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

Two particles $A$ and $B$,initially at rest,move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$,the speed of the center of mass of the system 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.

$30 \,kg$ 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 $\,m$)?

The equation $M\vec{A} = \vec{F}_{ext}$ denotes what?

$A$ body of mass $20 \ kg$ is moving with a velocity of $2v$ and another body of mass $10 \ kg$ is moving with velocity $v$ along the same direction. The velocity of their centre of mass is