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Ten small planes are flying at a speed of $150 k m h^{- 1}$ in total darkness in an air space that is $20 x 20 x 1.5 k m^{3}$ 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 safety region around the plane can be approximated by a sphere of radius $10 m$

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220 h

432 h

225 h

125 h

answer is D.

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Detailed Solution

Here, $v = 150 k m h^{- 1}$ , $N = 10$

$V = 20 x 20 x 1.5 k m^{3} .$

Diameter of plane, $d = 2 R = 2 x 10 = 20 m = 20 x 10^{- 3} k m$

$n = \frac{N}{V} = \frac{10}{20 \times 20 \times 1.5} = 0.0167 k m^{- 3}$

Mean free path of a plane $λ = \frac{1}{\sqrt{2} {πd}^{2} n}$

Time elapse before collision of two planes randomly,

$t = \frac{λ}{v} = \frac{1}{\sqrt{2} {πd}^{2} nv}$

$= \frac{1}{1.414 \times 3.14 \times {(20)}^{2} \times 10^{- 6} \times (0.0167) \times (150)} = \frac{10^{6}}{4449.5} = 224.74 h = 225 h$

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