Question

Two particles A and B initially at rest move towards each other under a mutual force of attraction. At the instant when velocity of $A$ is $v$ and that of $B$ is $2v$, the velocity of centre of mass of the system
$\begin{aligned} & \left( A \right)2v \\\ & \left( B \right)zero \\\ & \left( C \right)1.5v \\\ & \left( D \right)v \\\ \end{aligned}$

Answer 1

It is given that there is no external force hence no change in momentum of center of mass and it will remain zero. By the law of conservation of momentum, the momentum is zero. The initial velocity of the particles were $0$ because both $A$ and $B$ were at rest.

Formula used:
$F=ma$
Where:
$m$= mass of particle
$a$= acceleration

Complete step-by-step answer:
As given in problem that both the particles were at rest so at first, we calculate initial force on them:
Force ${{F}_{A}}$​ on particle $A$ is given by
${{F}_{A}}={{m}_{A}}aA=\dfrac{{{m}_{A}}v}{t}...\left( 1 \right)$
Similarly ${{F}_{B}}={{m}_{B}}{{a}_{B}}=\dfrac{{{m}_{B}}\left( 2v \right)}{t}...\left( 2 \right)$
Now:
$\dfrac{{{m}_{A}}v}{t}=\dfrac{{{m}_{B}}\left( 2v \right)}{t}(\because FA=FB)$
So,
$~{{m}_{A}}=2{{m}_{B}}$
For the centre of mass of the system
$v=\dfrac{{{m}_{A}}{{v}_{A}}+{{m}_{B}}{{v}_{B}}}{{{m}_{A}}+{{m}_{B}}}$
or
$~v=\dfrac{2{{m}_{B}}v-{{m}_{B}}\times 2v}{2{{m}_{B}}+{{m}_{B}}}=0$
Here we use negative signs because the particle is travelling in the opposite direction.

So, the correct answer is “Option B”.

Additional Information:
Law of conservation of momentum states that two or more bodies in an isolated system acting upon one another, their total momentum remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed. Some examples of law of conservation of momentum: Air-filled balloons, System of gun and bullet, Motion of rockets.
The law of conservation of momentum is a crucial consequence of Newton’s third law of motion.

Note: Alternatively, if we consider the 2 masses during a system then no external force is acting on the system. Mutual forces are internal forces. Since the centre of mass is initially at rest, it will be at rest until no external force applied on it.