Three particles $$\(P, Q\)$$ and $$\(R\)$$, of masses $$\(0.1 \mathrm{~kg}, 0.2 \mathrm{~kg}\)$$ and $$\(0.5 \mathrm{~kg}\)$$ respectively, are at rest in a straight line on a smooth horizontal plane. Particle $$\(P\)$$ is projected towards $$\(Q\)$$ at a speed of $$\(5 \mathrm{~m} \mathrm{~s}^{-1}\)$$. After $$\(P\)$$ and $$\(Q\)$$ collide, $$\(P\)$$ rebounds with speed $$\(1 \mathrm{~m} \mathrm{~s}^{-1}\)$$. $$\(Q\)$$ now collides with $$\(R\)$$. Immediately after the collision with $$\(Q, R\)$$ begins to move with speed $$\(V \mathrm{~m} \mathrm{~s}^{-1}\)$$. Given that there is no subsequent collision between $$\(P\)$$ and $$\(Q\)$$, find the greatest possible value of $$\(V\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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