Two identical particles move towards each other with velocity 2v and v, respectively. The velocity o

Q: Why is one velocity negative?

Because they are moving towards each other, meaning their velocity vectors point in opposite directions. If one is positive, the other must be negative.

Q: Does the mass matter if they are identical?

It cancels out in the end, but it's crucial for the weighting in the formula. You must include it to properly set up the equation.

Q: What if the particles had different masses?

You would just plug the different mass values into the same formula. The COM velocity would shift more towards the heavier or faster particle's momentum.

Q: Why isn't the answer just the average of the speeds $(2v+v)/2 = 1.5v$?

Because velocity is a vector! You must average their *velocities*, taking direction into account, not just their scalar *speeds*.

Q: What does the positive sign in the final answer mean?

It means the center of mass is moving in the same direction as the particle that initially had the velocity $2v$.

Question

Two identical particles move towards each other with velocity $2v$ and $v$, respectively. The velocity of centre of mass is

Options:

(A) $v$

(B) $\frac{v}{3}$

(C) $\frac{v}{2}$

(D) zero

Sigma Percentile · Accepted Answer

### The Vector Nature of Velocity

When dealing with the motion of multiple particles, the concept of the **Center of Mass (COM)** is an incredibly powerful tool. It allows us to simplify a complex system into a single point that represents the entire system's translational motion.

In this problem, we have two identical particles moving towards each other. The most common trap here is to simply average their speeds: $\frac{2v + v}{2} = 1.5v$. But remember, **velocity is a vector!** Direction matters immensely.

### Setting Up the Coordinate System

Let's define a coordinate system to keep track of directions. We can choose the direction of the first particle (moving with speed $2v$) as the positive $x$-direction. 
- Velocity of the first particle: $\vec{v}_1 = 2v \hat{i}$
- Velocity of the second particle: $\vec{v}_2 = -v \hat{i}$ (since it's moving *towards* the first particle, it must be moving in the opposite direction).

Since the particles are identical, their masses are equal: $m_1 = m_2 = m$.

### The Master Equation

The velocity of the center of mass for a system of particles is the mass-weighted average of their individual velocities:

$$ \vec{v}_{CM} = \frac{m_1\vec{v}_1 + m_2\vec{v}_2}{m_1 + m_2} $$

This equation is the heart of the problem. It tells us that the COM's velocity is determined by the total momentum of the system divided by the total mass.

### Final Calculation

Now, we simply substitute our values into the master equation:

$$ \vec{v}_{CM} = \frac{m(2v \hat{i}) + m(-v \hat{i})}{m + m} $$

Notice how the minus sign is crucial here. It represents the opposing momentum of the second particle. Let's simplify the numerator:

$$ \vec{v}_{CM} = \frac{2mv \hat{i} - mv \hat{i}}{2m} $$

$$ \vec{v}_{CM} = \frac{mv \hat{i}}{2m} $$

The mass $m$ cancels out perfectly:

$$ \vec{v}_{CM} = \frac{v}{2} \hat{i} $$

The positive sign in our final answer indicates that the center of mass moves in the $+x$ direction, which is the direction of the faster particle. This makes intuitive sense: the particle with more momentum "drags" the center of mass along with it!

Animated Solution for Physics - System of Particles: Two identical particles move towards each other with velocity $2 v$ and $v$ , respectively. The velocity of centre of mass is

Select Answer:

Visualized Solution

$System Setup$

$Velocity of Centre of Mass$

$Substituting Values$

$Simplifying the Expression$

$Final Velocity$

$Conclusion$

The Sigma Insight: Motion of Centre of Mass

The Vector Nature of Velocity

Setting Up the Coordinate System

The Master Equation

Final Calculation

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Animated Solution for Physics - System of Particles: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two identical particles move towards each other with velocity 2v and v, respectively. The velocity of centre of mass is