Animated Solution for Physics - System of Particles: Two particles $A$ and $B$ initially at rest, move towards each other by mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$, the speed of the centre of mass of the system is

A Tale of Mutual Attraction

Imagine you are floating in the vast emptiness of deep space. You and a friend are perfectly still, holding onto opposite ends of a long, invisible bungee cord. At this moment, neither of you is moving. If we were to calculate the velocity of your combined Center of Mass (CM), it would be exactly zero. This is the starting point of our problem: two particles, $A$ and $B$, initially at rest.

Suddenly, the mutual force of attraction kicks in. It could be gravity, or perhaps they hold opposite electrical charges. They begin to accelerate towards each other. The problem states that at a certain instant, particle $A$ is moving with a speed $v$, and particle $B$ is moving with a speed $2v$.

It is incredibly tempting to look at these speeds and try to calculate a new velocity for the center of mass. You might think, "Well, $B$ is moving faster, so the center of mass must be drifting towards $B$." But this is a classic trap designed to test your fundamental understanding of Newtonian mechanics.

To understand why the center of mass behaves the way it does, we must look at the forces causing the motion. Particle $A$ pulls on particle $B$, and particle $B$ pulls on particle $A$. According to Newton's Third Law, these forces are equal in magnitude and opposite in direction:

Because these forces originate from within the system itself, they are classified as internal forces. When we consider the system as a whole (particle $A$ + particle $B$), these internal forces perfectly cancel each other out.

The motion of the center of mass of any system is governed by one supreme rule: Newton's Second Law applied to a system of particles. It states that the net external force acting on a system is equal to the total mass of the system multiplied by the acceleration of its center of mass:

Since there is no outside entity pushing or pulling on our particles, the net external force is exactly zero ($\sum {\vec{F}}_{ext}=0$).

If the net external force is zero, the math leaves us with only one inescapable conclusion:

An acceleration of zero means that the velocity of the center of mass cannot change. It is a conserved quantity. Since the center of mass was initially at rest (${\vec{v}}_{cm,i}=0$), it must remain at rest forever, regardless of the chaotic, high-speed dance the individual particles are performing inside the system.

Therefore, the speed of the center of mass at the instant when the particles are moving at $v$ and $2v$ is still, beautifully and simply, zero.