Two particles of masses $1\ kg$ and $3\ kg$ move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is $2\ m/s,$ their centre of mass has a velocity of $0.5\ m/s$. When the relative velocity of approach becomes $3\ m/s,$ the velocity of the centre of mass is ........ $m/s$.

Two particles of masses $m_1$ and $m_2$ initially at rest start moving towards each other under their mutual force of attraction. The speed of the centre of mass at any time $t$,when they are at a distance $r$ apart,is

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 the center of mass of the system is:

$A$ $2\ kg$ body and a $3\ kg$ body are moving along the $x-$ axis. At a particular instant,the $2\ kg$ body has a velocity of $3\ m/s$ and the $3\ kg$ body has a velocity of $2\ m/s$. The velocity of the centre of mass at that instant is

Two masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and unstretched length $l$. The masses are placed on a frictionless straight channel,which we consider our $X$-axis. They are initially at rest at $x=0$ and $x=l$,respectively. At $t=0$,a velocity of $v_0$ is suddenly imparted to the first particle. At a later time $t$,the centre of mass of the two masses is at

The centre of mass of two masses $m$ and $m'$ moves by a distance $\frac{x}{5}$ when mass $m$ is moved by a distance $x$ and $m'$ is kept fixed. The ratio $\frac{m'}{m}$ is