A long plank of mass $2 \mathrm{kg}$ is placed on a smooth horizontal surface. Two blocks each of mass $1 \mathrm{kg}$ are placed in contact on its rough surface. The coefficient of friction between first block and plank is $0.2$ and that between second block and plank is $0.4$. The blocks are projected in the opposite direction on the plank with velocity $2 \mathrm{m} {\mathrm{s}}^{-1}$ and $6 \mathrm{m} {\mathrm{s}}^{-1}$ respectively. The velocity of the plank when both the blocks come to relative rest with respect to the plank and total work done by friction on the system (blocks + plank) will be:

A small block of mass $\text{'}m\text{'}$ is placed on bigger block of mass $M\mathit{,}$ which is placed on a frictionless horizontal surface. The two blocks are given equal speed $u,$ but opposite directions, as shown in the figure. After some time, it is observed that both the blocks are moving in the direction of motion of the lower block, with a
speed greater than $\frac{u}{2}.$ It can be concluded that -

A particle of constant mass $m$ moves in straight line under a time varying force. The change in momentum of the particle in a time interval $t_1$ to $t_2$ can be given as
$\mathrm{\Delta }p={\int }_{t_1}^{t_2}F\mathrm{d}t$.
The force $F$ acting on the particle as a function of time $t$ is shown in the graph. If velocity of the particle is zero at $t=0$, then

Two blocks initially at rest having masses $m_1$ and $m_2$ are connected by spring of spring constant $k$ (as shown in the figure). The block of mass $m_1$ is pulled by a constant force $F_1$ and the other block is pulled by a constant force $F_2\mathit{.}$ Find the maximum elongation of the spring.

A hemisphere of mass $3 \mathrm{m}$ and radius $R$ is free to slide with its base on a smooth horizontal table. A particle of mass $m$ is placed on the top of the hemisphere. If the particle is displaced with a negligible velocity, then find the angular velocity of the particle relative to the centre of the hemisphere at an angular displacement $\mathrm{\theta },$ when the velocity of the hemisphere is $v$.

Two blocks of masses $m_1$ and $m_2$ are connected by spring of constant $k.$ The spring is initially compressed and the system is released from rest at $t=0 \mathrm{s}$. The work done by spring on the blocks $m_1$ and $m_2$ be $W_1$ and $W_2$ respectively by time $t.$ The speeds of both the blocks at time $t$ are non-zero.
Then the value of $$ equals to

Ball $A$ of mass $m$ after sliding from an inclined plane strikes elastically to another ball $B$ of same mass at rest. Find the minimum height $H$ so that ball $B$ just completes the circular motion.