question_answer 43)

                  Ten small planes are flying at a speed of 150 km/li in total darkness in an air space that is \[20\times 20\times 1.5\text{ }k{{m}^{3}}\] 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 collision with your plane. Assume for this rough computation that a saftey region around die plane can be approximated by a sphere of radius 10 m.

Answer:

                  The problem is equivalent to the motion of molecules.                 Mean free path, \[\lambda =\,\frac{1}{\sqrt{2}\,\pi \,n\,{{d}^{2}}}\]                 Where, \[n=\]number of planes volume                 \[=\,\frac{10}{20\,\times 20\,\times 1.5}\,\,=\,0.0167\,\,k{{m}^{-3}}\]                 \[=0.0167\times {{10}^{9}}{{m}^{3}}\]                 \[d=20\,m\]                 \[\therefore \] \[\lambda \,=\,\frac{1}{1.414\,\times \,3.14\,\times \,400\,\times \,0.0167\,\times \,{{10}^{-9}}}\]                 \[=3.37\times {{10}^{7}}m.\]                 \[\upsilon =\,150\,km/h=150\,\times \,\frac{5}{18}=\,41.67\,m\,{{s}^{-1}}\]                 Time that will elapse between near collision,                 \[t=\,\frac{\lambda }{\upsilon }=\,\frac{3.37\,\times {{10}^{7}}m}{41.67\,m{{s}^{-1}}}\,=\,8.09\,\,\times \,{{10}^{5}}s\]                 = 224.7 hours.