$A$ bomb is kept stationary at a point. It suddenly explodes into two fragments of masses $1\,g$ and $3\,g$. The total kinetic energy ($K$.$E$.) of the fragments is $6.4 \times 10^4\,J$. What is the $K$.$E$. of the smaller fragment?

$A$ and $B$ are blocks of same mass $m$ exactly equivalent to each other. Both are placed on a frictionless surface connected by a spring. The natural length of the spring is $L$ and the force constant is $K$. Initially,the spring is at its natural length. Another equivalent block $C$ of mass $m$ travelling at speed $v$ along the line joining $A$ and $B$ collides with $A$. In an ideal condition,the maximum compression of the spring is:

$A$ ball is thrown vertically downwards from a height of $20 \, m$ with an initial velocity $v_0$. It collides with the ground,loses $50\%$ of its energy in the collision,and rebounds to the same height. The initial velocity $v_0$ is .................... $m/s$ (Take $g = 10 \, m/s^2$)

$A$ ball is falling freely from a height. When it reaches $10 ~m$ height from the ground, its velocity is $v_0$. It collides with the ground, loses $50 \%$ of its energy, and rises back to a height of $10 ~m$. Then the velocity $v_0$ is (in $~m / s$)

$A$ constant force of $5 N$ accelerates a stationary particle of mass $500 g$ through a displacement of $5 m$. The average power delivered is: (in $W$)

Two trolleys of mass $m$ and $3m$ are connected by a spring. They are compressed and released; once released,they move off in opposite directions and come to rest after covering distances $S_1$ and $S_2$ respectively. Assuming the coefficient of friction to be uniform,the ratio of distances $S_1:S_2$ is: