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Imagine a person and a weighing scale accelerating through empty space towards each other, with the person being pulled by a downward force $F$ (being exerted by some distant, unknown source) and the scale being pulled upward by a force of the same magnitude $F$ (being exerted by a different distant and unknown source).

Later they collide and the person ends up standing on the scale, but they stop moving since both are being pulled against each other with the same force $F$. In this case, do we say that the person's rest frame (and by extension the scale's rest frame since it's also at rest w.r.t. the person) is inertial or non-inertial? And what's the reasoning?

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Summary

If the initial frame in which you were observing the man and the weighing was an inertial frame, then the new frame of the man (or the weighing scale) after collision will be an inertial frame.

Explanation

From now on, I am assuming that the initial frame was inertial.

There are two possible cases. Either both the man and weighing scale come to rest after colliding (case 1), or they end up with some net velocity after the collision (case 2).

The case 1 will happen when the momentum of both the spaceship and the man would have been equal and opposite before the collision. In this case, the new frame of reference of the man after collision is the same as the original frame of reference before collision since both the frames are at rest with respect to each other.

The case 2 will be the general case and will happen when the momenta of both the bodies are unequal. In this case, after colliding, the net force on the system is zero (as you correctly noted) and thus by Newton's second law, their acceleration must also be zero. Thus after collision the frame of the man is another frame which is moving with a constant velocity (zero acceleration) with respect to the original frame. Thus, in this case as well, the man's frame of reference is inertial.

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  • $\begingroup$ Thanks a lot! Just one minor confusion: let's say you're the person in question (in the above thought experiment), then if you were to use an accelerometer before hitting the scale, you'd detect an acceleration and conclude that your rest frame is non-inertial. But after hitting the scale and coming to a stop, then on using the accelerometer you would no longer detect the acceleration because you're no longer undergoing proper acceleration. So wouldn't that be a scenario in which your initial rest frame was non-inertial, but later after hitting the scale, your rest frame became inertial? $\endgroup$
    – user9343456
    Commented Apr 25, 2020 at 14:41
  • $\begingroup$ @u23 Yes, that would be a case when my initial frame was a non inertial frame with respect to the space frame (which is assumed to be inertial) and my final frame would be an inertial frame, again, with respect to the space frame. $\endgroup$
    – user258881
    Commented Apr 25, 2020 at 15:11
  • $\begingroup$ but I thought there's supposed to be no preferred frame? Could you clarify what's meant by the "space frame"? $\endgroup$
    – user9343456
    Commented Apr 25, 2020 at 18:07
  • $\begingroup$ @u23 There isn't. But you need to define any particular frame as inertial to start with. You could have defined the accelerating man's frame as inertial and then in that case, the space would have become a non inertial frame. However, I can infer from your question that you chose a particular frame of reference (which I vaguely called space frame in my last comment) in which the man is accelerating with an acceleration $a$. Thus there is no preferred frame, however you have to start by assuming one frame as inertial and then describe everything with respect to that frame. $\endgroup$
    – user258881
    Commented Apr 25, 2020 at 18:15

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