DPP 1.2

Both the blocks shown in the given arrangement have a horizontal velocity towards right. If $a_{\mathrm{cm}}$ be the subsequent acceleration of the center of mass of the system of blocks, then $a_{\mathrm{cm}}$ equals,

Two blocks of masses $2 \mathrm{kg}$ and $1 \mathrm{kg}$, respectively, are tied to the ends of a string which passes over a light frictionless pulley. The masses are held at rest at the same horizontal level and then released. If $g=10 \mathrm{m} {\mathrm{s}}^{-2}$, the distance traversed by the combined center of mass of blocks in $2 \mathrm{s}$ is,

A system of two blocks $A$ and $B$ and a wedge $C$ is released from rest as shown in the figure. The masses of the blocks and the wedge are $m, 2m$ and $2m$, respectively. The displacement of wedge $C$ when block $B$ slides down the plane a distance $10 \mathrm{cm}$ is (neglect friction),

Two men $\mathrm{A}$ and $\mathrm{B}$ are standing on a plank. $\mathrm{B}$ is at the middle of the plank and $\mathrm{A}$ is at the left end of the plank. The system is initially at rest and masses are as shown in the figure. $\mathrm{A}$ and $\mathrm{B}$ start moving such that the position of $\mathrm{B}$ remains fixed with respect to ground, then $\mathrm{A}$ meets $\mathrm{B}$. Then, the point where $\mathrm{A}$ meets $\mathrm{B}$ is located at,

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,

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

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

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

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.

Which of the following are correct?

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),