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Instructors_Guide_ch02 - 2 Kinematics The Mathematics of...

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2 Kinematics, The Mathematics of Motion Recommended class days: 3 minimum, 4 preferred Of all the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me to have been the most amazing in character and the most stupendous in the scope of its consequences is the one relating to the problem of motion. Herbert Butterfield— The Origins of Modern Science Background Information Chapter 2 is a large and difficult chapter. Although to physicists the chapter says nothing more than v = dx / dt and a = dv / dt , these are symbolic expressions for difficult, abstract concepts. Student ideas about force and motion are largely non-Newtonian, and they cannot begin to grasp Newton’s laws without first coming to a better conceptual understanding of motion. As Butterfield notes in the above quote, the “problem of motion” was an immense intellectual hurdle. Galileo was perhaps the first to understand what it means to quantify observations about nature and to apply mathematical analysis to those observations. He was also the first to recognize the need to separate the how of motion—kinematics—from the why of motion—dynamics. These are very difficult ideas, and we should not be surprised that kinematics is also an immense intellectual hurdle for students. Student difficulties with kinematics have been well researched (Trowbridge and McDermott, 1980 and 1981; Rosenquist and McDermott, 1987; McDermott et al., 1987, Thornton and Sokoloff, 1990). Arons (1990) gives an excellent summary and makes many useful suggestions for teaching kinematics. Student difficulties can be placed in several categories. Difficulties with concepts: Students have a rather undifferentiated view of motion, without clear distinctions between position, velocity, and acceleration. Chapter 1 will have given them a start at making these distinctions, but they’ll need additional practice. In one study, illustrated in the figure above, students were shown two balls on tracks. Ball A is released from rest and rolls down an incline while ball B rolls horizontally at constant speed. Ball B overtakes ball A near the beginning, as the motion diagram shows, but later ball A overtakes ball B. Students were asked to identify the time or times (if any) at which the two balls have the same 2-1
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2-2 Instructor’s Guide speed. Prior to instruction, roughly half the students in a calculus-based physics class identify frames 2 and 4, when the balls have equal positions , as being times when they have equal speeds. Similarly (see references for details), students often identify situations in which two objects have the same speed as indicating that the objects have the same acceleration. Confusion of velocity and acceleration is particularly pronounced at a turning point, where a majority of students think that the acceleration is zero. McDermott and her co-workers found that roughly 80% of students beginning calculus-based physics make errors when asked to identify or compare accelerations, and that the error rate was still roughly 60% after conventional instruction. Thornton and Sokoloff
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