$A$ bullet fired into a fixed target loses half of its velocity after penetrating $1\,cm$. How much further will it penetrate before coming to rest,assuming that it faces constant resistance to motion?

$A$ ball of mass $m = 60 \text{ g}$ is shot with speed $v_0 = 22 \text{ m/s}$ into the barrel of a spring gun of mass $M = 240 \text{ g}$ initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. What fraction of the initial kinetic energy of the ball is now stored in the spring?

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

$A$ stationary object of mass $10 \ kg$ is subjected to two mutually perpendicular forces of $4 \ N$ and $3 \ N$. What will be its kinetic energy after $10 \ s$?

$A$ bullet of mass $25 \,g$ moving horizontally at a speed of $250 \,m/s$ is fired into a wooden block of mass $1 \,kg$ suspended by a long string. The bullet crosses the block and emerges on the other side. If the centre of mass of the block rises through a height of $20 \,cm$, find the speed of the bullet as it emerges from the block. (Take $g = 10 \,m/s^2$) (in $\,m/s$)

$A$ shell at rest on a smooth horizontal surface explodes into two fragments of masses $m_1$ and $m_2$. If just after explosion $m_1$ moves with speed $u$,then the work done by internal forces during the explosion is: