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Purposes: To study differential equations that model falling objects subject to air resistance -- for example, raindrops -- and to develop Euler's Method, a tool for approximating solutions of initial value problems.
Prerequisites: Discrete rate of change, derivative as slope of tangent line, slope field.
Timing: This module can be used in a differential calculus course -- we have used it many times in about the eighth or ninth week of a 14-week course. No prior knowledge of solutions of differential equations or of any form of integration is required. However, if it is used in a course in integral calculus or differential equations, students can be asked to complete the symbolic solutions. (We comment below about why this may not be a productive use of one's time.)
Use of CAS: Students (separately or in teams of 2 or 3) do their work in a CAS worksheet downloaded from the title page. The CAS window and the browser window should be open at the same time on each screen. All but the most basic commands are already entered in the worksheet -- but it won't compute much of anything until students fill in appropriate formulas. Thus, they need to know basic syntax for the selected CAS. When completed, the worksheet is the appropriate object to be submitted for grading.
Because of our timing for this module -- coming after a CAS tutorial and several other modules -- we assume students are reasonably comfortable with syntax. To get a sense of how this works and how the prerequisites get satisfied (even if not covered in the textbook), you may want to visit the CCP Differential Calculus page -- in particular, to view the Slope Fields module preceding Raindrops. (But please note that the CCP modules other than Raindrops are not part of this publication.)
Nature is not equipped with formulas -- although many students come to believe that it is. In particular, formulas for no-resistance falling objects (acceleration, velocity, position, time of fall) are staples in physics and mathematics textbooks. But these formulas don't apply to perhaps the most common falling object in our experience, a raindrop.
Nature is not linear either, often not even approximately -- although many students come to believe that it is. Assuming linear resistance is merely a mathematical convenience, whether we find the assumption in a mathematics book or a physics book. This assumption works (as an appropriate model) only for the very slowest-moving raindrops, and not at all for most falling objects of any size or shape. Thus, even the simplest objects in nature -- if they are to be modeled adequately by mathematics -- require models that are not easily solved or are impossible to solve in closed form. Our mathematics courses tend to concentrate on closed-form solutions and overlook the simplest approximating schemes -- such as Euler's Method -- that can convey far more insight into the nature of a solution than an elaborate formula can. (We are speaking here of a student's insight, not a mathematician's -- but working scientists often gain insight in the same way.)
In addition to dispelling misinformation about the "formulas of nature," we hope to counter a persistent myth about formulas themselves. Students learn repeatedly, from precalculus through differential equations, that asymptotic behavior (e.g., exponential decay) involves a limit that "can be approached but never reached." If the decay rate is sufficiently fast, then the limiting behavior is reached for all practical purposes very quickly -- an important message for budding scientists and engineers. Raindrops illustrate this very nicely.
Numbers. In general, we are not thrilled with models and numerical data for which we can offer the student no evidence. However, students are likely to be familiar with the constant-acceleration model -- perhaps even to have gathered experimental data with a motion detector -- and mistakenly believe that it applies to all falling bodies. They may also have encountered the linear-resistance model in a textbook -- and mistakenly believe it applies to all air-resistance situations. At the intended level of this module, we can't go into Reynolds numbers and other physical considerations. The data that exist would be meaningless, and this is not a situation for which students can gather their own data -- beyond their experience with various types of rain.
Units. We stress the units of our numerical constants (odd things such as "reciprocal seconds") because they are an important check on consistency within the calculations.
Orientation of the coordinate system. This is a matter of choice. Since this module is about the fall of raindrops from an arbitrary height (3000 feet), we consider the natural choice to be a downward orientation, with the origin at the point of fall, and with the splash at s = (or y =) 3000. If your students are not familiar with making this choice (e.g., if they think y = 0 always means ground level and "positive" always means "up"), then you may have to remind them of the arbitrariness of these choices and suggest that convenience is sometimes more important than consistency.
Strange words and phrases. We use the terminology that students will encounter later, and some patience is required in getting students to understand and use correct terminology. They usually have little difficulty with the idea that "differential equation" means an equation involving a derivative -- although "differential" may still be an unfamiliar term. It may help to stress that "Differential Calculus" is the name of a course in which we study derivatives. A much more serious point of confusion is "initial value problem." We try to get students to understand that this is a mathematician's shorthand for "a problem in which the given information includes both a differential equation and a starting value for the unknown function." If they have to say or write the whole thing a few times, they begin to appreciate the value of abbreviation. This issue comes up immediately in Part 1.
Closed-form solutions. In Part 2, we allow for the possibility that students may already know a method for solving first-order linear differential equations and IVP's. But we stress that this is not required. If this will be taken up at a later point in the course, the "drizzle drop" problem provides an example to which you can return.
At no point do we assume that students know a technique for finding a closed-form solution for the "thunderstorm drop" problem in Part 5. This can be done by a partial-fraction integration, and you may want to recall this problem if you happen to be teaching the same students at that stage of their calculus studies. But keep in mind the real message of this module: Closed-form solutions are really not worth significant effort in this case -- the Euler approximations (or the slope fields) show us that terminal velocity is reached almost immediately, and that's all you need to know. (See the Final Comments in Part 6.)
The secondary message in Part 1 is also important: If one applies a formula that "everyone knows" in a situation for which it is not applicable, the results are likely to be silly.
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