Two particles $A$ and $B$,initially at rest,move towards each other under a mutual force of attraction. At an instance when the speed of $A$ is $v$ and the speed of $B$ is $2v$,the speed of the centre of mass $(CM)$ is:

Two identical bodies $A$ and $B$ of equal masses have initial velocities $v_1 = 4\hat{i} \text{ m/s}$ and $v_2 = 4\hat{j} \text{ m/s}$ respectively. The body $A$ has acceleration $a_1 = 6\hat{i} + 6\hat{j} \text{ m/s}^2$ while the acceleration of the other body $B$ is zero. The centre of mass of the two bodies moves in . . . . . . path.

$A$ particle of mass $3m$ is projected from the ground at some angle with the horizontal. The horizontal range is $R$. At the highest point of its path,it breaks into two pieces of mass $m$ and $2m$. The smaller mass comes to rest,and the larger mass finally falls at a distance $x$ from the point of projection. The value of $x$ 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 $L$. When the man reaches the end $B$,the distance moved by the trolley with respect to ground is

$A$ cart of mass $M$ is tied to one end of a massless rope of length $10\, m$. The other end of the rope is held by a man of mass $M$. The entire system is on a smooth horizontal surface. The man is at $x = 0$ and the cart is at $x = 10\, m$. If the man pulls the cart by the rope,the man and the cart will meet at the point:

$A$ man inside a freely falling box throws a heavy ball towards a side wall. The ball keeps on bouncing between the opposite walls of the box. We neglect air resistance and friction. Which of the following figures depicts the motion of the centre of mass of the entire system (man,the ball and the box)?