A bomb of mass $3m$ kg explodes into two pieces of mass $m kg$ and $2m$ $kg$. If the velocity of m kg mass is $16 m/s$, the total kinetic energy released in the explosion is ................. $\mathrm{mJ}$

A bomb of mass $9\, kg$ explodes into two pieces of masses $3\, kg$ and $6\, kg$. The velocity of mass $3\, kg$ is $16\, m/s$. The $KE$ of mass $6\, kg$ is ............ $\mathrm{J}$

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x=0$, in a co-ordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $\mathrm{x}$ and the velocity is $\mathrm{v}$. At that instant, which of the following options is/are correct?

(image)

$[A]$ The $x$ component of displacement of the center of mass of the block $M$ is : $-\frac{m R}{M+m}$.

[$B$] The position of the point mass is : $x=-\sqrt{2} \frac{\mathrm{mR}}{\mathrm{M}+\mathrm{m}}$.

[$C$] The velocity of the point mass $m$ is : $v=\sqrt{\frac{2 g R}{1+\frac{m}{M}}}$.

[$D$] The velocity of the block $M$ is: $V=-\frac{m}{M} \sqrt{2 g R}$.

$Assertion$ : A helicopter must necessarily have two propellers.
$Reason$ : Two propellers are provided in helicopter in order to conserve linear momentum

Write the principle of conservation of mechanical energy for conservative force.

A space craft of mass $'M' $ and moving with velocity $ 'v' $ suddenly breaks in two pieces of same mass $m$. After the explosion one of the mass $ 'm'$  becomes stationary. What is the velocity of the other part of craft