Motion of Centre of Mass — Prelims Questions

Two particles of masses $m_1 = 1,\text{kg}$ and $m_2 = 2,\text{kg}$ are moving with velocities $\vec{v}_1 = (2\hat{i} - 3\hat{j}),\text{m/s}$ and $\vec{v}_2 = (\hat{i} + 2\hat{j}),\text{m/s}$ respectively. What is the velocity of the center of mass of the system?

A bomb initially at rest explodes into three fragments. If no external forces act on the bomb, what can be said about the motion of the center of mass of the fragments?

A $50,\text{kg}$ man stands at one end of a $100,\text{kg}$ boat of length $10,\text{m}$. The boat is at rest in still water. If the man walks to the other end of the boat, by what distance does the boat move relative to the shore?

A projectile is fired into the air. At its highest point, it explodes into two fragments of equal mass. One fragment falls vertically downwards. What is the path of the other fragment?

Three particles of masses $1,\text{kg}$, $2,\text{kg}$, and $3,\text{kg}$ are located at positions $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ respectively. What is the position vector of their center of mass?

A system of two particles has masses $m_1$ and $m_2$. If $m_1$ is at rest and $m_2$ is moving with velocity $\vec{v}$, what is the velocity of the center of mass of the system?