$A$ particle of mass $3 \ kg$ is moved slowly along the path $ABCDE$ from $A$ to $E$. The heights of $B$,$C$,and $D$ are $5 \ m$,$4 \ m$,and $6 \ m$ respectively. The total path length is $20 \ m$,the horizontal distance $AE = 10 \ m$,and the coefficient of friction $\mu = 0.6$. Calculate the work done in $J$ to slowly move the mass from $A$ to $E$. (Take $g = 10 \ m/s^2$)

In two separate collisions,the coefficients of restitution $e_1$ and $e_2$ are in the ratio $3:1$. In the first collision,the relative velocity of approach is twice the relative velocity of separation. Find the ratio between the relative velocity of approach and the relative velocity of separation in the second collision.

$A$ particle is projected vertically upwards with a speed of $16 \ m/s$. After some time,when it again passes through the point of projection,its speed is found to be $8 \ m/s$. It is known that the work done by air resistance is the same during upward and downward motion. Then the maximum height attained by the particle is ...................... $m$ $(g = 10 \ m/s^2)$.

Two blocks $A$ and $B$,each of mass $m$,are connected by a massless spring of natural length $L$ and spring constant $K$. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in the figure. $A$ third identical block $C$,also of mass $m$,moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with $A$. Then:

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.)

Consider a one-dimensional collision involving a body of mass $m_1$ moving in the positive $x$ direction with speed $v_0$ and a second body of mass $m_2$ initially at rest. The collision can be completely inelastic,completely elastic,or partially inelastic. After the collision,$m_1$ moves with velocity $v_1$ and $m_2$ moves with velocity $v_2$. Determine the range of final velocity $v_1$ for the following cases:
$(A)$ If $m_1 > m_2$
$(B)$ If $m_1 < m_2$