Constant Velocity Measuring and Graphing Velocity


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1 About this Lesson Constant Velocity Measuring and Graphing Velocity This lesson will provide students the opportunity to develop a quantitative understanding of the relationship between the motion of objects and the graphs that represent this motion. They will investigate position vs. time, velocity vs. time, and other aspects of motion by using a stopwatch and their own motion. Throughout the activity students are challenged to make connections between the motion they experience or observe and the graphs that are generated as well as give meaning to slopes and yintercepts. The graphs will then be used to develop the first of the kinematics equations for constant motion. This activity is included in LTF Physics Module 2. Objective Students will find the speed of a walker traveling at a constant velocity by finding the slope of a distance vs. time graph. Level Physics Common Core State Standards for Science Content LTF Science lessons will be aligned with the next generation of multistate science standards that are currently in development. These standards are said to be developed around the anchor document, A Framework for K 12 Science Education, which was produced by the National Research Council. Where applicable, the LTF Science lessons are also aligned to the Common Core Standards for Mathematical Content as well as the Common Core Literacy Standards for Science and Technical Subjects. Code Standard Level of Thinking Depth of Knowledge T E A C H E R P A G E S (Literacy) RST Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Understand ACED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. ACED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (Literacy) RST.9 Translate quantitative or technical information expressed in words in a text into visual form (e.g., a
2 10.7 table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. ACED.2 FIF.6 SID.6a SID.6c SID.8 SID.7 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Compute (using technology) and interpret the correlation coefficient of a linear fit. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Connections to AP Physics I. Newtonian mechanics A. Kinematics 1. Motion in one dimension T E A C H E R P A G E S Materials Each lab group will need the following: computer ruler, clear metric stopwatch Additional teacher materials: measuring tape 2 rolls tape, masking
3 Assessments The following types of formative assessments are embedded in this lesson: Assessment of prior knowledge Guided questions The following additional assessments are located on the LTF website: Physics Assessment: Kinematics 1D and 2D 2011 Physics Posttest, Free Response Question 1 Teacher Notes Students will record the distance and time for several 5m intervals as a walker travels at a constant velocity on a football field or other course. They will collect data and plot graphs of distance vs. time for three walkers, and relate the slope of each graph to the speed of the walker. The easiest way to perform this lab is to take the students out to the school football field because it is marked with lines 5 yards apart. You may want to tell the students they may approximate 5 yards as 5 meters, as the differences in these two distances should be very small for the purposes of this lab activity. Alternatively, you could mark a course under a breezeway or in a long hallway. You can use this lab to introduce motion at a constant velocity as well as reviewing good graphing technique with the students. T E A C H E R P A G E S
4 Answer Key Teacher Overview Constant Velocity Data and Observations Table 1. Walkers at Constant Velocity Time (s) Distance (m) First Walker Second Walker Third Walker T E A C H E R Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at i
5 Answer Key (continued) Teacher Overview Constant Velocity Analysis T E A C H E R Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at ii
6 Answer Key (continued) Teacher Overview Constant Velocity Conclusion Questions 1. Walker 3 traveled at the fastest speed, whereas Walker 1 traveled at the slowest speed. In the sample data, Walker 2 seems to have walked at the most constant velocity with the most data points lying on the best fit line. 2. On a distance vs. time graph, the slope of the line represents the velocity in meters per second. Each plot was a straight line because the velocity was constant for each walker. The greater the slope of the best fit line, the greater the velocity of the walker. 3. If the walker accelerated, the velocity would not be constant and the average velocity would appear to be greater than if the walker had walked from the beginning with a constant velocity. It is unlikely that all of the timers started their stopwatches at the exact time the teacher gave the signal. If many of the watches were started late, the time for the first 5 meters would appear to be less than it actually was, causing the average velocity of the walker to appear greater than it actually was. 4. The slope of a position vs. time graph is the velocity. The units would be meters per second because the y units are meters and the x units are seconds. If the slope is positive, the velocity is positive or the object is moving in the direction defined as positive. If the slope is negative, the object is moving in the direction defined as negative. 5. When the slope of a position vs. time graph is constant, the object is moving with a constant velocity. 6. When the slope of a position vs. time graph is changing, the object is either speeding up or slowing down. As the student walks the course, it is difficult to keep exactly the same pace without speeding up or slowing down at some points during the walk. These changes in velocity can be seen in the points lying to the side of the best fit lines. 7. As the student walks the course, it is difficult to keep exactly the same pace without speeding up or slowing down at some points during the walk. These changes in velocity can be seen in the points lying to the side of the best fit lines. 8. If the student started before or after the starter began the timing, this will result in a yintercept on your graph. If the walker starts at some point behind the zero line, this also results in a yintercept that indicates how far they were behind or in front of the zero line. T E A C H E R Extensions 1. This should produce a graph with a yintercept that has a negative value. This value should represent how far from the starting point the walker started when the timers began timing. The xintercept represents how much time it took the walker to reach the zero mark. 2. This should produce a graph with a yintercept that has a positive value. This value should represent how far past the starting point the walker started when the timers began timing. Extrapolating the xintercept should represent how much time it took the walker to reach the zero mark. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at iii
