$30 \,kg$ boy stands at the far edge of a floating plank, whose near edge is against the shore of a river. The plank is $10 \,m$ long and weighs $10 \,kg$. If the boy walks to the near edge of the plank, how far from the shore does the plank move (in $\,m$)?

Three particles $A, B$ and $C$ of masses $m, 2m$ and $3m$ are moving towards north, south and east respectively. If the velocities of the particles $A, B$ and $C$ are $6 \,ms^{-1}, 12 \,ms^{-1}$ and $8 \,ms^{-1}$ respectively, then the velocity of the centre of mass of the system of particles is (in $\,ms^{-1}$)

$A$ piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time,a bullet of mass $0.02\, kg$ is fired vertically upward,with a velocity $100\, ms^{-1}$,from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

In a system,two particles of masses $m_1 = 3 \text{ kg}$ and $m_2 = 2 \text{ kg}$ are placed at a certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance of $2 \text{ cm}$. In order to keep the center of mass of the system at the original position,the particle of mass $m_2$ should move towards the center of mass by a distance of . . . . . . $\text{cm}$.

Two blocks,each of mass $m$,are connected to each other by a massless string and placed on two inclined planes as shown in the figure. After releasing the system from rest,find the magnitude of the acceleration of the centre of mass of the two blocks. (Take $g = 10 \, m/s^2$)

Two particles of equal masses have velocities $\overrightarrow{v_1} = 2\hat{i} \ m/s$ and $\overrightarrow{v_2} = 2\hat{j} \ m/s$. The first particle has an acceleration $\overrightarrow{a_1} = (3\hat{i} + 3\hat{j}) \ m/s^2$,while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a