Peter D Murray Part 2 of Phy 690 Paper Developing the concept of the limit through calculating average velocity and acceleration of an object in motion The purpose of this paper is to develop one or more interdisciplinary projects that emphasize the strong connection of Calculus to Physics through an inquiry based application The project will demonstrate the use of Calculus in analyzing the mechanics of an object in free fall in order to anchor abstract Calculus concepts of the derivative and the limit to a physical observable context This is a very common activity conducted in introductory physics courses This project is intended for students who have completed or nearly completed an introductory calculus course as they will be expected to calculate derivatives as well as present their calculations using appropriate mathematical language and notation The project will be a discovery based learning experience where students will work in teams and present their findings to the larger group Students will derive the mathematical formulas by analyzing data collected on a free falling object Students will employ the use of calculus in their analysis to obtain an accurate model describing the motion of a free falling object Students will be required to present the detailed mathematical derivation of these calculations Students will keep a journal and be required to provide written descriptions of their observations Students will begin by taking measurements using the simple instruments of tape measure and stopwatch The hand collected data will be input to a TI 83 calculator for analysis Later these same measurements will be taken using Vernier motion sensors and analyzed with Logger Pro software Students will pursue the limit of an infinitely small time interval in measuring the instantaneous velocity of their object in motion Students will then derive from the data the average velocity and acceleration of the object at a particular moment in time then the function for position velocity and acceleration of the object Dropping a Ball How can we measure the precise position and velocity and acceleration of an accelerating object What measurements do you need to take to solve this problem These questions will be posed to the class to lead them to brainstorm ideas involving some method of marking position of the object at various moments in time Students should be placed in groups of four Each group will work through 3 stages of the activity Each stage leads to more precise measurement of velocity demonstrating the idea of taking the limit of a slope function as the time interval approaches zero Groups will gather raw data on the position of an object undergoing constant acceleration in free fall Stage 1 This activity is designed to put students through the rigor of old fashioned data collection In this activity the data collected will be crude in comparison to the data gathered by a motion sensor in stage 3 which takes more accurate and far greater number of measurements This stage is designed to put students through the rigor of calculating average velocity by hand calculating the slope of lines Stage 1 has students dropping a ball from increasing heights and recording clock readings at positions along the ball s path In our SHS activity we will be dropping a rubber ball from a second story balcony overlooking the atrium in the FA wing of SHS This could just as well be done in the classroom or on the athletic field bleachers or a stairwell Students will stretch a tape measure vertically along the path of the ball before they begin In each group one student will drop the ball while the other three record time measurements using stopwatches The ball will be allowed to fall increasing distances in each trail Students will first drop the ball from a height of one foot above the floor and measure the time it takes for it to land Next students will drop the ball from a height of 2ft then 4 ft 6ft 10ft using a step ladder 16ft 20ft and finally from 28ft the height of the atrium balcony Students will create a data table like the one shown here Figure 1 Figure 2 Figure 3 Above is a sample of data collected by a group of students Figure 1 shows raw data entered into L1 and L2 In column L1 represents clock readings after the ball was released from rest L2 represents the position of the ball where the ground level is designated as position zero 0 The TI 83 or newer model is standard instrument in today s math classroom and students should be well practiced in performing this simple derivation from the graphing calculator Students should already be familiar with writing mathematical functions from a given set of data Figure 2 shows the Quadratic Regression equation derived by the machine including the R2 value representing the accuracy of the equation to modeling the data Figure 3 shows a graph of the data where time sec is plotted along the horizontal axis and position plotted along the vertical axis Students will then calculate average velocity for each trail using the familiar slope formula y2 y1 position slope x2 x1 time d or in other words average velocity t drawing lines between each pair of data points as shown below to compile a table of average velocities along the ball s path Students will examine points along the parabolic curve of the position vs time graph to determine that the slope of the line connecting any two points is equal to the average rate of change in the object s position or in other words the average velocity of the object between those two points Some toil should be spent on trying to determine these values without the aid of the Calculus APC 6 APC 1 Students will be assigned to articulate written explanations of their derivation of this conclusion Position ft 0 28 Position ft 0 35 26 0 35 26 28 0 5 24 28 Time sec Figure 4 Time sec Figure 6 shows velocity vs time the average velocities calculated from the table in Figure 1 where L1 represents the average clock reading on each interval and L2 represents the average velocity on that interval Figure 6 Slopes should be calculated along large intervals such as 0 28 and 1 25 0 as well as between each two consecutive data points Students should be asked to reflect in their journals on the values and meaning of each of these slopes in comparison to one another Figure 5 At this point the class should gather for groups to present data and observations It should be concluded that velocity increases as the ball falls greater distance
View Full Document
Unlocking...