Section Summary

In this section we considered three initial-value-problem models for falling raindrops. The first, Galileo's Model, ignores air resistance. It is straightforward to solve symbolically, but it does not reflect our experience with rain.

The second model was based on Stokes' Law and incorporates air resistance proportional to the velocity in the velocity differential equation. With some work we were able to obtain a symbolic solution in this case as well — a change of variable transformed the equation to an exponential decay equation that we already knew how to solve. (We used the same solution-by-substitution technique in Chapter 3 to solve the murder victim problem, which is essentially the same mathematical problem in a different context.) The model fits reasonably well for drizzle drops, but it is not physically appropriate for larger rain drops.

The third model assumes air resistance proportional to the square of the velocity and is appropriate for large raindrops such as one finds in thunder storms. At this time, we are not ready to show how to determine the symbolic solution of this problem. Rather, we stated the solution formula and showed that it corresponded to empirical data. In the next section we will obtain a numerical solution for this initial value problem.