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I am looking at equations of colliding particles in a granular gas and wondering how to calculate them.

We assume that the grains are identical perfect spheres (in R^3) of diameter D>0, (x,v) and (x-Dn,w) are their states before a collision, where n ε S^2 is the unit vector along the centre of both spheres, and x the position vector of the centre of the first sphere, e is the restitution coefficient which relates the normal components of the particle velocities before and after collision, the post collisional velocities (v*,w*) then are such that

(v*-w*)n = -e((v-w)n)

I was wondering how from this equation do we calculate the change of velocity for the colliding particles:

v* = v- 1/2(1+e)((v-w)n)n,

w* = w+ 1/2(1+e)((v-w)n)n

Many thanks to anyone that can help!

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# Colliding Particles in Granular Material Flows

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