$A$ 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

Two wooden blocks of mass $M_1$ and $M_2$ rest on a frictionless table. $A$ bullet of mass $m$ is fired at $M_1$ with speed $v$,which gets embedded in it,and the two together finally collide with $M_2$. Find the velocity of $M_2$ after the collision. (Assume the collision between the $(M_1+m)$ system and $M_2$ is elastic and treat the problem as one-dimensional.)

$A$ bullet of mass $10 \,g$ is fired horizontally with a velocity $1000 \,ms^{-1}$ from a rifle situated at a height $50 \,m$ above the ground. If the bullet reaches the ground with a velocity $500 \,ms^{-1}$, the work done against air resistance in the trajectory of the bullet is : $(g=10 \,ms^{-2})$ (in $\,J$)

$A$ man is slipping on a frictionless inclined plane and a bag falls down from the same height. Then the velocity of both is related as

$A$ lift of mass $2000 \ kg$ starts from rest in the basement and moves to the fourth floor,which is at a height of $25 \ m$. As it passes the fourth floor,its speed is $3 \ ms^{-1}$. There is a constant frictional force of $500 \ N$ acting on it. Calculate the work done by the lift's motor in $kJ$.

Two balls,having linear momenta $\vec{p}_1 = p \hat{i}$ and $\vec{p}_2 = -p \hat{i}$,undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1^{\prime}$ and $\vec{p}_2^{\prime}$ be their final momenta. Which of the following option$(s)$ is (are) $NOT ALLOWED$ for any non-zero value of $p, a_1, a_2, b_1, b_2, c_1$ and $c_2$?
$(A)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j}$
$(B)$ $\vec{p}_1^{\prime} = c_1 \hat{k}$,$\vec{p}_2^{\prime} = c_2 \hat{k}$
$(C)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j} - c_1 \hat{k}$
$(D)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_1 \hat{j}$