Motion of Center of Mass

Since the center of mass has a well-defined position at every instant, it also has a well-defined velocity and acceleration, obtained simply by differentiating its position with respect to time once and twice respectively:

No net external force. When the net external force on a system of particles is zero, $\vec F_{ext}=0$, the individual particles making up the body can still move around relative to each other, driven purely by internal forces between them — but this internal rearrangement can never change the position of the center of mass. The center of mass simply stays at rest if it was at rest, or continues at whatever constant velocity it already had:

In either case, $\vec a_{CM}=0$.

The classic illustration is a person walking on an otherwise stationary boat floating on still water: the friction between the person's feet and the boat is internal to the person+boat system, so with no external horizontal force acting on the system as a whole, its center of mass stays exactly where it started — the boat must recoil in the direction opposite to the walker, with a speed set by $m_1v_1+m_2v_2=0$ (or, relative to the walker, by combining the two individual velocities), exactly analogous to the recoil of a gun firing a bullet. The same idea explains what happens in an explosion: if a moving or stationary body breaks apart purely due to internal forces (with no outside trigger), its center of mass keeps following exactly the trajectory (e.g. the same parabola, if the body was a projectile) that the original, unbroken body would have followed — even though the individual fragments fly off along entirely different paths of their own, the principle of moments can be used with the final resting position of the center of mass to work out where an unmeasured fragment must have landed, given where the other fragment(s) are known to have landed.

Net external force present. When a genuine external force does act, $\vec F_{ext}\ne0$, the center of mass accelerates in accordance with Newton's second law applied to the system as a single equivalent particle of the total mass $M$:

$\vec F_{ext}=\sum_im_i\vec a_i=M\vec a_{CM},\qquad \vec a_{CM}=\frac{\vec F_{ext}}{M}.$ …