Two identical blocks $P$ and $Q$ have mass $m$ each. They are attached to two identical springs (of spring constant $k$) initially unstretched. Both the blocks are initially in contact as shown. Now the left spring (attached with block $P$) is compressed by $A/2$ and the right spring (attached with block $Q$) is compressed by $A.$ Both the blocks are then released simultaneously. Then choose the correct option(s).

$N$ beads identical beads are resting on a smooth horizontal wire which is circular at the end with radius $r=0.5 \mathrm{m}$ as shown in the figure. Find the minimum velocity which should be imparted to the first bead such that $n^{\mathrm{th}}$ bead will fall in the tank after completing full circle in vertical plane as shown in the figure. Take all the collisions between the beads elastic $(e=1).$

An elevator platform is going up at a speed $20{ \mathrm{ms}}^{-1}$ and during its upward motion a small ball of $50 \mathrm{g}$ mass falling in downward direction strikes the platform elastically at a speed $5{ \mathrm{ms}}^{-1}.$ Find the speed (in ${\mathrm{ms}}^{-1}$) with which the ball rebounds.

A ball of mass $1 \mathrm{kg}$ moving with a velocity of $5 \mathrm{m}/\mathrm{s}$ collides elastically with rough ground at an angle $\theta$ with the vertical as shown in figure. What can be the minimum coefficient of friction if ball rebounds vertically after collision? (given $\tan \theta =2$)

Three balls $A, B$ and $C\left(m_A=m_C=4m_B\right)$ are placed on a smooth horizontal surface. Ball $B$ collides with ball $C$ with an initial velocity $v$ as shown in figure. Find the total number of collisions between the balls (all collisions are elastic).

A particle with a mass of $1 \mathrm{kg}$ is having a velocity of $10 \mathrm{m}/\mathrm{s}$ in +ve x -direction at $t=0.$ Forces ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ act on the particle whose magnitudes are changing with time according to the variation shown in figure. The magnitude of the velocity of the particle at $t=3\mathrm{s}$ (neglect gravity effect) is found to be $n\sqrt{5} \mathrm{m}/\mathrm{s}.$. Find the value of $n.$