If the kinetic energy of a body is directly proportional to time $t,$ the magnitude of force acting on the body is
$(i)$ directly proportional to $\sqrt{t}$
$(ii)$ inversely proportional to $\sqrt{t}$
$(iii)$ directly proportional to the speed of the body
$(iv)$ inversely proportional to the speed of the body

The quantities remaining constant in a general collision are

Equal force $F (> mg)$ is applied to the string in all the $3$ cases. Starting from rest,the point of application of force moves a distance of $2 \ m$ down in all cases. In which case does the block have maximum kinetic energy?

$A$ slide with a frictionless curved surface,which becomes horizontal at its lower end,is fixed on the terrace of a building of height $3h$ from the ground,as shown in the figure. $A$ spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0 = u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building,making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $e = 1 / \sqrt{3}$. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\vec{u}_0 = \sqrt{2gh} \hat{x}$
$(B)$ $\vec{v} = \sqrt{2gh} \hat{x} + \sqrt{2gh} \hat{z}$
$(C)$ $\theta = 60^{\circ}$
$(D)$ $d / h_1 = 2\sqrt{3}$

Two persons $A$ and $B$ are throwing a ball of mass $200 \ g$ at a wall as shown in the figure. The balls strike the wall perpendicularly at the same point at a height of $2 \ m$ from the ground. The balls strike the wall elastically at the same time and return back to $A$ and $B$ at the same time. They repeat this process. What is the average force exerted on the wall (in $N$)? (Take $g = 10 \ m/s^2$)

$A$ particle of mass $m$ moving with velocity $V_0$ strikes a simple pendulum of mass $m$ and sticks to it. The maximum height attained by the pendulum will be