Consider the following two statements: $[A]$ The linear momentum of a system is zero. $[B]$ The kinetic energy of the particles of the system is zero.

Two beads $A$ and $B$ of masses $m_1$ and $m_2$ respectively are placed on a smooth vertical circular wire of radius $R$ as shown in the figure. Bead $A$ is given a very gentle push so that it slides down and collides with bead $B$ and comes to rest. After the collision,bead $B$ reaches the height of the center of the circle. Then $m_1 : m_2$ is equal to:

Which of the following statements is true for a collision?

Two identical blocks $A$ and $B$, each of mass $m$, resting on a smooth floor, are connected by a light spring of natural length $L$ and spring constant $k$. The spring is at its natural length. $A$ third identical block $C$ (mass $m$) moving with a speed $v$ along the line joining $A$ and $B$ collides with $A$. The maximum compression in the spring is proportional to

$A$ lead bullet penetrates into a solid object and melts. Assuming that $40 \%$ of its kinetic energy is used to heat it,the initial speed of the bullet is ............ $m \, s^{-1}$.
(Given: Initial temperature of the bullet $= 127^{\circ} C$,
Melting point of the bullet $= 327^{\circ} C$,
Latent heat of fusion of lead $= 2.5 \times 10^{4} \, J \, kg^{-1}$,
Specific heat capacity of lead $= 125 \, J \, kg^{-1} K^{-1}$)

$A$ 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 ...... $ms^{-1}$.