Question

Two particles $A$ and $B$, initially at rest, move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$ the speed of center of mass is
A. Zero
B. $v$
C. $1.5v$
D. $3v$

Answer 1

In order to solve this question we need to understand the definition of center of mass which states that center of mass is that position where all the mass supposed to be centered means a path followed by a large body is same as trajectory followed by its center of mass. Also speed of center of mass of a set of particles is defined as summation of magnitude of momentum of all particles divided by total mass of the system.

Complete step by step answer:
Let us consider two particles to be the same: mass of particle $A$ is “$m$” and mass of particle $B$ is “$m$”. According to the question, the speed of $A$ is $v$ and the speed of $B$ is $2v$. Now according to formula for speed of center of mass, we have
$V = \dfrac{{{m_1}{v_1} + {m_2}{v_2}}}{{{m_1} + {m_2}}}$
Whereas $V$ is the speed of center of mass of the system.
Putting values we get
$V = \dfrac{{mv + 2mv}}{{m + m}}$
$\Rightarrow V = \dfrac{{3mv}}{{2m}}$
$\therefore V = 1.5v$

Hence, the correct option is C.

Note: It should be remembered that in the formula of center of mass direction of particles not considered because speed had been asked in question but if velocity asked then we must consider direction in which particles are moving. Also further collision between balls could be studied in terms of center of mass; it makes things quite easier then laboratory frame of reference.