One projectile moving with velocity $v$ in space, gets burst into $2$ parts of masses in the ratio $1 : 3$ . The smaller part becomes stationary. What is the velocity of the other part ?

A bob of mass $0.1\; kg$ hung from the celling of a room by a string $2 \;m$ long is set into oscillation. The speed of the bob at its mean position is $1\; m s ^{-1}$. What is the trajectory of the bob if the string is cut when the bob is

$(a) $ at one of its extreme positions,

$(b)$ at its mean position.

A bullet mass $10\, gm$ is fired from a gun of mass $1\,kg$. If the recoil velocity is $5\, m/s$, the velocity of the muzzle is ........ $m/s$

A buggy of mass $100\, kg$ is free to move on a frictionless horizontal track. Two men, each of mass $50\, kg$, are standing on the buggy, which is initially stationary. The men jump off the buggy with velocity $=10m/s$ relative to the buggy. In one situation, the men jump one after the other. In another situation, the men jump simultaneously. What is the ratio of the recoil velocities of the buggy in two cases?

A bomb is projected with $200\,m/s$ at an angle $60^o$ with horizontal. At the highest  point, it explodes into three particles of equal masses. One goes vertically upward with  velocity $100\,m/sec$, second particle goes vertically downward with the same velocity as  the first. Then what is the velocity of the third one-

A spacecraft of mass $M$ moves with velocity $V$ in free space at first, then it explodes breaking into two pieces. If after explosion a piece of mass $m$ comes to rest, the other piece of spacecraft will have a velocity