The physics
Momentum is mass times velocity, p = mv. It is a vector, so its direction counts.
Inside forces come in pairs, equal and opposite. They cannot change the total momentum. With no outside push, the total momentum stays the same.
m₁v₁ + m₂v₂ = m₁u₁ + m₂u₂
Bodies at rest start with zero momentum. When they push apart, their momenta stay equal and opposite. The lighter one moves faster.
The centre of mass moves only when an outside force acts. A man walking on a boat cannot move it. Two people pulling on a rope cannot move it either.
M vcm = Σ m v no outside force: vcm stays the same
A burst keeps the momentum, but it adds kinetic energy. A shell can burst in the air. Its centre of mass stays on the old path. Only gravity acts from outside.