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As the particles move with constant velocities their position vectors are
r1(t) = r1 + v1t
r2(t) = r2 + v2t
At the instant of collision, say at t = T,
r1(T) = r2(T)
r1 + v1T = r2 + v2T
r1 - r2 = (v2 - v1)T
Hence r1 - r2 and (v2 - v1) are parallel vectors. As they are parallel vectors, unit vectors along them must be same.
Hence,
r1 - r2/|r1 - r2| = (v2 - v1)/|(v2 - v1)|
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Moderation Team
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