A block of mass $m=10 \mathrm{kg}$ rests on a horizontal table. The coefficient of friction between the block and the table is $0.05$. When hit by a bullet of mass $50 \mathrm{g}$ moving with speed $v$, that gets embedded in it, the block moves and comes to stop after moving a distance of $2 \mathrm{m}$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H$, then neglecting energy losses and taking $g=10 \mathrm{m} {\mathrm{s}}^{-2}$, the value of $H$ is close to

 

A large number $\left(n\right)$ of identical beads, each of mass $m$ and radius $r$ are strung on a thin smooth rigid horizontal rod of length $L(L\gg r)$ and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic):

This question has $Statement\; -\; I$ and $Statement\; -\; II$ of the four choices given after the Statements, choose the one that best describes the two Statements.

$Statement\; -\; I:$ A point particle of mass $m$ moving with speed $\nu$ collides with stationary point particle of mass $M$. If the maximum energy loss possible is given as
$f\left(\frac{1}{2}m{\nu }^2\right)$ then $f=\left(\frac{m}{M+m}\right)$.

$Statement\; -\; II:$ Maximum energy loss occurs when the particles get stuck together as a result of the collision.

A ball of mass $100 \mathrm{g}$ is dropped from a height $h=10 \mathrm{cm}$ on a platform fixed at the top of a vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance $\frac{h}{2}$. The spring constant is _____ $\mathrm{N} {\mathrm{m}}^{-1}$
(Use $g=10{ \mathrm{m} \mathrm{s}}^{-2}$)