$A$ $1.0\, kg$ block collides with a horizontal weightless spring of force constant $2.75\, N/m$ as shown in the figure. The block compresses the spring $4.0\, m$ from the rest position. If the coefficient of kinetic friction between the block and the horizontal surface is $0.25$,the speed of the block at the instant of collision is ................. $m/s$.

$A$ 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?

$A$ 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. [$g = 10 \ m \ s^{-2}$]

$A$ child swings on a swing such that the minimum and maximum heights from the ground are $0.75 \ m$ and $2 \ m$ respectively. The maximum velocity of the swing is ...... $ms^{-1}$.

$A$ bullet of mass $10 \,g$ is fired horizontally with a velocity $1000 \,ms^{-1}$ from a rifle situated at a height $50 \,m$ above the ground. If the bullet reaches the ground with a velocity $500 \,ms^{-1}$, the work done against air resistance in the trajectory of the bullet is : $(g=10 \,ms^{-2})$ (in $\,J$)

$A$ tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $K$ with time $t$ most appropriately? The figures are only illustrative and not to the scale.