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 $m/s$)

In a one-dimensional collision between two particles, their relative velocity is ${\vec v_1}$ before the collision and ${\vec v_2}$ after the collision

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0 N m ^{-1}$ and the mass of the block is $2.0 kg$. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass $1.0 kg$ moving with a speed of $2.0 m s ^{-1}$ collides elastically with the first block. The collision is such that the $2.0 kg$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is. . . . . .

Four smooth steel balls of equal mass at rest are free to move along a straight line without friction. The first ball is given a velocity of $0.4 \,m/s$. It collides head on with the second elastically, the second one similarly with the third and so on. The velocity of the last ball is .......... $m/s$

A bullet when fired at a target with a velocity of $100\,\,m/sec$ penetrates one metre into it. If the bullet is fired at a similar target with a thickness $0.5\,\,metre,$  then it will emerge from it with a velocity of

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is $p_d $ ; while for its similar collision with carbon nucleus at rest, fractional loss of energy is $P_c$. The values of $P_d$ and $P_c$ are respectively