Two bodies of different masses $m_1$ and $m_2$ have equal momenta. Their kinetic energies $E_1$ and $E_2$ are in the ratio:

$A$ 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 $\,m/s$)

$A$ block of mass $m(=0.1 \ kg)$ is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force $F$ in the vertically downward direction. The linear momentum of the block increases by $2 \ kg \ m/s$ in $1 \ s$ after the block starts from rest. Then, (given $g=10 \ m/s^2$):

$A$ wooden block of mass $M$ rests on a horizontal surface. $A$ bullet of mass $m$ moving in the horizontal direction strikes and gets embedded in it. The combined system covers a distance $x$ on the surface. If the coefficient of friction between the wood and the surface is $\mu$,the speed of the bullet at the time of striking the block 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?
$(A)$ $\vec{u}_0 = \sqrt{2gh} \hat{x}$
$(B)$ $\vec{v} = \sqrt{2gh} \hat{x} + \sqrt{2gh} \hat{z}$
$(C)$ $\theta = 60^{\circ}$
$(D)$ $d / h_1 = 2\sqrt{3}$

$A$ man of mass $80 \ kg$ goes to the market on a scooter of mass $100 \ kg$ with a certain speed. On application of brakes, the stopping distance is $S_1$. The man returns home on the same scooter, with the same speed with a $60 \ kg$ bag of rice. If $S_2$ is the new stopping distance when the brakes are applied with the same force, then: