in the measured values about the graphical linear fit to the data. The equation of the line and the R 2 value should also be on the chart. If the points all fall along or close to the regression line, you can feel confident that the data is well modeled by the linear equation. In this case, the equation will be d = 1.8 t +.30 This equation indicates that the velocity was 1.8 m/sec, and the motion started at d =.30. The value of R 2 is a quantitative indicator of how good these values fit the theoretical straight line. Excel refers to R 2 as the coefficient of determination. It is similar to the correlation coefficient of graph theory. A value of 1.0 represents a perfect fit and you should expect to have a value above 0.99 or better for this data. Distance vs. Time Distance (m) y = 1.7 R 2 Time (sec) Motion Graphs: What Do They Tell Us? Key Concepts in Interpreting Motion Graphs: The slope at any point on a positiontime graph is the velocity. It may be negative, positive, or zero. If the slope is constant, the velocity is constant for that motion. The slope at any point on a velocitytime graph is the acceleration. It may be negative, positive, or zero. If the slope is constant, the acceleration is constant for that motion. A constant slope derived from a graph using Excel with a high R 2 value is a good indicator that the slope (velocity or acceleration) is truly constant, and that the equation obtained from the analysis is valid.
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