JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Conservation of Momentum

The particles move in opposite directions with speeds $v$ and $2v$. Their relative speed is $3v$. The circumference is $2\pi R$. They meet after time $t=\frac{2\pi R}{3v}$. In this time, the first particle travels $v\cdot t=\frac{2\pi R}{3}$ and the second travels $2v\cdot t=\frac{4\pi R}{3}$. After an elastic collision between equal masses, velocities are exchanged. The particle that had speed $v$ now has speed $2v$ and vice versa. After the first collision, they continue and meet again after another $\frac{2\pi R}{3v}$. After the second collision, velocities exchange again. The first particle will have traveled $\frac{2\pi R}{3}+\frac{4\pi R}{3}=2\pi R$, returning to point A. Therefore, 2 collisions occur before both return to A.