7 Constant Velocity Measuring and Graphing Velocity Kinematics is the study of how things move. Understanding the quantitative aspects of the motion of objects often involves making graphs of motion. These graphs may be of position vs. time, velocity vs. time, or perhaps other aspects of motion. Being able to construct graphs of motion as well as interpret graphs of motion are essential tools used in the development and understanding of kinematics. Graphs are often used either to derive or verify the equations of motion as well as to graphically illustrate the motion involved. Understanding kinematics ( how things move ) is necessary before the next step in physics, dynamics ( why things move ), can be mastered. Purpose In this lab, you will record the distance and time for several 5m intervals as a walker travels at a constant velocity on a football field. You will then plot graphs of distance vs. time for three walkers, and relate the slope of each graph to the speed of the walker. Materials Each lab group will need the following: computer ruler, clear metric stopwatch
8 PROCEDURE 1. The data gathering in this lab is performed as an entire class activity using: a. 3 Walkers b. 1 Data Recorder c. As many Timers as possible (the rest of the class) 2. Go out to the football field on your campus, or to the course your teacher has measured and marked for you. 3. Have two or more Timers stand on the same line of the football field. For this activity, we will use the approximation that 5 yards is about equal to 5 meters as our velocities need to be in m/s. 4. Place the other Timers on other 5m lines so that there will be at least two Timers on each of the lines from the 5m line to the 35m line of the field. 5. The Walker for the first trial will stand on the goal line (at 0 m). When your teacher gives the start signal, the Walker will begin walking the course in a straight line at a constant velocity, and all Timers will begin timing at the instant the walker begins walking. 6. As the Walker passes each 5m line, the timers on that line will stop their stopwatches. The Data Recorder for the first group will record the times for the two or more Timers at each line. Times should be measured to the nearest tenth of a second. 7. Each of the three Walkers will repeat Steps 5 and 6, and try to walk at a different speed than the previous Walker. This should generate a wide variety of data. 8. When all three Walkers have finished and the Data Recorders have obtained their data, return to the classroom. Each participant will record their own data as well as the data from two other groups. Record your values in the data table on your student answer page.
9 DATA AND OBSERVATIONS Distance (m) Time for First Walker (s) Time for Second Walker (s) Time for Third Walker (s)
10 ANALYSIS 1. Plot a graph of distance on the vertical axis versus time on the horizontal axis for your group s Walker. Be sure to use proper graphing techniques outlined in Foundation Lesson 4: Graphing Skills, including using a straightedge to draw the axes and scaling the axes so that the data uses most of the graph paper. 2. On the same axes, plot distance vs. time for the two other Walkers from your class data. You may want to plot the points for these Walkers in another color, or otherwise somehow distinguish the three sets of data from each other on the graph. 3. Using a ruler, draw the best fit straight line through each of the sets of your data points. Remember, the best fit straight line represents the trend of the data, so there should be some points above your line, below the line, and probably on the line. 4. The slope of each best fit straight line on the graph represents the speed of each respective Walker. Choose two points on each line and calculate the average speed of each walker. Show your calculation of each slope on your graph paper, and include the proper units for each slope.
11 CONCLUSION QUESTIONS 1. Which of the three Walkers traveled at the fastest speed? Which Walker traveled at the slowest speed? Which Walker traveled with the most consistent constant speed? 2. Write a general statement summarizing the concepts and results of this lab. 3. List two reasonable sources of error in this lab, and describe how each may have affected your results. 4. In general, explain the significance of the slope of a position vs. time graph. Include a discussion of positive and negative slope. 5. What type of motion is occurring when the slope of a position vs. time graph is constant?
12 CONCLUSION QUESTIONS (CONTINUED) 6. What type of motion is occurring when the slope of a position vs. time graph is changing? Relate this to the distribution of points around your best fit line for one of your Walkers. 7. When you plotted your data and drew the best fit line through the data, why were some of the data points not located directly on your best fit line? 8. When you drew your best fit line, did the line go through the origin at (0,0)? If not, what would the significance of a yintercept have on your data? If the Walker had started at some point other than the zero line, would this have affected the yintercept of your graph?
13 EXTENSIONS 1. Have one student rewalk the course at a constant speed but this time, have the student start 5 m behind the original starting line. This method will also require timers at the zero line as well as timers at the other locations. Graph the data, and explain the significance of the x and yintercepts. 2. Repeat having a student rewalk the course at a constant speed but this time, have the student start 5 m in front of the original starting line. This method will not require Timers at the zero line. Graph the data, and explain the significance of the x and yintercepts.
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