In Section 3.3 we studied a model for a dropped object falling under the influence of a constant gravitational force — and no other forces. Here we take a close look at possible assumptions about air resistance. First we observe that ignoring air resistance is an unrealistic assumption for a familiar falling object: a raindrop. Then we consider two different models that incorporate air resistance. We solve a differential equation that includes resistance proportional to velocity. Then we examine a proposed solution for a differential equation in which resistance is proportional to the square of velocity. We don't have the tools yet for deriving symbolic solutions for the second equation, but we will develop those tools later.

Section 5.1.1 Galileo's Model

Our discussion of the falling marble in Section 3.3 was based on Galileo's Model, the assumption that the only significant force acting on a falling object is the gravitational force. According to Newton's Second Law of Motion, such an object experiences a constant acceleration. In symbolic form, this model is the simplest initial value problem we have encountered so far:

Its solution, which can be expressed as "velocity is proportional to time," leads to our second simplest initial value problem:

And the solution of this initial value problem tells us that the distance fallen is a quadratic function of time:

In particular, if the raindrop falls, say, \(3000\) feet, then this last equation says that the time of fall would be

Then the velocity equation tells us that, after falling \(3000\) feet, the raindrop would be going $\mathrm{}$

$32.2\times 13.65=440$ feet per second,

or about \(300\) miles per hour.

Would you take a stroll in such a rain? Would you carry an umbrella? A raindrop striking the Earth (or anything it encountered sooner) at \(300\) miles per hour just doesn't fit with our experience about raindrops.

Note 1 – Sources