Q.A $60\ \text{kg}$ astronaut and a $15\ \text{kg}$ toolbox are initially at rest relative to each other, floating freely in space. The astronaut pushes the toolbox directly away, and it moves off at $8\ \text{m/s}$. Find the recoil speed of the astronaut.

From Newton's second law, $\vec{F}=\frac{d\vec{p}}{dt}$, it follows immediately that if the net force on a system is zero, its total linear momentum does not change -- it is conserved. This gives the principle of conservation of linear momentum: the total momentum of an isolated system (one with no net external force) is conserved during any interaction, whether a collision or an explosion. This holds because internal forces between parts of the system always occur as equal-and-opposite Newton's-third-law pairs, which exactly cancel when summed over the whole system, leaving only external forces able to change the system's total momentum.

This principle underlies the recoil of a gun when a bullet is fired, the scattering of fragments in an explosion (whose vector-summed momenta must still equal the pre-explosion momentum), and every collision analysis in mechanics -- it is always the first equation to write down when analysing any interaction between two or more bodies.