Animated Solution for Physics - System of Particles: Three particles $A,B$ and $C$ of equal mass move with equal speed $v$ along the medians of an equilateral triangle as shown in figure. They collide at the centroid $G$ of the triangle. After the collision, $A$ comes to rest, $B$ retraces its path with the speed $v$. What is the velocity of $C$?

A 3-Particle Collision

Imagine a perfectly symmetrical setup: three identical particles, $A$, $B$, and $C$, positioned at the vertices of an equilateral triangle. They are all moving along the medians directly towards the centroid $G$, and they all share the exact same speed $v$. This beautiful geometric arrangement is the stage for a fascinating physics problem that tests our understanding of conservation laws.

Before the collision even happens, we need to understand the state of the system. The three particles have equal masses and equal speeds, which means their momentum vectors have the exact same magnitude. Because they are moving along the medians of an equilateral triangle, these three momentum vectors are separated by exactly $120^{\circ }$.

What happens when you add three equal vectors separated by $120^{\circ }$? They perfectly cancel each other out! This is a classic result of vector addition. Therefore, the initial total linear momentum of the entire system is exactly zero:

When the particles collide at the centroid, it might be a messy, chaotic event. Energy might be lost as heat or sound. However, there is one crucial detail: there are no external forces acting on this system of three particles.

According to Newton's laws, in the absence of net external forces, the total linear momentum of an isolated system must remain strictly conserved. This means that whatever happens during the collision, the final momentum must equal the initial momentum. Since the initial momentum was zero, the final momentum must also be zero:

The problem gives us specific clues about what happens after the collision. First, we are told that particle $A$ comes to a complete rest. This simplifies our equation immensely, because its final momentum is now zero (${\vec{p}}_A^{\prime }=0$).

Next, we are told that particle $B$ "retraces its path" with the same speed $v$. This means it bounces back exactly along the line it came from, moving away from the centroid. Its final momentum is equal and opposite to its initial momentum.

Now, we must balance the books. Since the total final momentum must be zero, and $A$ is at rest, the momentum of $B$ and the momentum of $C$ must perfectly cancel each other out:

This simple equation tells us everything we need to know. Particle $C$ must have a momentum exactly equal and opposite to particle $B$'s final momentum. Since they have the same mass, particle $C$ must move with speed $v$ in a direction exactly opposite to the final velocity of $B$. It's a brilliant demonstration of how symmetry and conservation laws can effortlessly solve seemingly complex collisions!