Exercise 4.1. The Knudsen number, which plays an important role in low-
density flow problems, is defined as the dimensionless ratio $\lambda/L$, where $\lambda$ is the
mean free path and $L$ is some characteristic length of the boundaries. Flow for
which $\lambda/L \geq 1$ is sometimes called free-molecule flow. Consider a sphere
1 foot in diameter traveling through the atmosphere, and take the diameter of
the sphere as the characteristic length. Using the results of this section, find the
altitude above which free-molecule flow prevails, assuming that the density of
the atmosphere is given to a sufficient approximation by
$$\frac{\rho}{\rho_0} = e^{-\alpha H},$$
where $H$ is the altitude, $\alpha = 4.25 \times 10^{-5}/ft$, and $\rho_0$ is the sea-level density
$1.23 \times 10^{-3} gm/cm^3$. The molecular quantities required in the calculation can
