The motion of a point mass in a medium with a square law of drag

Chudinov, Peter

doi:10.1016/s0021-8928(01)00047-8

Cited by 16 publications

(27 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lamb [9] proposed an approximation of the trajectory y(x), whereas Chudinov gave a different approximation of the range x 0 [10]. our model with these previous studies.…”

Section: (C) Two Different Regimesmentioning

confidence: 68%

“…Owing to its application in the military context [5] and more recently in sports [6], this ballistic problem has also been studied in detail and geometrical constructions [7], numerical solutions [8] and theoretical discussions [9,10] have been proposed to account for the trajectory. One of the more famous works on the subject is probably the 'dialogues concerning the two new sciences' [2] first published in 1638, one century after Tartaglia.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The aerodynamic wall

et al. 2014

View full text Add to dashboard Cite

We study the trajectory of dense projectiles subjected to gravity and drag at large Reynolds number. We show that two types of trajectories can be observed: if the initial velocity is smaller than the terminal velocity of free fall, we observe the classical Galilean parabola: if it is larger, the projectile decelerates with an asymmetric trajectory first drawn by Tartaglia, which ends with a nearly vertical fall, as if a wall impeded the movement. This regime is often observed in sports, fireworks, watering, etc. and we study its main characteristics.

show abstract

“…Lamb [9] proposed an approximation of the trajectory y(x), whereas Chudinov gave a different approximation of the range x 0 [10]. our model with these previous studies.…”

Section: (C) Two Different Regimesmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

The aerodynamic wall

et al. 2014

View full text Add to dashboard Cite

show abstract

“…Another aspect we are willing to emphasize is that the solution of several problems in mechanics often does not require the knowledge of the complete solution of Newton's equations in the time domain, but rather the overall mathematical complexity of an apparent nontrivial problem can be considerably reduced by resorting to some "tricks" with which the student should become familiar to as soon as possible. 1 Among them the use of the so-called "chain rule" for derivatives appears to be a technical instrument of invaluable help to allow nontrivial problems to be addressed also at an elementary level of treatment. 2 To help the student to familiarize with this technique, the classical parabolic trajectory in vacuum is re-derived without touching the temporal laws of motion, but rather by removing the temporal variable from the Newton's law, written in the "natural" Cartesian reference frame, through the use of the chain rule.…”

Section: Introductionmentioning

confidence: 99%

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

Borghi¹

2013

Eur. J. Phys.

View full text Add to dashboard Cite

A didactical exposition of the classical problem of the trajectory determination of a body, subject to the gravity in a resistant medium, is proposed. Our revisitation is aimed at showing a derivation of the problem solution which should be as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates. A central role in our analysis is played by the so-called "chain rule" for derivatives, which is systematically used to remove the temporal variable from Newton's law in order to derive the differential equation of the Cartesian representation of the trajectory, with a considerable reduction of the overall mathematical complexity. In particular, for a resistant medium exerting a force quadratic with respect to the velocity our approach leads to the differential equation of the trajectory, which allows its Taylor series expansion to be derived in an easy way. A numerical comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented.

show abstract

“…We focus here on the role of the ball range, that is, the maximal distance that the ball can travel in one shot. Due to its application in the military context [2], and more recently in sports [3], this ballistic problem has been studied in detail for a long time, and geometrical constructions [4], numerical solutions [5] and theoretical discussions [6,7] have been proposed for approaching the actual trajectory.One famous early work on the subject is 'Dialogues Concerning Two New Sciences' [8] published in 1638 by Galileo, one century after 'triangular trajectories' were reported by Tartaglia [9, 10]. Here, we revisit the problem with a special focus on the maximal range and then study its correlation with the size of sports fields.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

On the size of sports fields

et al. 2014

View full text Add to dashboard Cite

The size of sports fields considerably varies from a few meters for table tennis to hundreds of meters for golf. We first show that this size is mainly fixed by the range of the projectile, that is, by the aerodynamic properties of the ball (mass, surface, drag coefficient) and its maximal velocity in the game. This allows us to propose general classifications for sports played with a ball.One could think that the size of a sports field is a function of the number of players, of the rules, of the ball shape, or of the way of launching the ball [1]. We focus here on the role of the ball range, that is, the maximal distance that the ball can travel in one shot. Due to its application in the military context [2], and more recently in sports [3], this ballistic problem has been studied in detail for a long time, and geometrical constructions [4], numerical solutions [5] and theoretical discussions [6,7] have been proposed for approaching the actual trajectory.One famous early work on the subject is 'Dialogues Concerning Two New Sciences' [8] published in 1638 by Galileo, one century after 'triangular trajectories' were reported by Tartaglia [9, 10]. Here, we revisit the problem with a special focus on the maximal range and then study its correlation with the size of sports fields. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 033039 B D Texier et al 2 New J. Phys. 16 (2014) 033039 B D Texier et al 6 Figure 2. Correlation between the size of the sports field L field for the different sports and the associated ballʼs maximum range, x max . The data are extracted from table 1. The solid line represents the equality between those two distances. New J. Phys. 16 (2014) 033039 B D Texier et al 9

show abstract

The motion of a point mass in a medium with a square law of drag

Cited by 16 publications

References 1 publication

The aerodynamic wall

The aerodynamic wall

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

On the size of sports fields

Contact Info

Product

Resources

About