Vacuum Trajectory Research Articles

Trajectory simulations by the numerical solution of the point-mass equations of motion for 7.62 mm/.308” rifle bullets

BACKGROUND: The understanding of the dynamics of the trajectory is important in ballistics to estimate the values of various flight variables accurately. The paper deals with the study of the fundamental principles of external ballistics, which allows to delve into the trajectory characteristics of the free flight trajectory of seven. 308 caliber bullets by numerically solving the point-mass equations of motion. Numerical solutions were performed by writing scripts in the Python programming language and using the Matplotlib library to plot simulated trajectories.
 AIM: the three aims of the study were to observe the variation of CD with Mach number (Ma) of flight and calculate an average CD for each bullet under consideration. Further, solving the 3-DoF (Degrees-of-Freedom) Point-Mass trajectory equations of motion for the given bullets (along side observing the effects of range winds on the trajectory behaviour as a variable). And finally, solving the flat-fire approximation with analysis of the effects of a crosswind.
 MATERIALS AND METHODS: Simulations of free-flight trajectories of seven different 7.62 mm/.308 rifle bullets (designated B0B6) have been carried out by the numerical solution of the equations of motion. The average drag force coefficients (CD) for B0B6 have been calculated by scaling the variation of CD with the Mach number of flight with reference to the G7 standard projectile. The Point-Mass trajectory model and its Flat-Fire approximation have been studied with and without the effect of range winds. The solutions of the systems of equations have been carried out by writing scripts in the Python programming language.
 RESULTS: It is observed that an increase in the bullet weight and consequently the sectional density lowers the CD. As expected, it is seen that the bullet with the highest drag (B0) has the shortest range and lowest apogee, while lower drag bullets fly further and higher. The crossover of trajectories is observed at ~30 angle of gun elevation, which implies that the maximum range is not achieved when fired at 45, as is the case with vacuum trajectories. Flat-fire approximation of the point-mass model was also solved to observe trajectories and crosswind deflections of the bullets when fired at 5 angles of elevation.
 CONCLUSION: This project presents the numerical solution of equations of motion of the Point-Mass model for a bullet fired from a gun to computationally simulate its trajectory. A group of seven 7.62 mm/.308 rifle bullets were chosen as samples to simulate free-flight trajectories. The programming language Python is well-equipped to carry out numerical solutions of systems of differential equations owing to its library of in-built functions which assists in writing an efficient script and reduces computational load. This method of solution can be applied with suitable modifications in the field of forensic ballistics for the reconstruction of bullet trajectories and to form a conclusion based on the available evidence from a crime scene.

Flight and bounce of spinning sports balls

Standard university or high-school physics teaching material on projectile motion is usually based on Newton's second law in vacuum, neglecting aerodynamics. We present a low-cost experiment for teaching projectile motion using the students' cell phones and sports equipment, which allows the students to test theory and numerical simulation against experimental data in the real world. For a shot put, theoretical predictions assuming projectile motion in vacuum agree with experimentally obtained trajectories in air to within a few centimeters. However, for a table tennis ball, vacuum trajectories can be almost three times as long as experimentally obtained trajectories. An equation of motion including the aerodynamic drag force has no analytic solution, but it is straightforward to integrate numerically for high-school or first-year university students. Accounting for aerodynamic drag substantially improves the match with experimental data for any ball. In a second experiment, balls are shot with spin resulting in curveball trajectories. Numerical simulations including the Magnus force can give accurate predictions of 3D curveball trajectories, both curving according to the normal and the inverse Magnus effect. Balls shot with topspin and backspin are also accurately modelled. Finally, we model the bounce of an arbitrarily spinning ball using linear and angular impulse-momentum theorems and coefficients of restitution in vertical and horizontal directions. We find agreement with experimental data to within centimeters.

A review of recent research into aerodynamics of sport projectiles

A review of aerodynamics research connected to sport projectiles is presented here. The review’s focus is on work conducted in the current millennium, though deference is made to some classic work still invaluable to modern research. Besides serving as a resource for seasoned scientists and engineers, this article is especially geared toward young investigators who are just beginning careers in sport science. Basic and sophisticated methods are discussed, including vacuum physics, air drag, lift, numerical approaches, trajectory analysis, wind tunnels, and computational fluid dynamics. Eighteen sports are discussed with an eye to future research.

Formulation of improved basis sets for the study of polymer dynamics through diffusion theory methods

In this work a new method is proposed for the choice of basis functions in diffusion theory (DT) calculations. This method, named hybrid basis approach (HBA), combines the two previously adopted long time sorting procedure (LTSP) and maximum correlation approximation (MCA) techniques; the first emphasizing contributions from the long time dynamics, the latter being based on the local correlations along the chain. In order to fulfill this task, the HBA procedure employs a first order basis set corresponding to a high order MCA one and generates upper order approximations according to LTSP. A test of the method is made first on a melt of cis-1,4-polyisoprene decamers where HBA and LTSP are compared in terms of efficiency. Both convergence properties and numerical stability are improved by the use of the HBA basis set whose performance is evaluated on local dynamics, by computing the correlation times of selected bond vectors along the chain, and on global ones, through the eigenvalues of the diffusion operator L. Further use of the DT with a HBA basis set has been made on a 71-mer of syndiotactic trans-1,2-polypentadiene in toluene solution, whose dynamical properties have been computed with a high order calculation and compared to the "numerical experiment" provided by the molecular dynamics (MD) simulation in explicit solvent. The necessary equilibrium averages have been obtained by a vacuum trajectory of the chain where solvent effects on conformational properties have been reproduced with a proper screening of the nonbonded interactions, corresponding to a definite value of the mean radius of gyration of the polymer in vacuum. Results show a very good agreement between DT calculations and the MD numerical experiment. This suggests a further use of DT methods with the necessary input quantities obtained by the only knowledge of some experimental values, i.e., the mean radius of gyration of the chain and the viscosity of the solution, and by a suitable vacuum trajectory, with great savings in computational time required. This offers a theoretical bridge between the experimental static and dynamical properties of polymers.

Effect of Solvent on Collective Motions in Globular Protein

Two molecular dynamics simulations on bovine pancreatic trypsin inhibitor (BPTI), have been made, one in vacuum, the other in water, in order to assess the effect of the solvent water on collective motions. Principal component analysis has been performed to determine collective modes, the principal components, which are assumed to behave as effectively independent harmonic oscillators. Projection of the protein's motion in water onto the plane defined by the first two principal components shows a clustering effect in the trajectory, absent in the vacuum trajectory. This is thought to be due to many local minima in the free energy surface caused by solute-solvent interactions. In order to assess the viscous effect of the solvent, friction coefficients for the principal components were determined by analyzing their velocity correlation functions in terms of the Langevin equation for an independent damped oscillator. Consistent with this analysis is that all modes have friction coefficients centered on the value of 47 cm -1 in a range of ± 10 cm -1. With this friction coefficient, all modes of effective frequencies below 23·5 cm -1 display overdamped motion. By assuming the harmonic approximation for the conformational energy surface for BPTI in vacuum to be valid for BPTI in water, and treating each mode as an independent damped oscillator with a friction coefficient of 47 cm -1, the shift to higher frequencies in the water spectrum relative to the vacuum spectrum could be almost exactly reproduced, indicating this shift is due solely to the viscous effect of the solvent. By analyzing the time correlation functions of the first four principal components it is found that they can be very well described as independent damped oscillators each with a friction coefficient of 47 cm -1.

Disaccharide conformational flexibility. II. Molecular dynamics simulations of sucrose

Molecular dynamics simulations have been used to study the motions in vacuum of the disaccharide sucrose. Ensembles of trajectories were calculated for each of the five local minimum energy conformations identified in the adiabatic conformational energy mapping of this molecule. The model sucrose molecules were found to exhibit a variety of motions, although the global minimum energy conformation was found to be dynamically stable, and no transitions away from this structure were observed to occur spontaneously. In all but one of these vacuum trajectories, the intramolecular hydrogen bond between residues was maintained, in accord with recent nmr studies of this molecule in aqueous solution. Considerable flexibility of the furanoid ring was found in the trajectories. No "flips" to the opposite puckering for this ring were found in the simulations starting from the global minimum, although such a transition was observed for a trajectory initiated with one of the higher local minimum energy conformations. Overall, the observed structural fluctuations were consistent with the experimental picture of sucrose as a relatively rigid molecule.

Topological Glueball Contributions to the Cylinder

Possible glueball contributions originated from the overlapping cluster configuration at the cylinder are studied in the dual topological unitarization. Two high-lying vacuum trajectories are obtained. One is the bare pomeron and the other has the same trajectory function as that of the phenomenological P'. It is argued that the result does not correspond to the cOIlventional P+ f scheme. 271 In the liNe expansion of QCD, the graph of the leading nonplanar term has a cylindrical shape with no quark loops. Color confinement hypothesis then leads us to expect the possible existence of glueballs as intermediate states in the q{j annihilation channel of this cylindrical amplitude. 1) Planar loop corrections convert the liNe cylinder into the so-called cylinder in the dual topological unitarization (DTU).2) The typical result of DTU is that the cylinder singu­ larities are obtained just through the cylinder renormalization of the planar poles. 3 ) It is improbable, however, that the insertion of quark loops extinguishes the glueball states which are inherent in the cylindrical topology. This suggests a possibility that the cylinder could generate its own singularity as a consequence of the glueball effect.4)~8) Therefore, it is important to examine whether physically acceptable pictures are realized when the cylinder retains the glueball contribution. The purpose of the present note is to investigate this problem in DTU. Especially, not only the intercept but the slope of the cylinder trajectories is examined in detail. In the following discussion, the glueball effect is intro­ duced through the overlapping cluster configuration in the s-channel cylinder discontinuity. 9) The effective random walk formalism recently proposed by Tan 10) is adopted as a calculative machinery. This formalism is fairly adequate to an investigation of the transverse structure of the multiperipheral (MP) process. Let us briefly introduce the effective random walk formalism. The standard MP integral equation for the reggeon-particle (Rp) amplitude in the J -plane reads

The flavouring of the pomeron and reggeon field theory with thresholds

The flavoured pomeron is introduced into reggeon field theory to show that, in the absence of a secondary vacuum trajectory P′, the critical solution is in fair agreement with experimental data.

The renormalization of the vacuum trajectories in dual unitarization

The pomeron trajectory (identified with the renormalized f-trajectory in dual unitarization) is evaluated for the range −1.0⩽ t⩽1.5. It is found to pass near the physical f-meson. When SU(3) symmetry is broken, the f′-trajectory is also obtained, together with mixing angles and couplings as functions of t. Renormalization effects are found to decrease with increasing t.

The pomeron-f identity and hadronic total cross sections at moderate energy

The topological expansion of the S-matrix implies that the pomeron and f are the same Regge trajectory. We confront this hypothesis with data by successfully fitting σ tot at moderate energies in a Regge model containing only one prominent vacuum trajectory, plus the ϱ and ω. Couplings and intercepts obey the constraints imposed by the topological expansion. The cylinder coupling is found to be 0.16. Existing data on ππ total cross sections are studied via FESR. The pomeron-f identity is consistent with the analysis. The conventional picture is not.

Evidence for a low-lying unrenormalized vacuum trajectory fromNNscattering

The apparently anomalous energy dependences of the $\mathrm{pp}$ and $\mathrm{pn}$ elastic polarizations at low energies are used to motivate the existence of a low-lying vacuum trajectory $\ensuremath{\sigma}$ with intercept ${\ensuremath{\alpha}}_{\ensuremath{\sigma}}<0$ coupling more strongly to baryons than to mesons. The $\ensuremath{\sigma}$ is identified with the medium-range attractive $\mathrm{NN}$ force in one-boson-exchange models at lower energies. We conjecture that the $\ensuremath{\sigma}$ trajectory undergoes some nondiffractive renormalization at ${p}_{\mathrm{lab}}\ensuremath{\gtrsim}30$ GeV/c in analogy with recent models of the Pomeron. The possible presence of other low-lying trajectories is briefly discussed.

Chiral magnetism (or magnetohadrochironics)

In the infinite-momentum frame a hadron may be viewed as a one-dimensional stringlike structure composed of many constituents. We consider the dynamics of such a system and show how a spontaneous breakdown of chiral symmetry may occur. The effect is very similar to ferromagnetism. We formulate the theory of the distribution of quantum numbers in hadrons and establish the relation between this theory, chiral magnetism, soft-pion theorems, Regge behavior, and duality. In particular we show how the Harari-Gilman-Weinberg theory of chiral representation mixing follows from an approximation analogous to the treatment of a magnetic impurity in a ferromagnetic system. The first stage of approximation, when applied to a quark-parton system leads to a $\overline{q}q$ meson model. We demonstrate the need for spin-orbit coupling in the infinite-momentum quark system. Higher approximations are expected to lead to exotic states. A necessary consequence of our theory of spontaneous chiral-symmetry breakdown is the existence of a Pomeron-like vacuum trajectory with unit intercept. The natural order of magnitude of high-energy meson-meson total cross sections turns out to be ${{f}_{\ensuremath{\pi}}}^{\ensuremath{-}2}=10$ mb, as conjectured by Pagels. A specific model included in an appendix yields ${\ensuremath{\sigma}}_{\mathrm{tot}}(\ensuremath{\pi}\ensuremath{\pi})=2{{f}_{\ensuremath{\pi}}}^{\ensuremath{-}2}$. We do not explicitly deal with strangeness, or S${\mathrm{U}}_{3}$.

