scholarly journals Extension of application field of analytical formulas for the computation of projectile motion in midair

2013 ◽  
Vol 35 (1) ◽  
Author(s):  
Peter Chudinov
Keyword(s):  
Initial Velocity ◽  
Analytical Approach ◽  
Initial Speed ◽  
Air Drag ◽  
Application Field ◽  
Projectile Motion ◽  
Classic Problem ◽  
Analytical Formulas ◽  
Drag Factor ◽  
Launch Angle

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. Application field of the previously obtained approximate analytical formulas has been expanded both in the upward launch angle and in the direction of increase of the initial speed of the projectile. The motion of a baseball is presented as an example. It is shown that in a sufficiently wide ranges of initial velocity and launch angle the relative error in calculating the distance of the ball does not exceed 1%.

scholarly journals Minimum and terminal velocities in projectile motion

2004 ◽  
Vol 26 (2) ◽  
pp. 125-127 ◽  
Author(s):  
E.N. Miranda ◽  
S. Nikolskaya ◽  
R. Riba
Keyword(s):  
Gravitational Field ◽  
Numerical Simulations ◽  
Drag Force ◽  
Initial Velocity ◽  
Terminal Velocity ◽  
Initial Speed ◽  
Projectile Motion ◽  
Minimum Speed ◽  
Minimum Velocity

The motion of a projectile with horizontal initial velocity V0, moving under the action of the gravitational field and a drag force is studied analytically. As it is well known, the projectile reaches a terminal velocity Vterm. There is a curious result concerning the minimum speed Vmin; it turns out that the minimum velocity is lower than the terminal one if V0 > Vterm and is lower than the initial one if V0 < Vterm. These results show that the velocity is not a monotonous function. If the initial speed is not horizontal, there is an angle range where the velocity shows the same behavior mentioned previously. Out of that range, the velocity is a monotonous function. These results comes out from numerical simulations.

Pirates of the Parametric

2011 ◽  
Vol 104 (9) ◽  
pp. 666-674
Author(s):  
Seán P. Madden ◽  
Dean Allison
Keyword(s):  
High School ◽  
High School Students ◽  
Initial Velocity ◽  
Digital Camera ◽  
Initial Speed ◽  
School Students ◽  
Projectile Motion ◽  
Motion Paths

Some children enjoy playing with the spring–loaded, plastic toy cannon that accompanies many model pirate ships. This cannon pivots in such a way that it can be fired at any angle from about −15 degrees to 195 degrees when viewed from a position directly in front of the cannon. The spring provides each fired projectile with approximately the same initial velocity. This toy, together with a digital camera, can be used with high school students to demonstrate the influence of angle on parabolic trajectories and, simultaneously, to explore the underlying parametric equations. In particular, students can discover and verify that the vertices of projectile motion paths for a given initial speed and a range of launch angles trace an ellipse.

scholarly journals Analytical Investigation of Projectile Motion in Midair

2017 ◽  
Vol 13 (4) ◽  
pp. 4919-4926 ◽  
Author(s):  
Peter Chudinov ◽  
Vladimir Eltyshev ◽  
Yuri Barykin
Keyword(s):  
Drag Force ◽  
Analytical Solutions ◽  
Analytical Investigation ◽  
Analytic Approach ◽  
Functional Dependencies ◽  
Elementary Functions ◽  
Air Drag ◽  
Projectile Motion ◽  
Classic Problem ◽  
Wide Range

Here is studied a classic problem of the motion of a projectile thrown at an angle to the horizon. The air drag force is taken into account as the quadratic resistance law. An analytic approach is used for the investigation. Equations of the projectile motion are solved analytically. All the basic functional dependencies of the problem are described by elementary functions. There is no need for to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball and a badminton shuttlecock are presented as examples.

Shape-programming of hyperelastic plates through differential growth: an analytical approach

Soft Matter ◽  
2019 ◽  
Vol 15 (11) ◽  
pp. 2391-2399 ◽  
Author(s):  
Jiong Wang ◽  
Qiongyu Wang ◽  
Hui-Hui Dai ◽  
Ping Du ◽  
Danxian Chen
Keyword(s):  
Plane Strain ◽  
Analytical Approach ◽  
Differential Growth ◽  
Shape Programming ◽  
Analytical Formulas

In this work, we study the plane-strain deformations of hyperelastic plates induced by differential growth, aiming to derive some analytical formulas for 2D shape-programming of hyperelastic plates.

scholarly journals Expression by Calculation of Correlation Between Initial Speed Losses by the Number ff Fires Shot With the 76 Mm, 1982 Model Cannon and Its Interpretation Depending on the Physical Wear by Pulling the Barrel

2018 ◽  
Vol 24 (3) ◽  
pp. 185-190
Author(s):  
Stelian Popescu ◽  
Mircea Vladu ◽  
Florin Ilie
Keyword(s):  
Initial Velocity ◽  
Regression Equation ◽  
Initial Speed ◽  
Real Situation ◽  
The Real ◽  
Inner Diameter ◽  
The Law ◽  
Material Wear

