$A$ uniform rod of length $L$ and mass $M$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ moving with velocity $v$ strikes the rod at one end perpendicularly and comes to rest. The velocity of the centre of mass of the rod after the collision is:

Two men $A$ and $B$ are standing on a plank of length $120 \ cm$ and mass $40 \ kg$. $B$ (mass $60 \ kg$) is at the middle of the plank and $A$ (mass $40 \ kg$) is at the left end of the plank. The system is initially at rest on a smooth horizontal surface. $A$ and $B$ start moving such that the position of $B$ remains fixed with respect to the ground until $A$ meets $B$. The point where $A$ meets $B$ is located at:

Separation of motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
$(a)$ Show $p = \sum p_{i}^{\prime} + M V$,where $p$ is the total momentum of the system,$p_{i}^{\prime} = m_{i} v_{i}^{\prime}$,and $v_{i}^{\prime}$ is the velocity of the $i^{th}$ particle relative to the centre of mass. Also,prove using the definition of the centre of mass that $\sum p_{i}^{\prime} = 0$.
$(b)$ Show $K = K^{\prime} + \frac{1}{2} M V^{2}$,where $K$ is the total kinetic energy of the system,$K^{\prime}$ is the total kinetic energy of the system relative to the centre of mass,and $\frac{1}{2} M V^{2}$ is the kinetic energy of the translation of the system as a whole.
$(c)$ Show $L = L^{\prime} + R \times M V$,where $L^{\prime} = \sum r_{i}^{\prime} \times p_{i}^{\prime}$ is the angular momentum of the system about the centre of mass. Note $r_{i}^{\prime} = r_{i} - R$.
$(d)$ Show $\frac{d L^{\prime}}{d t} = \sum r_{i}^{\prime} \times \frac{d p_{i}^{\prime}}{d t}$. Further,show that $\frac{d L^{\prime}}{d t} = \tau_{ext}^{\prime}$,where $\tau_{ext}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass.

$A$ system of $N$ particles is free from any external forces. Which of the following must be true for the sum of the magnitudes of the momenta of the individual particles in the system?

Consider the following two statements:
$1.$ Linear momentum of a system of particles is zero.
$2.$ Kinetic energy of a system of particles is zero.
Then:

Consider the following two statements for a system of particles:
$A$: The linear momentum of the system is zero.
$B$: The kinetic energy of the system is zero.