$A$ shell fired from a gun at an angle to the horizontal explodes in mid-air. Then the centre of mass of the shell fragments will move

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 blocks $A$ and $B$ are joined together with a compressed spring. When the system is released,the two blocks move with unequal speeds in opposite directions as shown in the figure. Select the correct statement.

Show that the total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of the centre of mass.

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 $v_2$.

$A$ cart of mass $M$ is tied to one end of a massless rope of length $10\, m$. The other end of the rope is held by a man of mass $M$. The entire system is on a smooth horizontal surface. The man is at $x = 0$ and the cart is at $x = 10\, m$. If the man pulls the cart by the rope,the man and the cart will meet at the point: