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

Concept: The velocity of the center of mass ($\vec{V}_{\text{cm}}$) for a discrete multi-particle system is defined as the mass-weighted average of the individual velocity vectors: $$\vec{V}_{\text{cm}} = \frac{m_1\vec{v}_1 + m_2\vec{v}_2 + \cdots + m_n\vec{v}_n}{m_1 + m_2 + \cdots + m_n}$$ Velocity is a vector quantity, so we must assign signs to represent the directions of motion.

Step 1: The problem mentions that the two particles are "identical," which means they have equal masses: $$m_1 = m_2 = m$$ They are moving "towards each other." Let us choose a coordinate axis where the first particle moves along the positive x-direction and the second particle moves along the negative x-direction: $$\vec{v}_1 = +2V \cdot \hat{i}$$ $$\vec{v}_2 = -V \cdot \hat{i}$$

Step 2: Substitute these values into the center of mass velocity vector equation: $$\vec{V}_{\text{cm}} = \frac{m(+2V \cdot \hat{i}) + m(-V \cdot \hat{i})}{m + m}$$ $$\vec{V}_{\text{cm}} = \frac{m(2V - V)\hat{i}}{2m}$$ $$\vec{V}_{\text{cm}} = \frac{mV \cdot \hat{i}}{2m} = \frac{V}{2} \cdot \hat{i}$$

Step 3: Taking the magnitude of this vector gives the speed of the center of mass: $$|\vec{V}_{\text{cm}}| = \frac{V}{2}$$ This matches Option (C).