Analytical construction of the projectile motion trajectory  in midair

Chudinov, Peter; Eltyshev, Vladimir; Barykin, Yuri

doi:10.15446/mo.n62.90752

Cited by 8 publications

(32 citation statements)

References 8 publications

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“…( 3) by the Runge-Kutta method of the 4th order. Dotted red lines on the same curves were obtained by using analytical approximations [8]. The complete coincidence of the solid and dotted curves testifies to the high accuracy of the approximations [8].…”

Section: Study Of the Effect Of Square-law Resistance On The Change In Anglesupporting

confidence: 54%

“…Dotted red lines on the same curves were obtained by using analytical approximations [8]. The complete coincidence of the solid and dotted curves testifies to the high accuracy of the approximations [8]. From the analysis of curves 1, 2, 3, 4 it can be seen that with an increase in the resistance coefficient k, angle 2 enlarges.…”

Section: Study Of the Effect Of Square-law Resistance On The Change In Anglesupporting

confidence: 52%

“…It should also be noted the usefulness of the formulas in Ref. [8], which made it possible to significantly reduce the amount of numerical calculations in this research.…”

Section: Discussionmentioning

confidence: 93%

“…(3). The second way makes use of high-precision analytical approximations [8] describing the movement of the projectile. In this research, both methods are used.…”

Section: Study Of the Effect Of Square-law Resistance On The Change In Anglementioning

confidence: 99%

“…( 3) by the Runge-Kutta method of the 4th order. Dotted red lines on the same curves were obtained by using analytical approximations [8].…”

Section: Investigation Of the Effect Of Square-law Resistance On The Change In Anglementioning

confidence: 99%

See 4 more Smart Citations

Some problems of the projectile motion with a square-law resistance

Chudinov¹,

Eltyshev²,

Barykin³

2021

Rev. Mex. Fis. E

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The influence of the force of the quadratic resistance of the medium on the change in some interesting characteristics of the motion of the projectile, which take place when the projectile moves in vacuum, is investigated. Loci are constructed numerically (and partly analytically) that ensures maximization of the arc length of the projectile trajectory and a non-decreasing of the length of the radius-vector. As examples, the motion of a baseball, a tennis ball and a badminton shuttlecock is studied.

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Section: Study Of the Effect Of Square-law Resistance On The Change In Anglesupporting

confidence: 54%

Section: Study Of the Effect Of Square-law Resistance On The Change In Anglesupporting

confidence: 52%

“…It should also be noted the usefulness of the formulas in Ref. [8], which made it possible to significantly reduce the amount of numerical calculations in this research.…”

Section: Discussionmentioning

confidence: 93%

“…(3). The second way makes use of high-precision analytical approximations [8] describing the movement of the projectile. In this research, both methods are used.…”

Section: Study Of the Effect Of Square-law Resistance On The Change In Anglementioning

confidence: 99%

“…( 3) by the Runge-Kutta method of the 4th order. Dotted red lines on the same curves were obtained by using analytical approximations [8].…”

Section: Investigation Of the Effect Of Square-law Resistance On The Change In Anglementioning

confidence: 99%

See 3 more Smart Citations

Some problems of the projectile motion with a square-law resistance

Chudinov¹,

Eltyshev²,

Barykin³

2021

Rev. Mex. Fis. E

Self Cite

View full text Add to dashboard Cite

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A modified linear drag induced deceleration using a transformation of Newton’s second equation of motion

Jobunga,

Odhiambo,

Mugambi

2024

AIP Advances

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The drag force exerted on a body moving in a fluid is empirically known to follow a linear drag law at low Reynolds numbers and a quadratic drag law at very high Reynolds numbers. The major challenge with the empirical description of the drag lies in the uncertainty of the description of the drag coefficient as a function of velocity. If the function is precisely known, it may be possible to find a general formula for the drag, which would, in principle, be applicable for all Reynolds numbers. The general law can consequently be tested using the linear and quadratic drag laws within their validity regimes to confirm its reliability and validity. The current absence of such a general law implies that the prediction of the dynamics of a body moving under drag forces in the intermediate regime of the Reynolds number can only be approximate and inaccurate since it is not guided by any valid law. In this study, we derive a general relation for the drag force, which may be applicable in all the Reynolds number regimes. We achieve this through a simple transformation of Newton’s second equation for uniformly decelerated motion. It turns out that our general relation is a modification of the linear drag law. The effect of the introduced correction term in the dynamics of a projectile is evidently promising, as can be seen in the more realistic results generated.

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Analytical velocity hodograph of projectile motion under polynomial drag

Saldaña

2022

J. Phys.: Conf. Ser.

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Velocity hodographs are of importance in weather analysis. A historically important case is the hodograph in the Kepler problem. Here, we explore the analytical solutions for the velocity hodograph for the projectile trajectories under a polynomial, in velocity, hydrodynamic dissipation. General solutions in terms of hypergeometric functions are found for odd powers in monomial terms. The even powers have analytical hodographs but we do not find a general expression for them. For completeness we include the constant or Coulomb case, as well as its sum with the linear and quadratic cases. Both shown explicit solutions as functions of the angle. Usually the linear case is discussed in textbooks and articles, here we find that more general cases admit analytical solutions, even when the position and time cannot be obtained explicitly and they need of numerical integration. Furthermore, motion with constant and, additional linear or quadratic drag could be used to design new university lab experiments.

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Analytical construction of the projectile motion trajectory in midair

Cited by 8 publications

References 8 publications

Some problems of the projectile motion with a square-law resistance

Some problems of the projectile motion with a square-law resistance

A modified linear drag induced deceleration using a transformation of Newton’s second equation of motion

Analytical velocity hodograph of projectile motion under polynomial drag

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