$A$ ball is thrown vertically down from a height of $40 \,m$ from the ground with an initial velocity $v$. The ball hits the ground, loses $\frac{1}{3}$ of its total mechanical energy, and rebounds back to the same height. If the acceleration due to gravity is $10 \,m/s^2$, then the value of $v$ is (in $\,m/s$)

$A$ body $x$ with a momentum $p$ collides with another identical stationary body $y$ one-dimensionally. During the collision,$y$ gives an impulse $J$ to body $x$. Then,the coefficient of restitution is

$A$ block of mass $5 \ kg$ is lifted to a height of $5 \ m$ by a force of $60 \ N$. Find:
$(1)$ Work done by the force.
$(2)$ Potential energy of the block at $5 \ m$.
$(3)$ Kinetic energy of the block at $5 \ m$.
$(4)$ Velocity of the block at $5 \ m$. (Take $g = 9.8 \ m/s^2$)

$A$ $1.0\, kg$ block collides with a horizontal weightless spring of force constant $2.75\, N/m$ as shown in the figure. The block compresses the spring $4.0\, m$ from the rest position. If the coefficient of kinetic friction between the block and the horizontal surface is $0.25$,the speed of the block at the instant of collision is ................. $m/s$.

An athlete throws a shotput of mass $25 \,kg$ with an initial speed of $4 \,ms^{-1}$ at an angle of $45^{\circ}$ with the horizontal from a height of $2 \,m$ above the ground. Assuming air resistance to be negligible,the kinetic energy of the shotput when it just touches the ground is . . . . . . $\left(g=10 \,ms^{-2}\right)$ (in $\,J$)

$A$ particle of mass $m$ moving horizontally with velocity $v_0$ strikes a smooth wedge of mass $M$,as shown in the figure. After the collision,the ball starts moving up the inclined face of the wedge and rises to a height $h$. Choose the correct statement related to the wedge $M$.