Water falls from a $40\,m$ high dam at the rate of $9 \times 10^{4}\,kg$ per hour. Fifty percent of the gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy, the number of $100\,W$ lamps that can be lit is (Take $g = 10\,m/s^2$)

$A$ ball of mass $m$ falls vertically from a height $h$ and collides with a block of equal mass $m$ moving horizontally with a velocity $v$ on a surface. The coefficient of kinetic friction between the block and the surface is $0.2$,while the coefficient of restitution $e$ between the ball and the block is $0.5$. There is no friction acting between the ball and the block. The velocity of the block decreases by:

$A$ bead of mass $m$ slides without friction on the wall of a vertical circular hoop of radius $R$ as shown in the figure. The bead moves under the combined action of gravity and a massless spring of constant $k$ attached to the bottom of the hoop. The natural (equilibrium) length of the spring is $R$. If the bead is released from the top of the hoop with negligible initial speed,what is the velocity of the bead when the length of the spring becomes $R$? ($g$ is the acceleration due to gravity)

$A$ toy of mass $20 \,g$ at rest acquires a velocity $(3 \hat{i}-2 \hat{j}) \,m/s$ in $2 \,s$. The power of the toy is: (in $\,W$)

$A$ pendulum of length $1 \,m$ and having a bob of mass $1 \,g$ is pulled aside through an angle $60^{\circ}$ with the vertical and then released. The power delivered by all the forces acting on the bob when the pendulum makes $30^{\circ}$ with the vertical is . . . . . . $\left(g=10 \,ms^{-2}\right)$ (in $\,mW$)

$A$ body $A$ moving with momentum $P$ collides one-dimensionally with another stationary body $B$ of same mass. During impact,$A$ gives impulse $J$ to $B$. Then which of the following is/are correct?
$(a)$ The total momentum of $A$ and $B$ is $P$ before and after impact and $(P-J)$ during the impact.
$(b)$ During the impact,$B$ gives impulse of magnitude $J$ to $A$.
$(c)$ The coefficient of restitution is $\left[\frac{2 J}{P}-1\right]$.
$(d)$ The coefficient of restitution is $\left[\frac{2 J}{P}+1\right]$.