For a collision between two particles,which of the following quantities is generally conserved?

$A$ body of mass $m_1$ moving with an unknown velocity $v_1 \hat{i}$ undergoes a one-dimensional collision with a body of mass $m_2$ moving with velocity $v_2 \hat{i}$. After the collision,the bodies $m_1$ and $m_2$ move with velocities $v_3 \hat{i}$ and $v_4 \hat{i}$ respectively. If $m_2 = 0.5\, m_1$ and $v_3 = 0.5\, v_1$,find $v_1$.

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.

$A$ ball of mass $10\, kg$ moving with a velocity $10 \sqrt{3} \, m/s$ along the $x$-axis,hits another ball of mass $20\, kg$ which is at rest. After the collision,the first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along the $y$-axis with a speed of $10 \, m/s$. The second piece starts moving at an angle of $30^{\circ}$ with respect to the $x$-axis. The velocity of the ball moving at $30^{\circ}$ with the $x$-axis is $x \, m/s$. The configuration of pieces after the collision is shown in the figure. The value of $x$ to the nearest integer is:

Three objects $A, B$,and $C$ are placed on a frictionless horizontal surface. Their masses are $m, 2m$,and $m$ respectively. Object $A$ moves towards $B$ with a speed of $9 \ m/s$ and undergoes an elastic collision with it. Subsequently,$B$ undergoes a perfectly inelastic collision with $C$. All motion occurs along the same straight line. What will be the final speed of object $C$ in $m/s$?

In the case of circular motion of a body,if a tangential force also acts on the body in addition to a centripetal force,then the work done: