A body of mass $2 \mathrm{kg}$ moving with a velocity $3 \mathrm{m} {\mathrm{s}}^{-1}$ collides with a body of mass $1 \mathrm{kg}$ moving with a velocity of $4 \mathrm{m} {\mathrm{s}}^{-1}$ in opposite direction, if the collision is head on and completely inelastic, then

Two masses $A$ and $B$ of $3 \mathrm{kg}$ and $2 \mathrm{kg}$ are connected by a long inextensible string which passes over a massless and frictionless pulley. Initially the height of both the masses from the ground is same and equal to $1$ metre. When the masses are released, mass $A$ hits the ground and gets stuck to the ground. Consider the length of the string as large, so that the pulley does not obstruct the motion of masses $A$ and $B. \left[g=10 \mathrm{m}/{\mathrm{s}}^2\right]$

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 $\tan \theta =2$)