Momentum taken from Wikipedia: Momentum: is the product of the mass and velocity of an object.
The sword or jo does not generate its own momentum, the momentum is generated by the person holding the sword or jo.
Velocity and mass are really analytic quantities, and not primary quantities -- momentum is the primary quantity. We know this because when either one is zero -- momentum is not really zero, even if we treat it that way for purpose of analysis.
Light has zero mass but has momentum, and can change the velocity of mass, and thus transfer its momentum. Inertia is simply resting momentum. Resting mass resists change in relative velocity but really is just the aspect of momentum at zero relative velocity. Sometimes reduction to analytic components makes sense-- sometimes it doesn't. The important thing is never to mistake the ruler for the world you mean to measure.
There are two basic ways to cut in terms of momentum transfer (though in variations). One way cuts in-phase with the blade -- stopping a forward rotation of the core and letting the angular momentum transfer and concentrate at the monouchi. The other way is out of phase with the cut, where the core counter-rotates to the rotation of the blade, but in stopping the core, propagation of the angular momentum to the monouchi is basically the same.
In nukitsuke, these are seen, respectively, in typical Tanimura-ha and Shimomura-ha (a gross simplification, I am aware). The same is true in variations of tai-jutsu. However, most people approach this with a natural bias toward the in-phase forms. It is fun to find the complement in the form where you can -- and usually makes people go --"Hm."
The point about analytic quantities applies here, because if you just think in terms of adding velocity it will seem confusing -- but in conserving momentum it is perfectly sensible. In terms of velocity it seems like you should "force the blade" to add "speed"(or "force")-- when in terms of conserving angular momentum you should just let the blade do its thing with the momentum it has been given, and just let the shortening arc of cut concentrate its angular momentum at the cut. You can't add any more "force" than that.