Motion of Centre of Mass

Given below are two statements:

Statement I: For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.

Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t. the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference. In the light of the above statements, choose the correct answer from the options given below:

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

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is _______ m. (Take $g = 10\text{ m/s}^2$)

In projectile motion two particles of masses $m_1$ and $m_2$ have velocities $\vec{V}_1$, and $\vec{V}_2$ respectively at time $t = 0$. Their velocities become$\overrightarrow{V_1'}$and $\overrightarrow{V_2'}$ at time 2 t while still moving in air. The value of $\left[\left(m_1\overrightarrow{V_1'} + m_2\overrightarrow{V_2'}\right) - \left(m_1\vec{V}_1 + m_2\vec{V}_2\right)\right]$ is ( $g =$ acceleration due to gravity)

Two particles A and B initially at rest, move towards each other under mutual force of attraction. At an instance when the speed of A is v and speed of B is 3v, the speed of centre of mass is:

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): When a fire cracker (rocket) explodes in mid air, its fragments fly in such a way that they continue moving in the same path, which the fire cracker would have followed, had it not exploded.

Reason (R): Explosion of cracker (rocket) occurs due to internal force only and no external force acts for this explosion.

In the light of the above statements, choose the most appropriate answer from the options given below

Four particles A, B, C and D with masses mA = m, mB = 2m, mC = 3m and mD = 4m are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

Two spherical bodies of mass M and 5M and radii R and 2R are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is:

A boy of mass 20 kg is standing on a 80 kg free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing 25 m from a wall. If he walks 10 m on the cart towards the wall, then the final distance of the boy from the wall will be

A projectile of mass M is fired so that the horizontal range is 4 km. At the highest point the projectile explodes in two parts of masses M/4 and 3M/4 respectively and the heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from point of firing) of the lighter part is:

Two persons of masses 55 kg and 65 kg respectively, are at the opposite ends of a boat. The length of the boat is 3.0 m and weights 100 kg. The 55 kg man walks up to the 65 kg man and sits with him. If the boat is in still water the centre of mass of the system shifts by:

Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be:

A body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two-parts; a body $B$ of mass $\frac{1}{3}M$ and a body $C$ of mass $\frac{2}{3}M.$ The center of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

Two identical particles move towards each other with velocity $2v$ and $v$ respectively. The velocity of center of mass is