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Assuming as true the phenomenological MOND law for low accelerations ($< 10^{-10} m s^{-2} = a_0$), and considering a small mass $m$ attached to a larger mass $M$ by a faint spring (let's think of it as one of those ping pong tabs with a ball attached to the center by a springy thread).

Now, the movement has two phases, when the ball is in the air and the ball acceleration points in the positive $X$ axis direction, the acceleration is always below $a_0$, so the MOND law says that the inertia (of $m$? or $M$ as well?) will be less than $m$, let's say it will be $\frac{m}{2}$. But when the ball hits the pad, it is bounced. The impulse during the impact lapse is presumably much higher than $a_0$, so the full inertia of the system applies to this phase.

Now, linear momentum is conserved thanks to the third law of Motion (action and reaction), but I seek clarification for exactly how MOND affects linear momentum conservation? Does it violate it? Is Linear momentum conserved independently of it? Can MOND be fixed to conserve it? How does it look the change?

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Momentum in MOND is conserved, when the theory is correctly interpreted in a general theoretical framework. An older possibility is AQUAL, which provides a generalized (AQUAdratic) Lagrangian from which the MOND law can be obtained. The conservation laws are derived from the Lagrangian by the usual methods. A more general approach is reported in this paper (by me). Again momentum is conserved.

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