Identify the type of collision

Since the two particles stick together after collision, this is a perfectly inelastic collision.

Apply conservation of linear momentum

Initial momentum:

• First particle: mass $m$, velocity $v$ $p_1 = mv$
• Second particle: mass $2m$, initially at rest $p_2 = 2m \cdot 0 = 0$

So total initial momentum is $p_{\text{initial}} = mv$

Find the total mass after collision

Since they stick together, combined mass is $m + 2m = 3m$

Let the common velocity after collision be $V$.

Then final momentum is $p_{\text{final}} = 3mV$

Equate initial and final momentum

$mv = 3mV$

Cancel $m$ from both sides: $v = 3V$

Therefore, $V = \frac{v}{3}$

Match with the options

The correct option is: $\boxed{\frac{v}{3}}$

So, Option B is correct.