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 spherical objects have masses $M$ and $5M$ and radii $R$ and $2R$ respectively. They are released in free space with their centers at a distance of $12R$. If they attract each other only due to gravitational force,the distance traveled by the smaller object before collision is: (in $R$)

$A$ small ball $B$ of mass $m$ is suspended with a light inelastic string of length $L$ from a block $A$ of the same mass $m$,which can move on a smooth horizontal surface as shown in the figure. The ball is displaced by an angle $\theta$ from the equilibrium position and then released. The displacement of the centre of mass of the $A+B$ system until the string becomes vertical is

Two masses $m_1$ and $m_2$ $(m_1 > m_2)$ are connected by a massless,flexible,and inextensible string passed over a massless and frictionless pulley. The acceleration of the center of mass of the system is

$A$ system consists of two particles of masses $m_1$ and $m_2$. If the particle of mass $m_1$ is moved towards the centre of mass through a distance $d$,then the distance the second particle should be moved,so as to keep the centre of mass at the same position is

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?