I am currently studying for a physics exam and I came across this question in my book:"An archer stands at rest on frictionless and fires a kg arrow horizontally at m/s by using a bow. The combined mass of the archer and the bow is kg. With what velocity does the archer move across the ice after firing the arrow?"
The solution given uses the law of conservation of momentum on the archer, the bow, and the arrow - and that's it. However, there are gravitational and normal forces that exist in this situation, so we can't do this without further reasoning. The book tries to justify using only the law of conservation of momentum by stating that no work is done by either of these forces. This doesn't make any sense to me seeing as total momentum is conserved when no external force acts on the system. I was thinking that one should use the law of conservation of momentum on the archer, the bow, the arrow, and the Earth in addition to the necessary 2D kinematics and the gravitational forces, etc.?
Any help would be appreciated.