Two blocks $A$ and $B$ of masses $1\, kg$ and $2\, kg$ are connected together by a spring and are resting on a horizontal surface. The blocks are pulled apart so as to stretch the spring and then released. The ratio of $K.E.$ of block $A$ to the $K.E.$ of block $B$ is:

An object of mass $2m$ is projected with a speed of $100 \ m/s$ at an angle $\theta = \sin^{-1}(3/5)$ to the horizontal. At the highest point, the object breaks into two pieces of same mass $m$ and the first one comes to rest. The distance between the point of projection and the point of landing of the second piece (in metres) is $(g = 10 \ m/s^2)$.

An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass $1 \, kg$ moves with a speed of $12 \, m s^{-1}$ and the second part of mass $2 \, kg$ moves with $8 \, m s^{-1}$ speed. If the third part flies off with $4 \, m s^{-1}$ speed,then its mass is ......... $kg$.

At high altitude,a body at rest explodes into two equal fragments,with one fragment receiving a horizontal velocity of $10 \, m/s$. The time taken by the two radius vectors connecting the point of explosion to the fragments to make an angle of $90^o$ is ............ $s$.

$A$ man of mass $100\,kg$ is standing on a platform of mass $200\,kg$ which is kept on a smooth ice surface. If the man starts moving on the platform with a speed $30\,m/s$ relative to the platform,then calculate with what velocity relative to the ice the platform will recoil.

An object of mass $3\,m$ at rest splits into three equal fragments. Two fragments have velocities $v\hat j$ and $v\hat i$. The velocity of the third fragment is