$A$ ball of mass $m$ is released from the top of a smooth movable wedge of mass $m$. When the ball collides with the floor,the velocity of the wedge is $v$. Then the maximum height attained by the ball after an elastic collision with the floor is: (Neglect any edge at the lower end of the wedge).

$A$ spring-block system is placed on a rough horizontal floor. The block is pulled towards the right to give the spring some elongation and then released.

Two blocks $1$ and $2$ of equal mass $m$ are connected by an ideal string over a frictionless pulley. The blocks are attached to the ground by springs having spring constants $k_1$ and $k_2$ such that $k_1 > k_2$. Initially,both springs are unstretched. Block $1$ is slowly pulled down a distance $x$ and released. Just after the release,the possible values of the magnitudes of the accelerations of the blocks $a_1$ and $a_2$ can be

$A$ block of mass $15 \;kg$ is placed on a long trolley. The coefficient of static friction between the block and the trolley is $0.18$. The trolley accelerates from rest with $0.5 \;m s^{-2}$ for $20 \;s$ and then moves with uniform velocity. Discuss the motion of the block as viewed by
$(a)$ a stationary observer on the ground,
$(b)$ an observer moving with the trolley.

Two pulley arrangements shown in the figure are identical. The mass of the rope is negligible. In figure $(a)$,the mass $m$ is lifted by attaching a mass $2m$ to the other end of the rope. In figure $(b)$,$m$ is lifted up by pulling the other end of the rope with a constant downward force $F = 2mg$. The accelerations of $m$ in the two cases are respectively:

Two beads connected by a massless inextensible string are placed over a fixed ring as shown in the figure. The mass of each bead is $m$,and there is no friction between bead $B$ and the ring. Find the minimum value of the coefficient of friction between bead $A$ and the ring so that the system remains in equilibrium. ($C$ is the center of the ring,and the line $AC$ is vertical.)