Small smooth spheres $$\(A\)$$ and $$\(B\)$$, of equal radii and of masses $$\(4 \mathrm{~kg}\)$$ and $$\(2 \mathrm{~kg}\)$$ respectively, lie on a smooth horizontal plane. Initially $$\(B\)$$ is at rest and $$\(A\)$$ is moving towards $$\(B\)$$ with speed $$\(10 \mathrm{~m} \mathrm{~s}^{-1}\)$$. After the spheres collide $$\(A\)$$ continues to move in the same direction but with half the speed of $$\(B\)$$. A third small smooth sphere $$\(C\)$$, of mass $$\(1 \mathrm{~kg}\)$$ and with the same radius as $$\(A\)$$ and $$\(B\)$$, is at rest on the plane. $$\(B\)$$ now collides directly with $$\(C\)$$. After this collision $$\(B\)$$ continues to move in the same direction but with one third the speed of $$\(C\)$$. $$\(A\)$$ and $$\(B\)$$ coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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