In the figure shown, the system is at rest, initially. Two persons, $A$ and $B$, of masses $40 \mathrm{kg}$ each, move with speeds $v_1$ and $v_2$, respectively, towards each other on a plank lying on a smooth horizontal surface as shown in the figure. The plank travels a distance of $20 \mathrm{m}$ towards right direction in $5 \mathrm{s}$. (Here, $v_1$ and $v_2$ are given with respect to the plank). Then,

 

$\left(\mathrm{i}\right)$ $v_1=10 \mathrm{m} {\mathrm{s}}^{-1}, v_2=0 \mathrm{m} {\mathrm{s}}^{-1}$

$\left(\mathrm{ii}\right)$ $v_1=15 \mathrm{m} {\mathrm{s}}^{-1}, v_2=5 \mathrm{m} {\mathrm{s}}^{-1}$

$\left(\mathrm{iii}\right)$ $v_1=20 \mathrm{m} {\mathrm{s}}^{-1}, v_2=10 \mathrm{m} {\mathrm{s}}^{-1}$

A small sphere of radius $R$ is held against the inner surface of a smooth spherical shell of radius $6R$ as shown in the figure. The masses of the shell and the small sphere are $4M$ and $M$, respectively.

This arrangement is placed on a smooth horizontal table. The small sphere is now released. The $x$-coordinate of the center of the shell when the smaller sphere reaches the other extreme position is,

Two particles of equal mass $m$ are projected from the ground with speeds $v_1$ and $v_2$ at angles ${\theta }_1$ as ${\theta }_2$ shown in the figure.

Two identical rods are joined at one of their ends by a pin. The joint is smooth and rods are free to rotate about the joint. The rods are released in vertical plane on a smooth surface as shown in the figure. The displacement of the joint from its initial position to the final position is (i.e., when the rods lie straight on the ground),

A rail wagon of mass $95\mathrm{ }\mathrm{kg}$ is moving with velocity $10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ on frictionless rails. A body of mass $5\mathrm{ }\mathrm{kg}$ is thrown into the wagon with velocity $12.5\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ as shown in the figure. The body collides inside with the walls of the wagon and due to internal losses, it comes to rest with respect to the wagon. Find the velocity of the wagon when the body comes to rest w.r.t. the wagon. [Assume that the velocity of the wagon and the object are in the same plane.]

Two men of masses $80 \mathrm{kg}$ and $60 \mathrm{kg}$ are standing on a wooden plank of mass $100 \mathrm{kg}$ that has been placed over a smooth surface. If both the men start moving towards each other with speeds $1 \mathrm{m} {\mathrm{s}}^{-1}$ and $2 \mathrm{m} {\mathrm{s}}^{-1}$, respectively, the velocity of the plank by which it starts moving is,