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 $m/s$?

Two balls $A$ and $B$ having masses $1\, kg$ and $2\, kg$,moving with speeds $21\, m/s$ and $4\, m/s$ respectively in opposite directions,collide head-on. After the collision,$A$ moves with a speed of $1\, m/s$ in the same direction. Which of the following statements is correct?

In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-$II$, five physical quantities of the particle are mentioned: $\overrightarrow{p}$ is the linear momentum, $\overrightarrow{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.

List-$I$	List-$II$
$P$. $\vec{r}(t) = \alpha t \hat{i} + \beta t \hat{j}$	$1$. $\overrightarrow{p}$
$Q$. $\vec{r}(t) = \alpha \cos \omega t \hat{i} + \beta \sin \omega t \hat{j}$	$2$. $\overrightarrow{L}$
$R$. $\vec{r}(t) = \alpha(\cos \omega t \hat{i} + \sin \omega t \hat{j})$	$3$. $K$
$S$. $\vec{r}(t) = \alpha t \hat{i} + \frac{\beta}{2} t^2 \hat{j}$	$4$. $U$
	$5$. $E$

$A$ block of mass $1.9\, kg$ is at rest at the edge of a table of height $1\, m$. $A$ bullet of mass $0.1\, kg$ collides with the block and sticks to it. If the velocity of the bullet is $20\, m/s$ in the horizontal direction just before the collision,then the kinetic energy just before the combined system strikes the floor is $....J$. (Take $g = 10\, m/s^2$. Assume there is no rotational motion and loss of energy after the collision is negligible.)

$A$ block of mass $m$ slides from rest at a height $H$ on a frictionless inclined plane as shown in the figure. It travels a distance $d$ across a rough horizontal surface with coefficient of kinetic friction $\mu$ and compresses a spring of spring constant $k$ by a distance $x$ before coming to rest momentarily. Then the spring extends and the block travels back attaining a final height of $h$. Then,

Given below are two statements:
Statement $I$: $A$ truck and a car moving with the same kinetic energy are brought to rest by applying brakes which provide equal retarding forces. Both come to rest in equal distance.
Statement $II$: $A$ car moving towards the east takes a turn and moves towards the north, the speed remains unchanged. The acceleration of the car is zero.
In the light of the given statements, choose the most appropriate answer from the options given below.