Two objects of masses $200 \ g$ and $500 \ g$ have velocities $10\hat{i} \ m/s$ and $3\hat{i} + 5\hat{j} \ m/s$ respectively. The velocity of their center of mass (in $m/s$) is:

$A$ body $A$ of mass $M$ falls vertically downwards under gravity. It breaks into two parts: a body $B$ of mass $1/3 M$ and a body $C$ of mass $2/3 M$. The center of mass of bodies $B$ and $C$ taken together shifts from the direction of $A$ by:

Four particles $A, B, C$ and $D$ with masses $m_A=m, m_B=2m, m_C=3m$ and $m_D=4m$ are at the corners of a square. They have accelerations of equal magnitude $a$ with directions as shown. The acceleration of the centre of mass of the particles is

The figure shows a man of mass $m$ standing at the end $A$ of a trolley of mass $M$ placed at rest on a smooth horizontal surface. The man starts moving towards the end $B$ with a velocity $u_{rel}$ with respect to the trolley. The length of the trolley is $L$. The distance moved by the man with respect to ground is

Two equal rods joined at one end are kept on a smooth surface as shown and released. The trajectory of the centre of mass of both rods is:

$A$ block of mass $M$ is tied to one end of a massless rope. The other end of the rope is in the hands of a man of mass $2M$ as shown in the figure. Initially,the block and the man are resting on a rough plank of mass $2M$ as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. The pulley is massless and frictionless. What is the magnitude of the displacement of the plank when the block meets the pulley? (The man does not leave his position on the plank during the pull).