Ten small planes are flying at a speed of $150 \ km/h$ in total darkness in an air space that is $20 \times 20 \times 1.5 \ km^3$ in volume. You are in one of the planes, flying at random within this space with no way of knowing where the other planes are. On the average, about how long a time will elapse between near collisions with your plane? Assume for this rough computation that a safety region around the plane can be approximated by a sphere of radius $10 \ m$.

(D) We can model the motion of the airplanes as the random motion of gas molecules.
$(i)$ At the time of a near collision, the distance between the centers of two airplanes is $d = 2 \times 10 \ m = 20 \ m = 0.02 \ km$.
$(ii)$ The number density of airplanes in the given volume is $n = \frac{N}{V} = \frac{10}{20 \times 20 \times 1.5} = \frac{10}{600} = 0.0167 \ km^{-3}$.
$(iii)$ The time interval between two successive near collisions is given by the mean free time, $t = \frac{\bar{l}}{v}$, where $\bar{l} = \frac{1}{\sqrt{2} \pi n d^2}$ is the mean free path.
Thus, $t = \frac{1}{\sqrt{2} \pi n d^2 v}$.
Substituting the values:
$t = \frac{1}{1.414 \times 3.14 \times 0.0167 \times (0.02)^2 \times 150}$
$t = \frac{1}{1.414 \times 3.14 \times 0.0167 \times 0.0004 \times 150}$
$t = \frac{1}{0.00444} \approx 225 \ hours$.