Two particles, $$\(A\)$$ and $$\(B\)$$, have masses $$\(3 m\)$$ and $$\(4 m\)$$ respectively. The particles are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately after the collision, $$\(A\)$$ and $$\(B\)$$ are moving in the same direction with speeds $$\(\frac{u}{3}\)$$ and $$\(u\)$$ respectively. In the collision, $$\(A\)$$ receives an impulse of (a) Find the coefficient of restitution between $$\(A\)$$ and $$\(B\)$$. (6) When $$\(A\)$$ and $$\(B\)$$ collide they are at a distance $$\(d\)$$ from a smooth vertical wall, which is perpendicular to their direction of motion. After the collision with $$\(A\)$$, particle $$\(B\)$$ collides directly with the wall and rebounds so that there is a second collision between $$\(A\)$$ and $$\(B\)$$. This second collision takes place at distance $$\(x\)$$ from the wall. Given that the coefficient of restitution between $$\(B\)$$ and the wall is $$\(\frac{1}{4}\)$$ (b) find $$\(x\)$$ in terms of $$\(d\)$$. (6)

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