Two small balls, $$\(A\)$$ and $$\(B\)$$, are moving in opposite directions along the same straight line on smooth horizontal ground. The mass of $$\(A\)$$ is $$\(2 m\)$$ and the mass of $$\(B\)$$ is $$\(3 m\)$$. The balls collide directly. Immediately before the collision, the speed of $$\(A\)$$ is $$\(2 u\)$$ and the speed of $$\(B\)$$ is $$\(u\)$$. The coefficient of restitution between $$\(A\)$$ and $$\(B\)$$ is $$\(e\)$$, where $$\(e> 0\)$$ By modelling the balls as particles, (a) show that the speed of $$\(B\)$$ immediately after the collision is $$\(\frac{1}{5} u(1+6 e)\)$$. (6) After the collision with ball $$\(A\)$$, ball $$\(B\)$$ hits a smooth fixed vertical wall which is perpendicular to the direction of motion of $$\(B\)$$. The coefficient of restitution between $$\(B\)$$ and the wall is $$\(\frac{5}{7}\)$$ Ball $$\(B\)$$ rebounds from the wall and there is a second direct collision between $$\(A\)$$ and $$\(B\)$$. (b) Find the range of possible values of $$\(e\)$$. (4)

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