Some Real Analysis behind Optimization of Projectile Motion

Kantrowitz, Robert; Neumann, Michael

doi:10.1007/s00009-013-0379-5

Cited by 14 publications

(12 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Perhaps somewhat surprisingly, it turns out that the characterization of the optimal angle provided in part (e) of Theorem 1 carries over from the case without air resistance, while most of the other characterizations given in Corollary 4.1 of [12] are bound to fail in the present setting. In the archetypical case of shooting from ground level to ground level in a nonresistive medium, the characterization of the optimal angle in terms of (e) is, of course, clear from the fact that this angle is π/4, a result that dates back to Nicolò Tartaglia and Galileo Galilei; see also [3,5].…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…Theorem 1 may be viewed as the counterpart of Theorem 1.1 and Corollary 4.1 of [12] for the case of projectile motion without air resistance. Perhaps somewhat surprisingly, it turns out that the characterization of the optimal angle provided in part (e) of Theorem 1 carries over from the case without air resistance, while most of the other characterizations given in Corollary 4.1 of [12] are bound to fail in the present setting.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…For related results, special cases, and alternative methods, we refer to [2][3][4][5][6][7]11,12]. Another important issue is, of course, the case of projectile motion under non-linear air resistance, as discussed, for instance, in [8,9].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To address this issue, we need to impose a canonical condition on the impact function which is motivated by the recent discussion of the corresponding simpler optimization problem for motion without air resistance in [11,12]. To formulate this condition, it is beneficial to eliminate the parameter t in (1) and express the flight curves as functions of the variable x.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 3 more Smart Citations

Optimization of projectile motion under linear air resistance

Kantrowitz

Neumann

2015

Rend. Circ. Mat. Palermo

Self Cite

View full text Add to dashboard Cite

This article concerns the problem of maximizing the horizontal range of a projectile that is launched from atop a tower and is subject only to gravity and linear air resistance. Here the surface to which the projectile is launched is represented by a continuous function whose intersection with every flight path is a single point. In this general setting, tools from real analysis need to be employed to gain detailed information about the shape of the graph and the maximum of the distance function. In particular, the article provides geometric insight into the maximal lateral displacement of the projectile and the corresponding optimal launch angle.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 87%

Section: Introduction and Main Resultsmentioning

confidence: 98%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Optimization of projectile motion under linear air resistance

Kantrowitz

Neumann

2015

Rend. Circ. Mat. Palermo

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Optimization of Projectile Motion Under Air Resistance Quadratic in Speed

Kantrowitz

Neumann

2016

Mediterr. J. Math.

View full text Add to dashboard Cite

Some Real Analysis behind Optimization of Projectile Motion

Cited by 14 publications

References 5 publications

Optimization of projectile motion under linear air resistance

Optimization of projectile motion under linear air resistance

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

Optimization of Projectile Motion Under Air Resistance Quadratic in Speed

Contact Info

Product

Resources

About