$A$ body of mass $(4m)$ is lying in the $x-y$ plane at rest. It suddenly explodes into three pieces. Two pieces,each of mass $(m)$,move perpendicular to each other with equal speeds $(v)$. The total kinetic energy generated due to the explosion is ................. $mv^2$.

$A$ shell is fired from a cannon with velocity $v \text{ m/s}$ at an angle $\theta$ with the horizontal direction. At the highest point in its path,it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed in $\text{m/s}$ of the other piece immediately after the explosion is:

$A$ 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?
$(A)$ $\vec{u}_0 = \sqrt{2gh} \hat{x}$
$(B)$ $\vec{v} = \sqrt{2gh} \hat{x} + \sqrt{2gh} \hat{z}$
$(C)$ $\theta = 60^{\circ}$
$(D)$ $d / h_1 = 2\sqrt{3}$

Consider two carts,of masses $m$ and $2m$,at rest on an air track. If you push both the carts for $3\,s$ exerting equal force on each,the kinetic energy of the light cart is

In the diagram shown,there is no friction at any contact surface. Initially,the spring has no deformation. What will be the maximum deformation in the spring? Consider all the strings to be sufficiently large. Consider the spring constant to be $K$.

$A$ ball is thrown vertically down from a height of $40 \,m$ from the ground with an initial velocity $v$. The ball hits the ground, loses $\frac{1}{3}$ of its total mechanical energy, and rebounds back to the same height. If the acceleration due to gravity is $10 \,m/s^2$, then the value of $v$ is (in $\,m/s$)