$A$ particle is released from a height $h$. The particle is given a constant horizontal velocity. Assuming $g$ remains constant everywhere, which graph correctly represents the kinetic energy $E$ of the particle with respect to time $t$?

$A$ ball is thrown vertically downwards from a height of $20 \, m$ with an initial velocity $v_0$. It collides with the ground,loses $50\%$ of its energy in the collision,and rebounds to the same height. The initial velocity $v_0$ is .................... $m/s$ (Take $g = 10 \, m/s^2$)

$A$ target is made of two plates,one of wood and the other of iron. The thickness of the wooden plate is $4\,cm$ and that of the iron plate is $2\,cm$. $A$ bullet fired goes through the wood first and then penetrates $1\,cm$ into the iron. $A$ similar bullet fired with the same velocity from the opposite direction goes through the iron first and then penetrates $2\,cm$ into the wood. If $a_1$ and $a_2$ are the retardations offered to the bullet by the wood and iron plates respectively,then:

$A$ bullet of $10\, \text{g}$, moving with velocity $v$, collides head-on with the stationary bob of a pendulum and recoils with velocity $100\, \text{m/s}$. The length of the pendulum is $0.5\, \text{m}$ and the mass of the bob is $1\, \text{kg}$. The minimum value of $v$ in $\text{m/s}$ so that the pendulum completes a vertical circle is: (Assume the string to be inextensible and $g=10\, \text{m/s}^2$)

$A$ bullet of mass $0.02 \ kg$ travelling horizontally with velocity $250 \ ms^{-1}$ strikes a block of wood of mass $0.23 \ kg$ which rests on a rough horizontal surface. After the impact,the block and bullet move together and come to rest after travelling a distance of $40 \ m$. The coefficient of sliding friction of the rough surface is $\left(g=9.8 \ ms^{-2}\right)$

$A$ particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $dK/dt = \gamma t$,where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?
$(A)$ The force applied on the particle is constant
$(B)$ The speed of the particle is proportional to time
$(C)$ The distance of the particle from the origin increases linearly with time
$(D)$ The force is conservative