Raindrops : Models and Reality

Model Reality Check

Its articles will emphasize the oft-neglected modeling step of returning the model to the world of reality. For example, it will provide an opportunity to examine models commonly found in mathematics textbooks and assess them in the light of the original context that was to be modeled. In other words, it will try to fulfill the oft-neglected step in modeling of reflecting on what insight a model offers into the original phenomenon. This issue contains such an article by me about the speed of falling raindrops. The article was occasioned by teaching a course in differential equations, in the course of which I came across various models for raindrops in assorted textbooks and papers. Other standard models in textbooks for calculus, differential equations, and difference equations include

• parachuting,
• projectile motion in air,
• logistic growth with harvesting (fisheries being a popular application),
• other biological growth models (such as the Beverton-Holt model and the Ricker model), and
• predator-prey models.

Beyond textbook models, I am also looking forward to the Journal publishing a model of the skydive of Felix Baumgartner from 120,000 feet in October 2012-one that agrees with the data taken in the jump (see [Tierney 2012]).

I invite readers to contribute to this new department and new endeavor, and to suggest topics for it.

Introduction

Rain sometimes falls gently and sometimes falls hard. A drop of rain can fall thousands of feet before hitting us; it doesn't hurt us (unless it falls as hail). Araindrop tends to reach a terminal velocity; moreover, it is highly viscous, with a highly elastic impact that diffuses over a surface area larger than the drop's cross section (i.e., it tends to splatter) [Why doesn't rain hurt. . . 2012].

But what is that terminal velocity, and how does it depend on other factors?

Several models in textbooks and in the literature of mathematical modeling have given answers. We compare them with one another, with models used by meteorologists, and with data.