Two identical uniform discs roll without slipping on two different surfaces $AB$ and $CD$ (see figure) starting at $A$ and $C$ with linear speeds $v _1$ and $v _2$, respectively, and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_1=3 \ m / s$, then $v_2$ in $m / s$ is $\left(g=10 \ m / s ^2\right)$

A particle of mass $0.1 \,kg$ is subjected to a force which varies with distance as shown. If it starts its journey from rest at $x=0$, then its velocity at $x=12 \,m$ is .............. $m / s$

A bomb of $12 \,kg$ explodes into two pieces of masses $4 \,kg $ and $8 \,kg$. The velocity of $8\,kg$  mass is $6 m/sec$. The kinetic energy of the other mass is ............. $\mathrm{J}$

A body of mass $10\,kg$ moving with speed of $3 \,ms ^{-1}$ collides with another stationary body of mass $5 \,kg$. As a result, the two bodies stick together. The $KE$ of composite mass will be .......... $J$

If a body of mass $200\, g$ falls from a height $200 \,m$ and its total $P.E.$ is converted into $K.E.$ at the point of contact of the body with earth surface, then what is the decrease in $P.E.$ of the body at the contact $(g = 10\,m/{s^2})$ ............ $\mathrm{J}$

Two bodies of different masses ${m_1}$ and ${m_2}$ have equal momenta. Their kinetic energies ${E_1}$ and ${E_2}$ are in the ratio