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How does conservation of momentum change in moving frames ( constant velocity ) and non-inertial frames?

In this question's accepted answer, it says that if the time period of application of pseudo force is negligible, then the conservation of momentum holds. But I have learnt that momentum is always conserved? Where does this aspect of time period come into the picture?

Firefox1921
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tryst with freedom
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2 Answers2

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Momentum conservation applies to systems that are translationally invariant, so if your system is accelerating, there is an omnipresent pseudo-force, $\vec g$, that breaks translational invariance. If the acceleration is uniform, you can consider a potential energy, $\phi$, such that:

$$ \vec g = -\vec{\nabla}\phi(\vec x) $$

The system now has a position dependent energy; it is not invariant under all translations, and momentum is not conserved. For example, if you drop a ball: it gains momentum in the direction of $\vec g$.

Regarding the linked answer; here is an example. If you are studying collisions on a frame with position dependent potential energy, say 2 billiard balls on a slightly tilted table (or perhaps golf putting on a difficult green), you are going to have to consider $\phi({\vec x})$ in estimating the run-out of your shot (aka: reading the green), but it does not factor into the collision.

If you want a certain speed and direction for your billiard ball, you can use your "level-table" shot skill set. As the other answer points out, the collision forces are much, much, greater than $m\vec{g}$, and the quicker the collision, the greater the ratio.

JEB
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The reason we require a small impact/collision time for momentum conservation is quite simple. If we want to believe we can neglect external forces like friction or drag for the instant that the collision happens, then the collision time has to be very small so that these forces don't have enough time to dissipate the momentum of the colliding bodies. We can therefore, safely assume that only the mutual internal forces between the bodies act during the short-impact time. It's important however, to understand that impact time will not matter if external forces can actually be neglected.

EMMANUEL CHIDERA
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