Two small particles of equal masses start moving in opposite directions from a point $A$ in a horizontal circular orbit. Their tangential velocities are $v$ and $2v$, respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at $A$, these two particles will again reach the point $A$?

Step 1: Analyze the collision dynamics.

For two particles of equal mass undergoing a head-on elastic collision, they exchange their velocities. This velocity exchange is mathematically equivalent to a "pass-through" scenario, where the two particles continue along their paths with their original speeds as if they passed through each other.

Step 2: Calculate the time and position of the first collision.

Let the radius of the circular orbit be $R$. The total circumference of the orbit is $2\pi R$. Since the particles move in opposite directions with speeds $v$ and $2v$, their relative speed is:

$$v_{\text{rel}} = v + 2v = 3v$$

The time $t_1$ taken for the first collision to occur is:

$$t_1 = \frac{2\pi R}{3v}$$

The angular position $\theta_1$ of the first collision from point $A$ is determined by the distance traveled by the slower particle:

$$\theta_1 = \frac{v \cdot t_1}{R} = \frac{2\pi}{3} = 120^\circ$$

Step 3: Determine the positions of subsequent collisions.

Using the pass-through equivalence, subsequent collisions occur at regular intervals of $\Delta t = t_1$.

The second collision occurs at time $t_2$:

$$t_2 = 2t_1 = \frac{4\pi R}{3v}$$

The angular position of the second collision is:

$$\theta_2 = \frac{4\pi}{3} = 240^\circ$$

The third meeting occurs at time $t_3$:

$$t_3 = 3t_1 = \frac{2\pi R}{v}$$

The angular position of the third meeting is:

$$\theta_3 = 2\pi \equiv 0^\circ$$

This third meeting occurs exactly at the starting point $A$.

Step 4: Count the collisions between point $A$.

The particles collide at $120^\circ$ and $240^\circ$ before meeting again at point $A$ ($360^\circ$). Thus, there are exactly $2$ elastic collisions between their departure from $A$ and their return to $A$.

Answer: 2