Abstract In the rifled barrels of each type of armament, the material wear of the rifled bore is closely related to the number of fires shot, this being expressed by the variation of the inner diameter of the barrel in the sections where the measurements were made and evidenced by the decreases in the initial velocity, (ΔVo) relative to the number of fires shot (N). As the number of fires shot increases, the initial velocity of the fired projectiles decreases. In the present paper, the regression equation of the initial velocity of the 76 mm, 1982 model cannon was determined and it was emphasized that the differences between the initial velocity determined by the regression calculation (Voir) and the experimentally determined initial velocity (Voi) are very small. Therefore, it can be appreciated that the law of regression of the determined initial velocity reproduces faithfully the real situation of the decrease in the initial velocity when firing with this type of cannon

scholarly journals Analytical approximations of diving-wave imaging in constant-gradient medium

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. S131-S140 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Initial Velocity ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Image Point ◽  
Practical Applications ◽  
Analytical Approximations ◽  
Wave Imaging ◽  
Analytical Formulas ◽  
Constant Gradient ◽  
Gradient Medium

Full-waveform inversion (FWI) in practical applications is currently used to invert the direct arrivals (diving waves, no reflections) using relatively long offsets. This is driven mainly by the high nonlinearity introduced to the inversion problem when reflection data are included, which in some cases require extremely low frequency for convergence. However, analytical insights into diving waves have lagged behind this sudden interest. We use analytical formulas that describe the diving wave’s behavior and traveltime in a constant-gradient medium to develop insights into the traveltime moveout of diving waves and the image (model) point dispersal (residual) when the wrong velocity is used. The explicit formulations that describe these phenomena reveal the high dependence of diving-wave imaging on the gradient and the initial velocity. The analytical image point residual equation can be further used to scan for the best-fit linear velocity model, which is now becoming a common sight as an initial velocity model for FWI. We determined the accuracy and versatility of these analytical formulas through numerical tests.

DAMPING TIME AND STABILITY OF DENSITY FERMION PERTURBATIONS IN THE EXPANDING UNIVERSE

2001 ◽  
Vol 10 (05) ◽  
pp. 663-679 ◽  
Author(s):  
COSTANTINO SIGISMONDI ◽  
SIMONETTA FILIPPI ◽  
REMO RUFFINI ◽  
LUIS ALBERTO SÁNCHEZ
Keyword(s):  
Boltzmann Equation ◽  
Lower Limit ◽  
Linear Approximation ◽  
Analytical Approach ◽  
Method Of Characteristics ◽  
Classic Problem ◽  
Maximum Value ◽  
Free Particles ◽  
Density Perturbations ◽  
Classical Particles

The classic problem of the growth of density perturbations in an expanding Newtonian universe is revisited following the work of Bisnovatyi-Kogan and Zel'dovich. We propose a more general analytical approach: a system of free particles satisfying semidegenerate Fermi–Dirac statistics on the background of an exact expanding solution is examined in the linear approximation. This differs from the corresponding work of Bisnovatyi-Kogan and Zel'dovich where classical particles fulfilling Maxwell–Boltzmann statistics were considered. The solutions of the Boltzmann equation are obtained by the method of characteristics. An expression for the damping time of a decaying solution is discussed and a zone in which free streaming is hampered is found, corresponding to wavelengths less than the Jeans one. In the evolution of the system, due to the decrease of the Jeans length, those perturbations may lead to gravitational collapse. At variance with current opinions, we deduce that perturbations with λ≥λ J Max /1.48 are able to generate structures and the lower limit for substructures mass is M=M J max /(1.48)3≈M J max /3, where M J max is the maximum value of the Jeans mass.

An Analytical Approach to Tooth Friction Losses in Spur and Helical Gears—Influence of Profile Modifications

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
P. Velex ◽  
F. Ville
Keyword(s):  
Numerical Simulations ◽  
Analytical Approach ◽  
Tooth Profile ◽  
Design Stage ◽  
Power Losses ◽  
Helical Gears ◽  
Friction Power ◽  
Wide Range ◽  
Analytical Formulas ◽  
Displacement Based

An original displacement-based formulation of tooth friction power losses in spur and helical gears is established, which can account for the influence of tooth profile modifications. Several analytical formulas are derived enabling friction losses to be easily estimated for a wide range of gears at the design stage. Numerous comparisons with both the classic formulas in the literature and the results of numerical simulations are presented, which confirm the accuracy of the proposed approach.

scholarly journals Analytical construction of the projectile motion trajectory in midair

MOMENTO ◽  
2021 ◽  
pp. 79-96
Author(s):  
Peter Chudinov ◽  
Vladimir Eltyshev ◽  
Yuri Barykin
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Resistance Force ◽  
Tennis Ball ◽  
Projectile Motion ◽  
Classic Problem ◽  
Analytical Construction ◽  
Air Resistance ◽  
Wide Range ◽  
Large Period

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. An analytic approach is mainly applied for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal, that is, they can be used for any initial conditions of throwing. As a limit case of motion, the vertical asymptote formula is obtained.  The value of the vertical asymptote is calculated directly from the initial conditions of motion. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball, a tennis ball, and a shuttlecock of badminton are presented as examples.

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