$A$ and $B$ are blocks of same mass $m$ exactly equivalent to each other. Both are placed on a frictionless surface connected by a spring. The natural length of the spring is $L$ and the force constant is $K$. Initially,the spring is at its natural length. Another equivalent block $C$ of mass $m$ travelling at speed $v$ along the line joining $A$ and $B$ collides with $A$. In an ideal condition,the maximum compression of the spring is:

An object of mass $M$ is at rest on a smooth horizontal surface. Objects of different masses collide head-on elastically with the object of mass $M$. All colliding objects have the same fixed kinetic energy $E$,and in each case,mass $M$ is initially at rest. The kinetic energy transferred to the stationary mass $M$ depends on the linear momentum $P$ of the incoming colliding mass. How does the energy transferred to $M$ vary with the linear momentum $P$?

Which of the following statements is true?

$A$ rain drop of radius $2 \; mm$ falls from a height of $500 \; m$ above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height,it attains its maximum (terminal) speed,and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is $10 \; m s^{-1}$?

$A$ 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 $60 \ kg$ gets $10^5 \ calories$ of energy from food. If his efficiency is $28\%$,up to what height in $m$ can he climb?