Two particles, $$\(A\)$$ and $$\(B\)$$, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. The mass of $$\(A\)$$ is $$\(2 m\)$$ and the mass of $$\(B\)$$ is $$\(3 m\)$$. Immediately after the collision, $$\(A\)$$ and $$\(B\)$$ are moving in opposite directions with the same speed $$\(v\)$$. In the collision, $$\(A\)$$ receives an impulse of magnitude $$\(5 \mathrm{mv}\)$$. (a) Find the coefficient of restitution between $$\(A\)$$ and $$\(B\)$$. (6) After the collision with $$\(A\)$$, particle $$\(B\)$$ strikes a smooth fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of the particles. The coefficient of restitution between $$\(B\)$$ and the wall is $$\(f\)$$. As a result of its collision with $$\(A\)$$ and with the wall, the total kinetic energy lost by $$\(B\)$$ is $$\(E\)$$. As a result of its collision with $$\(B\)$$, the kinetic energy lost by $$\(A\)$$ is $$\(2 E\)$$. (b) Find the value of $$\(f\)$$. (4)

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