$A$ particle of mass $m$ is thrown horizontally from the top of a tower and another particle of mass $2m$ is thrown vertically upward. The acceleration of the centre of mass is .............

Three blocks $A$,$B$ and $C$ are arranged as shown in the figure such that the distance between two successive blocks is $10 \ m$. Block $A$ has a mass of $10 \ kg$,block $B$ has a mass of $25 \ kg$,and block $C$ has a mass of $15 \ kg$. Block $A$ is displaced towards block $B$ by $2 \ m$ and block $C$ is displaced towards block $B$ by $3 \ m$. The distance through which the block $B$ should be moved so that the centre of mass of the system does not change is

Given below are two statements: one is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
Assertion $(A)$: When a firecracker (rocket) explodes in mid-air,its fragments fly in such a way that the centre of mass of the fragments continues to move along the same parabolic path that the firecracker would have followed had it not exploded.
Reason $(R)$: The explosion of a firecracker (rocket) occurs due to internal forces only,and no external force acts to cause this explosion.

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

$A$ system consists of two particles of masses $m_1$ and $m_2$ separated by a distance $r$. If the particles move towards each other with velocities $v_1$ and $v_2$ respectively,the velocity of the centre of mass of the system is:

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 . . . . . . $\text{cm}$.