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 body of mass $1000 \mathrm{~kg}$ is moving horizontally with a velocity $6 \mathrm{~m} / \mathrm{s}$. If $200 \mathrm{~kg}$ extra mass is added, the final velocity (in $\mathrm{m} / \mathrm{s}$ ) is:

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 rifle man, who together with his rifle has a mass of $100\,kg$, stands on a smooth surface and fires $10$ shots horizontally. Each bullet has a mass $10\,g$ and a muzzle velocity of $800\,ms ^{-1}$. The velocity which the rifle man attains after firing $10$ shots is $..........\,ms^{-1}$

A ball is projected at $60^o$ from horizontal at $200\, m/s$. At maximum height during its  flight it explodes into $3$ equal fragments. Out of them one part travel at $100\, m/s$  vertically up while other at $100\, m/s$ vertically down, then third part will have  speed just after explosion :-