Velocity of a particle of mass $2 \mathrm{kg}$ changes from ${\overrightarrow{v}}_1=(-2\hat{\mathrm{i}}-2\hat{\mathrm{j}}) \mathrm{m} {\mathrm{s}}^{-1}$ to ${\overrightarrow{v}}_2=(\hat{\mathrm{i}}-\hat{\mathrm{j}}) \mathrm{m} {\mathrm{s}}^{-1}$ after colliding with
a plane surface.

A ball of mass $1 \mathrm{kg}$ is dropped from a height of $3.2 \mathrm{m}$ on smooth inclined plane. The coefficient of restitution for the collision is $e=1/2.$ The ball's velocity become horizontal after the collision.

A pendulum bob of mass $m$ connected to the end of an ideal string of length $l$ is released from rest from horizontal position as shown in figure. At the lowest point, the bob makes an elastic collision with a stationary block of mass $5 \mathrm{m},$ which is kept on a frictionless surface. Mark out the correct statement(s) for the instant just after the impact.

A string of length $3l$ is connected to a fixed cylinder whose top view is shown in the figure. The string is initially slack. The other end of the string (connected to a marble) is moving at a constant velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ as shown. The string will get stretched at some instant and impulsive tension occurs in the string. If hinge is exerting a force of $40000 \mathrm{N}$ for $0.25 \mathrm{ms}$ on the cylinder to bear up the impact of impulsive tension, then mark the correct statements. (Take string to be light, breaking tension of the string is $2\times {10}^5 \mathrm{N}.$)

In the figure, the block $B$ of mass $m$ starts from rest at the top of a wedge $W$ of mass $M.$ All surfaces are without friction. $W$ can slide on the ground. $B$ slides down onto the ground, moves along it with a speed $v,$ has an elastic collision with the wall, and climbs back on to $W.$

Two blocks $A$ and $B$ of masses $m$ and $2m$ respectively placed on a smooth floor are connected by a spring. A third body $C$ of mass $m$ moves with velocity $v_0$ along the line joining $A$ and $B$ and collides elastically with $A.$ At a certain instant of time after collision it is found that the instantaneous velocities of $A$ and $B$ are same then

The wall is at extreme right is fixed as shown in figure. Coefficient of restitution for collision between two balls is $1/2$ and between ball and wall is $1.$ Then speed of A and B after all possible collisions are