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If I have a particle moving radially away from me in a straight line and I extrapolate time to infinity, I could see the particle in a particular direction (given enough time for the light to reach me).

Now, imagine a particle starts at my location and moves in an outward spiral with a constant velocity. Now, if I extrapolate time to infinity, in which direction should I look to find the particle?

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    $\begingroup$ 1. "moves in an outward spiral with a constant velocity". That's contradictory. A particle with constant velocity moves in a straight line. But even if you meant constant speed, there are numerous types of spiral path. 2. Why do you expect that the direction towards the particle could converge at infinity to a particular angle? $\endgroup$
    – PM 2Ring
    Commented Jun 18 at 7:59

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in which direction should I look to find the particle?

To determine this you need to write the equation for the position of the particle in polar coordinates. There is not a shortcut here as “outward spiral” is too vague and could refer to any number of different paths.

Once you have the equation in terms of polar coordinates you take the limit $$\lim_{t\rightarrow \infty} \theta(t)$$ If that limit exists then that is the direction to look. There is no guarantee that this limit exists. If it does not exist then there is no specific direction that it tends to approach.

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