In free space,a shell moving with velocity $60\,m/s$ along the positive $x$-axis passes through the origin and explodes into two pieces with a mass ratio of $1:2$. After the explosion,the velocity of the centre of mass is ....... $m/s$.

Two blocks of masses $10\,kg$ and $4\,kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of $14\,m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is ........ $m/s$.

On a smooth horizontal plane,a uniform string of mass $M$ and length $L$ is lying in a state of rest. $A$ man of the same mass $M$ is standing next to one end of the string. Now,the man starts collecting the string. Finally,the man collects all the string and puts it in his pocket. What is the displacement of the man with respect to the Earth in the process of collection?

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 $v_2$.

Two bodies of masses $2\,kg$ and $4\,kg$ are moving with velocities $2\,m/s$ and $10\,m/s$ respectively towards each other due to mutual gravitational attraction. What is the velocity of their centre of mass,given that the bodies were at rest initially?

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