Question

Ten small planes are flying at a speed of 150 km/h in total darkness in an air space that is 20 × 20 × 1.5 km³ 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 the plane can be approximated by a sphere of radius 10 m.

Answer 1

We can consider the planes as the motion of molecules in confined space. The distance between the two planes travelled between the collision or just to avoid an accident is the time of relaxation for the mean free path λ.

`Time = (Distance)/(Speed)`

= `λ/v`

= `1/(sqrt(2)n.πd^2.v)`

N = number of particles per unit volume V = `N/(Volume)`

`n = 10/(20 xx 20 xx 1.5 Km^3)`

= `0.0167  km^-3`

`d = 2 xx 10  m`

= `20  m`

= `20 xx 10^-3  km`

`v = 150  kmh^-1`

∴ `Time = 1/(sqrt(2)nπd^2v)`

= `1/(1.414 xx 0.0167 xx 3.14 xx 20 xx 20 xx 10^-6 xx 150)`

`t = 10^6/4448.8`

= 255 hrs