First slide

Mean free path

Question

Ten small planes are flying at a speed of 150kmph in total darkness in an air space that is $(20 \times 20 \times 1.5) k m^{3}$ 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 an average, about $25 \times N$ hours time will elapse between collision of a plane with your plane. Assume a safety region as a sphere of radius 10 m around every plane. Find N.

Moderate

Solution

Using kinetic theory of ideal gases,
Mean collision time = $t = \frac{m e a n f r e e p a t h}{a v e r a g e v e l o c i t y} = \frac{λ}{v} = \frac{1}{\sqrt{2} π d^{2} n v}$
Where, n = number density of planes and d = safe-region diameter=2R=2(10)m=20m= $20 \times 10^{- 3} k m$
Here, $n = \frac{n u m b e r o f p l a n e s}{v o l u m e} = \frac{10}{(20 \times 20 \times 1.5) k m^{3}} = 0.0167 k m^{- 3}$
$\Rightarrow t = \frac{1}{\sqrt{2} \times 3.14 \times (20^{2} \times 10^{- 6}) 0.0167 \times 150} = 225 h$ .

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