How the Centre of Mass Moves

You cannot lift yourself by your own bootstraps. Physics puts it more precisely: internal forces can never move the centre of mass.

Two particles pulling on each other#

Two particles on a line attract each other and nothing else acts on them. Their centre of mass is at $X = (m_1x_1 + m_2x_2)/(m_1 + m_2)$, so its velocity and acceleration are mass-weighted averages too:

With mutual force F, $a_1 = F/m_1$ and $a_2 = -F/m_2$, so the numerator is $F - F = 0$. The centre of mass does not accelerate. If the particles started at rest, the centre of mass stays put while they rush together.

Any system of particles#

For particle i, Newton’s second law reads $m_i\vec a_i = \sum_{j\ne i}\vec F_{ij} + \vec F_i^{\,ext}$. Add these up over all particles. The internal forces come in third-law pairs ($\vec F_{ij} + \vec F_{ji} = 0$) and cancel, leaving only the external forces. Since $\sum m_i\vec a_i = M\vec a_{cm}$:

The exploding shell#

A projectile’s only external force is gravity, so its centre of mass follows the parabola $x = u_xt$, $y = u_yt - \tfrac12gt^2$ (9.10). Now let it burst in mid-air. The pieces fly off on different parabolas, but the burst is internal: the centre of mass of the fragments carries on along the original curve, exactly as if nothing had happened, until one piece hits the ground and an external force joins in.