Trajectory of a body in a resistant medium: an elementary derivation

Borghi, Riccardo

doi:10.1088/0143-0807/34/2/359

Cited by 10 publications

(17 citation statements)

References 25 publications

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“…This trajectory is also derived in an alternative interesting way by Borghi [5]. The two predetermined points on the trajectory are y(r 1 ) = h 1 , y(r 2 ) = h 2 and employing the dimensionless quantities (6), the equations are…”

Section: Linear Dragmentioning

confidence: 99%

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Determining the velocities and angles for a free kick problem

Pakdemi̇rli̇

Aksoy

2015

Can. J. Phys.

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The free kick problem is considered for three distinct cases: (i) no air drag or lift, (ii) linear drag, and (iii) linear drag and lift. For the first case, closed form formulas are derived for the initial velocities and angles. For the second case, two coupled algebraic equations written from the trajectory equation are given and solved numerically for the initial velocities and angles. For the third case, the equations of motion are solved approximately using perturbation techniques assuming the lift coefficient to be small compared to the drag coefficient. Because the time variable cannot be eliminated between the equations, four coupled sets of algebraic equations are solved numerically for the initial velocities and angles. All three results are compared with each other and the influences of drag and lift coefficients on the velocities and angles are outlined. PACS Nos.: 45.10.Hj, 45.20.D, 45.40.Gj. Résumé : Nous étudions la dynamique du coup franc pour trois différents cas : (i) sans trainée ni poussée aérodynamique, (ii) avec trainée linéaire, (iii) avec poussée et trainée en régime linéaire. Dans le premier cas, nous développons des formules analytiques pour différentes valeurs de la vitesse initiale et de l'angle initial. Dans le second cas, nous obtenons deux équations algébriques couplées pour la trajectoire et elles sont solutionnées numériquement pour différentes conditions initiales de vitesse et d'angle. Dans le troisième cas, les équations du mouvement sont solutionnées approximativement par techniques perturbatives en supposant que le coefficient de poussée est faible devant celui de trainée. Puisque la variable de temps ne peut pas être éliminée des équations, quatre équations algébriques couplées sont solutionnées numériquement pour différentes conditions initiales. Nous comparons les trois cas et soulignons l'effet des coefficients de trainée et de poussée. [Traduit par la Rédaction]

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Section: Linear Dragmentioning

confidence: 99%

“…Hu et al [4] expressed solutions for the range and optimal angle of elevation in terms of the Lambert W function. Borghi [5] derived the trajectories for no air resistance, linear drag, and quadratic drag in an interesting direct way.…”

Section: Introductionmentioning

confidence: 99%

Determining the velocities and angles for a free kick problem

Pakdemi̇rli̇

Aksoy

2015

Can. J. Phys.

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“…These formulas are advantageous even for first-year undergraduates. The question of the motion of a projectile in midair has stirred up interest of authors (Cohen et al, 2014;Kantrowitz & Neumann, 2013;Borghi, 2013). For the construction of the separative solutions various methods are application approximate (Benacka, 2010;Vial, 2007;Parker, 1977;Erlichson, 1983 The converse formulas make it possible to carry out an resolvent examination of the motion of a projectile in a medium with resistance in the highway it is done spherical object, quadratic drag force, ula, relative error…”

Section: Introductionmentioning

confidence: 99%

“…This is why the description of the projectile course by means of a single approximate analytical formula under the square mien resistance is of powerful methodological and instructive significance. In (Chudinov, 2002;2013) comparatively harmless approximate analytical formulas have been possess to ponder the motion of the projectile in a medium with a quadratic comfit violence. In this article, these formulas are interest to solve the humanistic problem of maximizing the projectile distance.…”

Section: Introductionmentioning

confidence: 99%

A Case Study on Projectile Motion

Vivek¹,

Rohini.²,

Dhanya.³

2018

IJTSRD

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In this paper, the question of the motion of a projectile thrown at an angle to the horizon is learned. With zero air drag force, the analytic resolution is well understood. The course of the projectile is a parabola. In situations of practical interest, such as jaculatory a ball with the event of the strike of the medium the square resistance law is generally used. In that casing the question likely does not have an true analytic resolution and therefore in most expert publications it is solution numerically. Analytic advance to the resolution of the question are not sufficiently professional. Meanwhile, resolve solutions are very handy, for a honest preparedness to win problems and are expressly costly for a qualitative analysis. That is why the representation of the missile summon with a uncombed rough resolve formula under the quadratic air resistance instant big methodological interest. Lately these formulas have been procure. These formulas bestow us to obtain a finished divisive narration of the problem. This delineation perfect divisive formulas for shape the basic eight parameters of missile movement. Analytical formulas have been infer for the six fundamental official dependences of the statement including the trajectory equality in Cartesian coordinates. Also this description contains the judgment of the optimal throwing angle and maximum frequent of the motion. In the destitution of publicity resistance, all these relations transfer into well-understood formulas of the hypothesis, speculation of the parabolic motion of the projectile. The proposed analytical discharge oppose from other solutions by clearness of formulas, comfortableness of use and high fidelity (relation hallucination is near 1 %). The motion of a baseball is instant as an example. The converse formulas make it possible to carry out an resolvent examination of the motion of a projectile in a medium with resistance in the highway it is done in the casing of no drag.

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“…The problem of the motion of a projectile in midair has aroused interest of authors (Cohen et al, 2014;Kantrowitz & Neumann, 2013;Borghi, 2013). For the construction of the analytical solutions various methods are usedboth the traditional approaches (Benacka, 2010;Vial, 2007;Parker, 1977;Erlichson, 1983;Tan, Frick & Castillo, 1987)], and the modern methods (Yabushita, Yamashita & Tsuboi, 2007).…”

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Description of the Projectile Motion with a Quadratic Drag Force

Chudinov¹

2014

AJS

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In this paper, the problem of the motion of a projectile thrown at an angle to the horizon is studied. With zero air drag force, the analytic solution is well known. The trajectory of the projectile is a parabola. In situations of practical interest, such as throwing a ball with the occurrence of the impact of the medium the quadratic resistance law is usually used. In that case the problem probably does not have an exact analytic solution and therefore in most scientific publications it is solved numerically. Analytic approaches to the solution of the problem are not sufficiently advanced. Meanwhile, analytical solutions are very convenient for a straightforward adaptation to applied problems and are especially useful for a qualitative analysis. That is why the description of the projectile motion with a simple approximate analytical formula under the quadratic air resistance represents great methodological interest. Lately these formulas have been obtained. These formulas allow us to obtain a complete analytical description of the problem. This description includes analytical formulas for determining the basic eight parameters of projectile motion. Analytical formulas have been derived for the six basic functional dependences of the problem, including the trajectory equation in Cartesian coordinates. Also this description includes the determination of the optimum throwing angle and maximum range of the motion. In the absence of air resistance, all these relations turn into well-known formulas of the theory of the parabolic motion of the projectile. The proposed analytical solution differs from other solutions by simplicity of formulas, ease of use and high accuracy (relative error is about 1-2 %). The motion of a baseball is presented as an example. The proposed formulas make it possible to carry out an analytical investigation of the motion of a projectile in a medium with resistance in the way it is done in the case of no drag.

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Trajectory of a body in a resistant medium: an elementary derivation

Cited by 10 publications

References 25 publications

Determining the velocities and angles for a free kick problem

Determining the velocities and angles for a free kick problem

A Case Study on Projectile Motion

Approximate Analytical Description of the Projectile Motion with a Quadratic Drag Force

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