Two bodies of masses $2 \ kg$ and $4 \ kg$ are moving with velocities $20 \ m/s$ and $10 \ m/s$ towards each other due to mutual gravitational attraction. What is the velocity of their centre of mass (in $m/s$)?

$A$ wooden plank of mass $90 \ kg$ and length $3.3 \ m$ is floating on still water. $A$ girl of mass $20 \ kg$ walks from one end to the other end of the plank. The distance through which the plank moves is (in $cm$)

$A$ projectile of mass $3m$ explodes at its highest point into three equal fragments. One fragment returns to the point of projection. The second fragment comes to rest. The distance of the third fragment from the point of projection when it lands on the ground is ......... $m$. (The range of the projectile without explosion was $100 \ m$)

$A$ 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:

There are some passengers inside a stationary railway compartment. The track is frictionless. The centre of mass of the compartment itself (without the passengers) is $C_1$,while the centre of mass of the 'compartment plus passengers' system is $C_2$. If the passengers move about inside the compartment along the track,then:

$A$ boat of mass $90\ kg$ is floating in still water. $A$ boy of mass $30\ kg$ walks from the stern to the bow. The length of the boat is $3\ m$. The distance through which the boat moves is .......... $m$.