Which physical quantity is conserved in both elastic and inelastic collisions?

$A$ man of mass $60$ $kg$ wants to lose $5$ $kg$ of mass by climbing up and down stairs. Assume that twice the amount of fat is burned while climbing up compared to climbing down. If burning $1$ $kg$ of fat provides $7000$ $kcal$ of energy, how many times must he climb up and down the stairs to lose $5$ $kg$ of mass (in $\times$)? (Assume the energy spent in climbing up is $20$ $kcal$ and climbing down is $10$ $kcal$ per trip).

$A$ bob of mass $m$,suspended by a string of length $l_1$,is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point,it collides elastically with another bob of mass $m$ suspended by a string of length $l_2$,which is initially at rest. Both the strings are massless and inextensible. If the second bob,after collision,acquires the minimum speed required to complete a full circle in the vertical plane,the ratio $\frac{l_1}{l_2}$ is

$A$ dumbbell consisting of two masses $m$ each,connected by a light rigid rod of length $l$,falls from a height $h$ onto two pads of equal height,one steel and the other brass. The coefficients of restitution are $e_1$ and $e_2$ $(e_1 < e_2)$. To what maximum height will the centre of mass of the dumbbell rise after bouncing off the pads?

$A$ $0.5 \, kg$ block moving at a speed of $12 \, ms^{-1}$ compresses a spring through a distance of $30 \, cm$ when its speed is halved. The spring constant of the spring in $N m^{-1}$ is:

In the figure shown,two identical balls of mass $M$ and radius $R$ each are placed in contact with each other on a frictionless horizontal surface. $A$ third ball of mass $M$ and radius $R$ is coming down vertically and has a velocity $v_0$ when it simultaneously hits the two balls and comes to rest. Then,each of the two bigger balls will move after the collision with a speed equal to: