So I just started reading Goldstein's classical dynamics and I'm having a brain fart trying to understand something. I found a slide that paraphrases a part of the section, which is about systems of particles and center of mass. It is attached to this post. Particularly, Goldstein says: "In order that the motion of the center of mass be unaffected, the ejection of the exhaust gases at high velocity must be counterbalanced by the forward motion of the vehicle." So how is the motion of the center of mass unaffected by internal forces? It seems to me that motion of the center of mass is directly affected by these internal forces. Specifically, the center of mass moves or undergoes motion because of the force of thrust that emerges from the rocket, which by Newton's third law, there is an equal and opposite force that causes the rocket (and the center of mass) to move forward. Goldstein appears to consider the center of mass as a system of the exhaust and the rocket itself. Could someone clarify this?
Goldstein considers the centre of mass of the rocket and the exhaust system because all he would need to do is to consider the forces which are external to that system as those external forces are the only ones which will change the motion of the centre of mass of the system.
If there are no external forces then the rate of change of momentum of the system is zero and the motion of the centre of mass of the rocket and exhaust system does not change.
So the change in momentum of the rocket plus the change in momentum of the exhaust gases is zero.
The derivation of the Tsiolkovsky rocket equation is shown here.
The rocket and exhaust system is chosen because the mass of that system does not change and so there is no difficulty in using Newton's second law in the derivation.