$A$ bouncing ball of mass $200 \,g$ falls from a height of $5 \,m$ onto a horizontal ground. After every impact with the ground, the velocity of the ball decreases by $\frac{1}{2}$ times. The total momentum the ball imparts to the ground after $3$ impacts is (Let $g=10 \,m/s^2$):

In the figure,two blocks $M$ and $m$ are tied together with an inextensible and light string. The mass $M$ is placed on a rough horizontal surface with coefficient of friction $\mu$ and the mass $m$ is hanging vertically against a smooth vertical wall. The pulley is frictionless. Choose the correct statement$(s)$ related to the tension $T$ in the string.

$A$ $1\; kg$ block situated on a rough incline is connected to a spring of spring constant $100\; N m^{-1}$ as shown in the figure. The block is released from rest with the spring in the unstretched position. The block moves $10\; cm$ down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

$A$ bead of mass $m$ can slide on a frictionless fixed ring of radius $r$. With the help of two identical springs of force constant $k$,it is connected to two diametrically opposite nails $A$ and $B$,each of which is at a distance $0.5r$ from the centre $O$ of the ring. The relaxed length of each spring is negligible compared to the radius of the ring. The bead is given a small velocity. What can you predict for the further motion of the bead before any of the springs strikes a nail?

An ideal spring is connected between two blocks of masses $M$ and $m$. This system can move on a smooth horizontal table. The blocks are brought closer to compress the spring and then released. In the subsequent motion,select the true statement$(s)$:
$(a)$ They move in opposite directions with speeds inversely proportional to their masses.
$(b)$ The ratio of their speeds remains constant.
$(c)$ Linear momentum and energy of the system remain conserved.

$10,000$ small balls,each weighing $1 \, g$,strike $1 \, cm^2$ of area per second with a velocity of $100 \, m/s$ in a normal direction and rebound with the same velocity. The value of pressure on the surface will be: