Question

Two identical particles move towards each other with velocities $2V$ and $V$ respectively. The velocity of the center of mass of this system is:

• A. $V$
• B. $\frac{V}{3}$
• C. $\frac{V}{2}$
• D. $\frac{V}{4}$

Hint:

Always pay attention to directional phrases like "towards each other" or "opposite directions." Neglecting the negative sign for the opposing velocity vector would lead to an incorrect calculation of $\frac{2V + V}{2} = \frac{3V}{2}$.

Answer 1

The Correct Option is C

Step 1: Understand the system.
Two identical particles move toward each other. One has velocity $2V$ and the other has velocity $V$. Since they are identical, they have the same mass $m$.

Step 2: Write the centre of mass velocity formula.
The velocity of the centre of mass is the mass weighted average of the velocities: \[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

Step 3: Choose a direction sign.
Velocity has direction. Let motion to the right be positive. The first particle moves right with $+2V$. The second moves toward it, so it moves left with $-V$.

Step 4: Put in the masses.
Both masses are $m$: \[ V_{cm} = \frac{m(2V) + m(-V)}{m + m} \]

Step 5: Simplify the top and bottom.
The top becomes $m(2V - V) = mV$. The bottom becomes $2m$: \[ V_{cm} = \frac{mV}{2m} \]

Step 6: Cancel and get the answer.
Cancel $m$: \[ V_{cm} = \frac{V}{2} \] The centre of mass moves at half of $V$, in the direction of the faster particle. \[ \boxed{\frac{V}{2}} \]