$A$ large block of wood of mass $M = 5.99 \, kg$ is hanging from two long massless cords. $A$ bullet of mass $m = 10 \, g$ is fired into the block and gets embedded in it. The (block $+$ bullet) then swing upwards,their centre of mass rising a vertical distance $h = 9.8 \, cm$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is: (Take $g = 9.8 \, ms^{-2}$) (in $m/s$)

Two masses $m_1$ and $m_2$ are connected by a string of length $l$. They are held in a horizontal plane at a height $H$ above two heavy plates $A$ and $B$ made of different materials placed on the floor. Initially,the distance between the two masses is $a < l$. When the masses are released under gravity,they collide with $A$ and $B$ with coefficients of restitution $e_1 = 0.8$ and $e_2 = 0.4$ respectively. Find the time after the collision when the string becomes tight. (Assume $H >> l$)

$A$ 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).

An engine pumps up $100 \ kg$ of water through a height of $10 \ m$ in $5 \ s$. Given that the efficiency of the engine is $60\%$. If $g = 10 \ m \ s^{-2}$,the power of the engine is .............. $kW$.

To simulate car accidents,auto manufacturers study the collisions of moving cars with mounted springs of different spring constants. Consider a typical simulation with a car of mass $1000 \; kg$ moving with a speed $18.0 \; km/h$ on a rough road having $\mu = 0.5$ and colliding with a horizontally mounted spring of spring constant $6.25 \times 10^{3} \; N m^{-1}$. What is the maximum compression of the spring in $m$?

$A$ ball moving with speed $v$ hits another identical ball at rest. The two balls stick together after the collision. If the specific heat of the material of the balls is $S$,the temperature rise resulting from the collision is