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Modeling the linear drag on falling balls via interactive fuzzy initial value problem

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Abstract

In this paper, we consider the linear drag of a falling ball, which can be well described using an interactive fuzzy initial value problem. The solution of the interactive fuzzy initial value problem gives us two types of solutions. One solution is when the uncertainty increases as time evolves, that is to say when the diameter of the fuzzy velocity increases exponentially. Hence, we ignore this solution, because we cannot expect this type of behavior for a ball that drags on a specific fluid. After all, experimentally, the ball must reach a well-known terminal velocity. The other branch of solution behaves as expected, the time-dependent fuzzy velocity converges to a well-known terminal velocity; meaning that, the diameter of the fuzzy velocity converges to terminal velocity. Therefore, we explore several conditions of fuzzy initial velocity and conclude that, for any fuzzy initial velocity, the fuzzy terminal velocity always converges to the classical terminal velocity, which is well known in the literature. We also present the corresponding time-dependent fuzzy acceleration, which becomes null for a sufficiently long time.

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Notes

  1. The symbol \(*\) denotes the usual convolution product, that is, given \(f_{1}(t)\) and \(f_{2}(t)\), then \((f_{1} *f_{2})(t)=\int _{0}^{t}f_{1}(\xi )f_{2}(t-\xi )d \xi \).

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Acknowledgements

Onofre Rojas and Sérgio Martins de Souza would like to thank CNPq and FAPEMIG for partial support.

Funding

This research was supported by CNPq (Grant 310497/2018-3) and FAPEMIG (Grant APQ-0302617).

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Authors and Affiliations

  1. Institute of Applied Social Sciences, Federal University of Alfenas, Varginha, Minas Gerais, Brazil

    Silvio Antonio Bueno Salgado, Danilo Machado Pires & Leandro Ferreira

  2. Physics Department, Federal University of Lavras, Lavras, Minas Gerais, Brazil

    Onofre Rojas & Sérgio Martins de Souza

Authors
  1. Silvio Antonio Bueno Salgado
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  2. Onofre Rojas
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  3. Sérgio Martins de Souza
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  4. Danilo Machado Pires
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  5. Leandro Ferreira
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Correspondence to Silvio Antonio Bueno Salgado.

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Communicated by Leonardo Tomazeli Duarte.

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Salgado, S.A.B., Rojas, O., de Souza, S.M. et al. Modeling the linear drag on falling balls via interactive fuzzy initial value problem. Comp. Appl. Math. 41, 43 (2022). https://doi.org/10.1007/s40314-021-01736-8

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  • DOI: https://doi.org/10.1007/s40314-021-01736-8

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