Motion of Center of Mass

Two bodies of $2 \mathrm{kg}$ and $4 \mathrm{kg}$ are moving with velocities $20 \mathrm{m}/\mathrm{s}$ and $10 \mathrm{m}/\mathrm{s}$ respectively towards each other under mutual gravitational attraction. Find the velocity of their centre of mass in $\mathrm{m}/\mathrm{s}$.

In free space, a shell moving with velocity $60 \mathrm{m} {\mathrm{s}}^{-1}$ along the positive $x$-axis. When it passes through origin, it explodes into two peices of mass ratio $1:2$. After the explosion, the velocity of the centre of mass is :-

Two bodies of different masses $2 \mathrm{kg}$ and $4 \mathrm{kg}$ are moving with velocities $2 \mathrm{m} {\mathrm{s}}^{-1}$ and $1 \mathrm{m} {\mathrm{s}}^{-1}$ towards each other due to mutual gravitational attraction. Then the velocity of the centre of mass is

Two bodies of masses $10 \mathrm{kg}$ and $2 \mathrm{kg}$ are moving with velocity $\left(-2\hat{\mathrm{i}}+7\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ and $\left(10\hat{\mathrm{i}}-35\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ respectively. The velocity of the centre of mass will be

Blocks$A$ and $B$ are resting on a smooth horizontal surface given equal speeds of $2 \mathrm{m} {\mathrm{s}}^{-1}$ in opposite sense as shown in the figure.

At $t=0$ the position of blocks are shown, then the co-ordinates of center of mass at $t=3\mathrm{s}$ will be :-

When an explosive shell travelling in a parabolic path under the effect of gravity explodes in the mid air, the centre of mass of the fragments will move:-

If particles moves under mutual gravitational attraction force then find the ratio of their acceleration just after releasing from rest:-

A small ball is moving inside a box which is placed on a smooth horizontal surface. During motion, the ball makes collisions with the walls of the box then the position of the centre of mass:

Find the linear momentum of a disc having mass $2 \mathrm{kg}$ and radius $15 \mathrm{cm}$  revolving around the central axis by  $4 \mathrm{rad} {\mathrm{s}}^{-1}$.

Which of the following is possible if a non-zero external force acts on a system of particles while at a particular time $t$ the velocity and acceleration of the center of mass are found to be $v_0$ and $a_0$.

If the external forces acting on a system have zero resultant, the centre of mass

Two blocks A and B of mass m and 2m respectively are connected together by a light spring of stiffness K and natural length $\mathcal{l}$. The system is lying on a smooth horizontal surface with the block A in contact with a fixed vertical wall as shown in the figure. The block B is pressed towards the wall by a distance $x_0$ from the natural position of spring and then released. There is no friction anywhere.

If block B is released at time $t=0$ and A just loses contact with the wall at time $t=\mathrm{\Delta }t$ then the average acceleration of the centre of mass of the system during the time $t=0$ to $t=\mathrm{\Delta }t$ is:

Two particles of equal mass have velocities ${\overrightarrow{v}}_1=2\hat{i} \mathrm{m} {\mathrm{s}}^{-1}$ and ${\overrightarrow{v}}_2=2\hat{j} \mathrm{m} {\mathrm{s}}^{-1}$. First particle has an acceleration ${\overrightarrow{a}}_1=(3\hat{i}+3\hat{j}) \mathrm{m} {\mathrm{s}}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

Two particles $A$ and $B$ (both initially at rest) start moving 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 centre of mass is

A bomb of mass 1 kg is thrown vertically upwards with a speed of $100\frac{m}{s}$ . After 5 seconds it explodes into two fragments. One fragment of mass 400 gm is found to go down with a speed of $25\frac{m}{s}$ . What will happen to the second fragment just after the explosion? $g=10\frac{m}{s^2}$

Two bodies A and B of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ respectively are connected by a massless spring of force constant k. A constant force F starts acting on the body A at $\mathrm{t}=0$ . Then

A solid cylinder of mass m and radius R rolls down an inclined plane of height h without slipping. The speed of its centre of mass when it reaches the bottom is

Two bodies of $6\text{kg}$ and $4\text{kg}$ masses have their velocity (in $\text{m/s}$) $5\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+10\widehat{\mathrm{k}}$ and $10\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+5\widehat{\mathrm{k}}$ respectively. Then the velocity of their centre of mass (in $\text{m/s}$) is

A loaded spring gun of mass $M$ fires a shot of mass $m$ with a velocity $v$ at an angle of elevation $\theta$. The gun was initially at rest on a horizontal frictionless surface. After firing, the centre of mass of gun-shot system:

Two blocks of masses $10 \mathrm{kg}$ and $4 \mathrm{kg}$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulsive force gives a velocity of $14 \mathrm{m} {\mathrm{s}}^{-1}$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass of the system at that very moment is