This experiment features several basic physics, algebra and data handling concepts. Students use the motion detector to investigate the free-fall motion of large coffee filters with the goal of finding the relationship between the number of filters in free-fall and their terminal velocities. 
The beauty of using coffee filters is that they can produce downward motions of all varieties, from free-fall (air resistance negligible) to floating downward at constant speed (downward force due to Earth's gravity balanced by the upward directed air resistance or drag force, when filters fall at constant or terminal speed.).
We introduce the mathematical concept of piecewise functions in this experiment, because students must extract a subset of the data corresponding to the terminal velocity of the coffee filters. Although the falling filters do not actually generate piecewise d(t) functions, the selection of the relevant portion of the collected data lists is a useful technique commonly encountered in CBL experiments. Thus this module includes concepts from the physics of free-fall in air, to understanding piecewise functions to improving students' data handling skills.
The module begins with a homework assignment on piecewise functions which should be done by the students the previous night, and discussed in class before beginning the module. After discussing the homework, the experiment is preceded by a story problem about the height vs time function (piecewise) of a cup of coffee. The reason for stressing piecewise functions is to improve students' interpretation of motion graphs (distance vs time, velocity vs. time and acceleration vs. time) which are often piecewise functions, and to hone their data handling skills using graphing calculators.
The experiment begins by challenging the students to minimize the effects of air resistance for the coffee filters. A single, wadded-up coffee filter simulates free-fall quite nicely. Also 10 nested coffee filters fall essentially at free-fall rate. (Free-fall means that an object falls under the influence of Earth's gravitational force, and that air resistance is negligible.) The students should discover by exploration the fastest way to get one or more filters to fall to the ground. Allow enough time for them to produce well-defined parabolic d(t) curves (indicative of accelerated motion).
The class discussion following this exploratory part of the module should bring to light that the speed of the filter depends on its cross-sectional area presented to the air. 
The discussion will also serve to remind students what free-fall d(t) graphs look like in the absence of significant air resistance. If you want to really minimize air resistance, then a heavy object such as a tennis ball or a book can be dropped toward the motion detector. In this case you should protect the motion detector by placing a wire "in" basket over the detector.
The object of the second part of the experiment is for the students to find a mathematical function which relates the number of the filters to the terminal velocity of the filters. Terminal velocity is the maximum (and constant) speed attained by the filter when the upward drag force due to air resistance equals that of the downward force of gravity, F(drag) = -F(gravity). If the net force acting on an object is zero, then (by Newton's second law, F=ma) the net acceleration on the object must also be zero. Hence the filter has a constant speed when the two forces become equal. As more filters are nested together, the weight of the filters changes, so that the F(gravity) changes. The dependence of the terminal velocity is a function of the drag force (See Derby, Fuller and Gronseth, The Ubiquitous Coffee Filter, The Physics Teacher, vol 35, p. 168, March 1997.) The experiment will show that the relationship between the number of coffee filters and the terminal velocity is N~vT2. The TI-83/CBL is used to measure the d(t) graphs using the program BOUNCEIT. This program is used because CHOOSE can be used to select parts of the d(t) data which are transferred to lists L3 and L4 in the TI-83 (doesn't overwrite the data collected). Students should first check that the data collected have a flat segment corresponding to constant velocity (zero acceleration). Then they should CHOOSE that part of their data, plot it, and determine the terminal velocity. Students will need to set up data tables for number of filters(N) and terminal velocity (vterm). It is this N vs v(terminal) function to which they finally fit a regression curve which can be used to predict the terminal velocity for a given number of filters.
Mathematics: We introduce the mathematics concept of piecewise functions both as homework and as a story problem to begin this experiment. We note that the coffee filter motion is not itself a piecewise function. The reason we introduce piecewise functions in this module is that the experiment requires students to recognize which portion of the graphed data corresponds to the coffee filter fallling at terminal velocity. To select only those relevant data points, we suggest using the CHOOSE program to limit the domain to the linear part of the distance vs time graphs. The BOUNCEIT program places the extracted data in the correct lists (L3 and L4) in the TI-83 calculators. The procedure for fitting a linear regression curve should be useful for teaching curve fitting.
Because this technique can be used for many different kinds of experiments using the TI-83/CBL equipment, this module serves several valuable purposes from the physics of free-fall in air, to understanding piecewise functions to improving students' data handling skills.
The mathematics homework to accompany this experiment is quite traditional in its comparison of the function definition to the graph. The use of the TI-83 to graph piecewise functions may not be familiar to all instructors. The following is offered as a brief explanation of how the TI-83 can be used to draw the piecewise function graphs.
The TI-83 TEST menu has 6 choices: = 
> < < . All of these will return a 1 or 0 when used to compare two values. 7=5 returns a 0 because the statement 7=5 is false.
6<12 returns a 1 because the statement 6<12 is true.
For graphing, a domain can be defined using these TESTs.
(x>3) returns a 0 for all x values less than or equal to 3 and returns a 1 for all x values greater than 3.
(x>3)(x<9) will return 1
1 for x values between 3 and 9 and returns 01 or 10 (both equal 0) for all other values of x.
So, y1 = (3x + 2)/((x > 3)(x < 9)) uses division by 1 (which produces a graph) within the domain and division by 0 (which doesn't produce a graph) beyond the domain.
Combining a number of these "y =" graphs will produce a piecewise function.
(Note: It is possible to add the separate "y =" graphs into one and get the same results.)
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