1. Let the masses of the two identical particles be $m$ each.

2. Choose a positive direction.
   Take the direction of the first particle (speed $2v$) as positive.

   Then:

   • Velocity of particle 1: $u_1 = +2v$
   • Velocity of particle 2: since it is moving towards the first particle, its velocity is in the opposite direction: $u_2 = -v$

3. Use the formula for center of mass velocity: $V_{cm} = \frac{m u_1 + m u_2}{m+m}$

4. Substitute the values: $V_{cm} = \frac{m(2v) + m(-v)}{2m}$

5. Simplify: $V_{cm} = \frac{2mv - mv}{2m} = \frac{mv}{2m} = \frac{v}{2}$

6. Therefore, the velocity of the center of mass is $\boxed{\frac{v}{2}}$

7. Option check:

   • A: $v$ ❌
   • B: $v/3$ ❌
   • C: $v/2$ ✅
   • D: zero ❌

Hence, the correct option is C.