$A$ ball is thrown vertically downwards from a height of $20 \, m$ with an initial velocity $v_0$. It collides with the ground,loses $50\%$ of its energy in the collision,and rebounds to the same height. The initial velocity $v_0$ is .................... $m/s$ (Take $g = 10 \, m/s^2$)

$A$ block of weight $10 \ N$ slides down a curved track $AB$ and then onto a rough horizontal surface. The coefficient of kinetic friction between the block and the rough surface is $0.20$. If the block starts sliding from a height of $1.0 \ m$ above the horizontal surface,calculate the distance $S$ it travels on the rough surface before coming to rest. [$g = 10 \ m \ s^{-2}$]

$A$ lift of mass $2000 \ kg$ starts from rest in the basement and moves to the fourth floor,which is at a height of $25 \ m$. As it passes the fourth floor,its speed is $3 \ ms^{-1}$. There is a constant frictional force of $500 \ N$ acting on it. Calculate the work done by the lift's motor in $kJ$.

$A$ locomotive of mass $m$ starts moving so that its velocity varies according to the law $v = k \sqrt{S}$,where $k$ is a constant and $S$ is the distance covered. Find the total work performed by all the forces acting on the locomotive during the first $t$ seconds after the beginning of motion.

Two cars,both of mass $m$,collide and stick together. Prior to the collision,one car had been traveling north at speed $2v$,while the second was traveling at speed $v$ at an angle $\phi$ south of east (as indicated in the figure). The magnitude of the velocity of the two-car system immediately after the collision is:

Two blocks of masses $m_1$ and $m_2$ are connected by a spring of spring constant $k$,initially at their natural length as shown. $A$ sharp impulse is given to mass $m_2$ so that it acquires a velocity $v_0$ towards the right. If the system is kept on a smooth floor,find the maximum elongation that the spring will suffer.