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(C) Since the collision is elastic and the masses are equal,the velocities of the particles will be interchanged after each collision.Let the particle

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$2$

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(C) Since the collision is elastic and the masses are equal,the velocities of the particles will be interchanged after each collision.Let the particles collide at time $t$ at an angle $	heta$ from point $A$.The distance traveled by the first particle is $R	heta = vt$.The distance traveled by the second particle is $R(2\pi - 	heta) = 2vt$.Dividing the two equations: $\frac{	heta}{2\pi - 	heta} = \frac{vt}{2vt} = \frac{1}{2}$.This gives $2	heta = 2\pi - 	heta$,so $3	heta = 2\pi$,which means $	heta = \frac{2\pi}{3} = 120^{\circ}$.After the first collision at $120^{\circ}$,the velocities are interchanged. The particle that had speed $v$ now has speed $2v$,and the particle that had speed $2v$ now has speed $v$.They will collide again after traveling another $120^{\circ}$ relative to each other,which occurs at an angle of $240^{\circ}$ from point $A$.After this second collision,the velocities are interchanged again. The particles will then travel the remaining $120^{\circ}$ to reach point $A$ simultaneously.Thus,they reach point $A$ after $2$ collisions.

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