# Fractional Derivatives and Projectile Motion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Properties of Fractional Calculus, L-Derivative, and Λ-Derivative

_{1}-transform, we convert any function f(x) from the initial space to the intermediate space as follows:

_{2}), F(x) is transformed into F(X) as follows:

_{2}) as

_{1}

^{−1}and Λ

_{2}

^{−1}, we restore the solution F(X) of the equation in Λ-space to f(x) in the initial space. In our case

## 3. Analysis of Projectile Motion with Least Air Resistance via L-Fractional Derivative

_{x}(t) (x-axis velocity of the projectile), and U

_{y}(t) (y-axis velocity of the projectile):

_{x}(0) = U(0) cosφ and U

_{y}(0) = U(0) sinφ, while x(0) = y(0) = 0.

## 4. Analysis of Projectile Motion with Least Air Resistance via Λ-Fractional Derivative

^{2}= 32.2 ft/s

^{2}.

_{1}and Λ

_{2}[27,29]. First, we impose the Λ

_{1}transformation in the initial function, defined as

_{2}transformation on F(t):

_{1}and Λ

_{2}on the Λ-fractional derivative converts it to itself, i.e.,

_{1}and Λ

_{2}at the equations and take the following system of equations:

_{1}and B

_{1}, are the integrations constants to be determined by the initial conditions.

_{2}and B

_{2}are integration constants to be determined by the initial conditions.

_{2}= Β

_{2}= 0 and A

_{1}= υ

_{0}∙ cosθ, B

_{1}= υ

_{0}∙ sinθ.

#### Time of Flight and Range

_{b}.

## 5. Application of Projectile Motion with Least Resistance

_{0i}in Table 1, and $x{\u2019}_{i}$ is the range that responds to the same U

_{0i}for any particular case studied (e.g., L-derivative, Λ-derivative, classical case). The measure d of the experimental case compared to the classical case was found to be 5768.975, which is considered relatively high. The minimum measure of the case we studied with this projectile motion through using the Caputo derivative was found to be 1209.38 for γ = 1.99. Nevertheless, this measure is also quite high.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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U(0) (ft/s) | Range (ft) | Flight Time (s) | φ (Degrees) |
---|---|---|---|

334 | 3189 | 14.4 | 45° |

368 | 3804 | 15.7 | 45° |

400 | 4425 | 17.0 | 45° |

431 | 5049 | 18.2 | 45° |

U(0) (ft/s) | Range (ft) | Flight Time (s) | φ (Degrees) |
---|---|---|---|

334 | 3464.67 | 14.67 | 45° |

368 | 4205.71 | 16.17 | 45° |

400 | 4968.944 | 17.57 | 45° |

431 | 5768.975 | 18.93 | 45° |

**Table 3.**Range and time of flight for various γ, with φ = 45° and g = 32.2 ft/s

^{2}[31] (Caputo derivative).

U_{0}(ft/s) | Range of Projectile (ft) | Time of Flight (s) | ||||
---|---|---|---|---|---|---|

γ = 1.95 | γ = 1.97 | γ = 1.99 | γ = 1.95 | γ = 1.97 | γ = 1.99 | |

334 | 3803.31 | 3659.29 | 3526.79 | 16.1 | 15.5 | 14.9 |

368 | 4640.66 | 4455.55 | 4285.56 | 17.8 | 17.1 | 16.5 |

400 | 5506.94 | 5277.71 | 5067.55 | 19.4 | 18.6 | 17.9 |

431 | 6418.76 | 6141.62 | 5887.89 | 21.1 | 20.2 | 19.3 |

**Table 4.**Range and time of flight for various γ, with φ = 45° and g = 32.2 ft/s

^{2}(L-fractional derivative).

U_{0} (ft/s) | Range of Projectile (ft) | Time of Flight (s) | ||||
---|---|---|---|---|---|---|

γ = 1.95/2 | γ = 1.97/2 | γ = 1.99/2 | γ = 1.95/2 | γ = 1.97/2 | γ = 1.99/2 | |

334 | 2545.7 | 2874.44 | 3249.3 | 12.42 | 13.26 | 14.16 |

368 | 3289.21 | 3506.41 | 3957.91 | 13.68 | 14.6 | 15.64 |

400 | 3662.23 | 4129.17 | 4663.22 | 14.88 | 15.88 | 16.96 |

431 | 4245.81 | 4791.27 | 5413.79 | 16.04 | 17.12 | 18.28 |

**Table 5.**Range and time of flight for various γ, with φ = 45° and g = 32.2 ft/s

^{2}(Λ-fractional derivative).

U_{0} (ft/s) | Range of Projectile (ft) | Time of Flight (s) | ||||
---|---|---|---|---|---|---|

γ = 1.95/2 | γ = 1.97/2 | γ = 1.99/2 | γ = 1.95/2 | γ = 1.97/2 | γ = 1.99/2 | |

334 | 2903.88 | 3112 | 3341 | 12.99 | 13.62 | 14.31 |

368 | 3500.86 | 3762 | 4050 | 14.24 | 14.97 | 15.75 |

400 | 4111.61 | 4429 | 4779 | 15.42 | 16.23 | 17.1 |

431 | 4748.23 | 5125 | 5543 | 16.6 | 17.45 | 18.42 |

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Lazopoulos, A.K.; Karaoulanis, D.
Fractional Derivatives and Projectile Motion. *Axioms* **2021**, *10*, 297.
https://doi.org/10.3390/axioms10040297

**AMA Style**

Lazopoulos AK, Karaoulanis D.
Fractional Derivatives and Projectile Motion. *Axioms*. 2021; 10(4):297.
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**Chicago/Turabian Style**

Lazopoulos, Anastasios K., and Dimitrios Karaoulanis.
2021. "Fractional Derivatives and Projectile Motion" *Axioms* 10, no. 4: 297.
https://doi.org/10.3390/axioms10040297