Two particles which are initially at rest,move towards each other under the action of their internal attraction. If their speeds are $v$ and $2v$ at any instant,then the speed of the centre of mass of the system will be

Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the velocity of $A$ is $v$ and that of $B$ is $2v$,the velocity of the centre of mass of the system is:

Two identical bodies $A$ and $B$ of equal masses have initial velocities $v_1 = 4\hat{i} \text{ m/s}$ and $v_2 = 4\hat{j} \text{ m/s}$ respectively. The body $A$ has acceleration $a_1 = 6\hat{i} + 6\hat{j} \text{ m/s}^2$ while the acceleration of the other body $B$ is zero. The centre of mass of the two bodies moves in . . . . . . path.

Two persons of masses $55\, kg$ and $65\, kg$ respectively,are at the opposite ends of a boat. The length of the boat is $3.0\, m$ and it weighs $100\, kg$. The $55\, kg$ man walks up to the $65\, kg$ man and sits with him. If the boat is in still water,the center of mass of the system shifts by ....... $m$.

Two blocks of masses $5 \, kg$ and $2 \, kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse provides a velocity of $7 \, m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is ............ $m/s$.

Two particles of masses $2 \ kg$ and $1 \ kg$ are moving along the same line with speeds of $2 \ m/s$ and $5 \ m/s$ respectively. If both particles are moving in the same direction,the speed of the centre of mass is $v_1$. If both particles are moving in opposite directions,the speed of the centre of mass is $v_2$. Find $v_1$ and $v_2$.