Motion of the Centre of Mass and Conservation of Linear Momentum

Differentiating the defining formula $\vec R = \sum_i m_i \vec r_i / M$ (Section 5.3) once with respect

to time, and using $M = \sum_i m_i$ as a constant (no mass enters or leaves the system),

The right-hand side is nothing but the total linear momentum $\vec P$ of the system -- the vector sum

of the momenta of every individual particle. So $\vec P = M\vec V_{cm}$: the total momentum of any system

of particles equals the total mass times the velocity of its own centre of mass, exactly as though the

whole system were one single particle of mass $M$ moving with velocity $\vec V_{cm}$.

Differentiating once more,

Now, the force $\vec F_i$ on each particle $i$ can always be split into an external part (from outside the

system) and an internal part (from every other particle within the system). By Newton's third law, every

internal force comes in an equal-and-opposite pair between two particles of the system, so when the sum

$\sum_i \vec F_i$ is taken over the WHOLE system, every internal force pair cancels exactly, leaving only

the net external force:

This is the central result of this section: the centre of mass of any system of particles moves exactly as a single particle of mass $M$ would move under the net external force alone -- entirely independent

of whatever internal forces (a string pulling two blocks together, an explosive charge, the mutual

gravitational pull between the particles) may also be acting within the system. An explosion that splits a

shell into several fragments, for instance, involves only internal forces between the fragments at the

instant of the explosion; the shell's centre of mass therefore continues, completely undisturbed, along

the very same trajectory the shell was already following, right through the instant of the explosion,

exactly as illustrated in this chapter's worked example of a shell bursting into two pieces mid-flight.

Conservation of linear momentum follows immediately as the special case $\vec F_{ext} = 0$: if no net …