As Homer Powley (I think it was) put it, the aerodynamic forces "conspire" to point the bullet into the trajectory path. There is a constant translation of overturning forces to the opposite side by spin that makes this true. It is why a bullet yawing most extremely after clearing the muzzle has its nose describe a helical path around the mean trajectory path (the coning or precession I mentioned earlier). The degree to which the bullet resists that aerodynamic conspiracy is dependent on how fast it spins, its mass, and how hard the air resistance reaction force is that it experiences. The degree of success they have in resisting the conspiracy is indicated by how far their average nose-up yaw above the line tangent to the trajectory at any given moment in flight. That is a function of the gyroscopic stability factor which takes spin, velocity, atmospheric density, bullet length and mass all into account in one number.
The tendency to nose over does, in fact, occur. Just look at the angled ramp on the back of a Springfield '03 ladder sight to see how much the sight windage has to move to the side to compensate over a long distance. Typically, with normal gyroscopic stability factors, you see around one foot of windage drift at 1000 yards due to this effect.
If you want to calculate change in stability with distance, you can. Velocity loss is in any ballistic tables. Loss of spin is slower (which is why s
increases with distance) but there is surface friction slowing it. Geoffry Kolbe has a spin decay approximation of:
Where current spin rate is N, and the initial spin rates is Nm (spin rate units are your choice, but must be consistent), and t is the flight time in seconds and and d is bullet diameter in inches:
N=Nm exp[-0.035 t / d]
Momentum is not a good indicator of stability because it is the product of velocity and mass which don't have the same effect on stability. The gyroscopic stability factor, which takes all the elements of stability into account, increases directly with mass for a given velocity. So, if you keep the same bullet shape and and add enough powder to keep the same velocity, but increase momentum 67% by increasing the mass 67% by replacing the lead alloy core with tungsten, you increase the stability factor 67%. But if you keep the original bullet and increase momentum 67% by adding enough powder to increase velocity 67% you do not get a 67% increase in stability factor. Only about 19%. The reason is that at the higher velocity, air pressure on the bullet nose is greater, so it takes more spin and momentum to keep the bullet from tumbling under those greater aerodynamic forces. That cancels out almost 70% of the gains from the higher momentum and spin.
For example, we take a 168 grain Sierra MatchKing at 2600 fps in a 12" twist barrel. The gyroscopic stability factor is:
We now increase momentum 67% by increasing mass 67% and keeping velocity the same, so the same size bullet now weighs 280.6 grains:
=2.871 a gain of 1.152
We now increase momentum 70% for the original 168 grain bullet by increasing velocity 67% to 4342 fps:
=2.040 a gain of only 0.321
So, momentum was increased 67% in both cases, but did not bring about the same effect on stability. It is still bullet length and rate of twist that matter most, which is why the old Greenhill formula only used those two factors.