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Zeno’s Paradoxes and the Viscous Friction Force

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Abstract

In this paper, we connected Zeno’s paradoxes and motions with the viscous friction force \(F=-bv\). For the progressive version of the dichotomy paradox, if the body speed is constant, the sequences of positions and instants are infinite, but the series of distances and time variations converge to finite values. However, when the body moves with force \(F=-bv\), the series of time variations becomes infinite. In this case, the body crosses infinite points, approximating to a final position forever, as the progressive version of the dichotomy paradox describes. The same procedures with constant speed and motion with force \(F =-bv\) result in different spatial sequences and the same series for the regressive version of the dichotomy paradox. Nonetheless, the regressive version of the dichotomy paradox does not describe a body that approximates a final position forever. Finally, for the Achilles paradox, we find the positional and temporal sequences of the man and the tortoise at constant speeds. Analogously to the progressive version of the dichotomy paradox, the positional and temporal sequences are infinite, but the spatial and temporal series converge to finite values. If Achilles and the tortoise are identical points that move with force \(F=-bv\), and the turtle’s position in function of time presents a temporal advantage, the man and the animal will cross successive infinite places forever, and the quick hero will never overtake the slow reptile, as the paradox described. We conclude that the progressive version of the dichotomy and Achilles paradoxes can describe motions when the force is \(F =-bv\).

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  1. Universidade Federal de São Paulo, Diadema, SP, Brazil

    Leonardo Sioufi Fagundes dos Santos

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dos Santos, L.S.F. Zeno’s Paradoxes and the Viscous Friction Force. Found Phys 52, 67 (2022). https://doi.org/10.1007/s10701-022-00589-3

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