There is a simple solution to your question. You must analyze what belongs to the system and what doesn't. In both cases, the ground isn't a part of the system, hence if the ground has any interactions with the system, then you may not conserve momentum. However, in the second case, the spring is a part of the system, i.e. the forces on the blocks due to springs are internal forces, whose effects don't affect the force in Newton's second law, or just that there is no net unbalanced force. Thus you may conserve momentum in the second case but not in the first case.
So here are the conclusions
- Conservation of Momentum may be applied only if the net external force is equal to zero.
- If any source of force is in the system, it is considered to be an internal force, and hence the net effect of these forces cancel out.
Hope this helps
I am assuming by $G$ you actually mean $g$, local gravitational acceleration ($9.8 \,\mathrm{m/s^2}$ on Earth's surface) and not the universal gravitation constant, usually denoted $G$ ($6.7\cdot 10^{-11} \mathrm{N~m^2~kg^{-2}}$), which does not change.
If that's the case, your question is
a change in $g$, then, means a change in acceleration, which means a change in force which then means a change in momentum.
There is a little bit of confusion here. Even a constant force will cause a constant acceleration; so a constant, uniform gravity field will cause continuous acceleration (constantly increasing speed v). Since momentum is mv, this means momentum is changing constantly even under a constant force (and accompanying constant acceleration).
The fact that the local value of $g$ changes in your scenario does not add anything significant, except that the acceleration and force will increase instead of being constant, and the momentum will increase more quickly than it would have in a constant gravity field.
I can see that g is indeed changing in time and so momentum must also be changing
Acceleration does not have to change in order for momentum to change. As discussed, a constant g will cause a continuous increase in speed and momentum.
but I don't see how g is changing without there being an external force
The external force is gravity.
Best Answer
Gravity is a non impulsive force. it means it takes enough time to cause action and does not change momentum in an instant. So in time just before and after collision, momentum is conserved and gravity is neglected for such small time. it will work in all horizontal collisions like
BUT
if ball does not strike horizontally, then Normal and/or friction are impulsive so you cannot conserve momentum even for short durations