Factorization and the couplings of the vacuum trajectory

The decoupling theorems associated with an isolated factorizable pomeron pole of unit intercept are re-examined. It is found that the coupling of three such poles, Γ( t, t, 0), need not vanish, precisely at the point t = 0. This is demonstrated by summing only over states in the appropriate unitarity sum, and sum rule, which are consistent with the M 2, s/ M 2 → ∞ limit. The triple-Regge region then makes a constant contribution to σ total, insteadsb of the ln ln s result obtained if the isolated pole is assumed to couple also to states such that s/ M 2 = constant. The physical implications regarding factorization and the pole-cut relationship are discussed. The relationship between higher order optical theorems (Mueller discontinuities) and particular terms in the unitarity sum for the two → two absorptive part A 22 is exploited. Consistent contributions to the triple-Regge region contribute constant vertex corrections to pure pole behaviour in A 22. There is no cut contribution and the magnitude of the vertex corrections reflects the relative amount of diffractive production. The analysis is extended to multiple fireball production where pure multipole structures emerge. The series naturally terminates if the diffractive component is sufficiently small. The implications for the behaviour of the total cross section at machine energies are discussed.

Inclusive Production of Vector Mesons in the Dual-Resonance Model

The vector-meson inclusive distribution is considered by making use of the eight-point dual amplitude for which two of the external lines are regarded as the decay products of a single vector meson. We find that the vector-meson vertex is factorizable both in the fragmentation and central regions only for large values of $\ensuremath{\kappa}={{p}_{\ensuremath{\perp}}}^{2}+{m}^{2}$. Nonfactorizability for low values of $\ensuremath{\kappa}$ is due to the correlations between the beam-target system and the decay products. Explicit and yet simple expressions for the vertex function as well as the vectormeson production cross section are given to the leading order in $\ensuremath{\kappa}$ for the triple-Regge and pionization limits. They show that the vector-meson decays, in either of the kinematic regions, along the preferred beam direction with a ${cos}^{2}\ensuremath{\beta}$ dependence, in the rest frame of the vector meson, provided that the intercept of the leading vacuum trajectory takes the value unity. Furthermore, it is found that in the central region the states with helicity \ifmmode\pm\else\textpm\fi{}1 contribute dominantly to the decay correlations, while in the triple-Regge region the state with helicity zero is favored.

Spin structure of high-energy multiple scattering

A multichannel K-matrix formalism is used to unitarize the amplitudes for elastic scattering and diffraction dissociation written down in the impact-parameter representation. The inelastic intermediate states in the unitarity relations are effectively represented by a set of quasi-two-particle states. The complications due to nonvanishing spins are considered and the constraints resulting from definite (natural or unnatural) parity in the s- and t-channel are investigated. It is found that the Pomeranchuk contribution can be built up from multiple exchanges of lower lying trajectories. The Pomeranchuk coupling in helicity flip transitions turns out in this model to be logarithmically suppressed with increasing energy compared to the corresponding helicity nonflip couplings of the “vacuum trajectory”. The coupling determining the imaginary parts of the helicity nonflip elastic scattering and diffraction dissociation amplitudes is found to correspond to natural parity exchanged in the t-channel only. Contrary to this the real parts of the amplitudes for helicity nonflip elastic scattering and diffraction dissociation receive in this model at large energies contributions from both natural and unnatural parity exchange in the t-channel. The same is true for the Pomeranchuk contributions to helicity flip elastic scattering and diffraction dissociation processes which are, already in lowest order rescattering, determined by a mixture of natural and unnatural parity exchanged in the t-channel.

Short-range correlations in two-particle productions

Based on the analysis of the eight-point dual amplitude, we show that the two-particle distribution function exhibits a transverse momenta cutoff of the form, exp −4(p 1 ⊥ 2+p 1 ⊥ 2+2p 1 ⊥p 1 ⊥cosβ) , in the pionization region, where β depends on the relative rapidity y and the relative azimuthal angle φ between p 1 ⊥ and p 2 ⊥. Explicit correlation function is obtained to the exp (− y) order. As it is expected from a Mueller analysis, we find the correlation length in y to be ( α V − α V ′) −1 in terms of the two leading vacuum trajectories with positive intercepts.

A new vacuum trajectory for pion-nucleon scattering

It is shown that the asymptotic behaviour of the isospin-zero part A (+) of the invariant amplitude A in πN scattering is determined by a vacuum trajectory with α(0) = −0.47 ± 0.17. With duality arguments and continuous-moment sum rules we confirm the assumption that s-channel helicity is conserved in πN scattering. Because of the smallness of the trajectory, the influence of strong contributions of Δ(1236) and possible complications by wrong-signature nonsense points the slope cannot be satisfactorily determined.

Secondary vacuum trajectories and scalar meson exchange

A new second vacuum trajectory passing throughf*(1515) and σ(548), α(t)=− 0.3+t, is proposed, which, together with αp″(t=0)=− 0.5, overcomes the failure of Igi's original superconvergent πN sum rule and avoids the ghost-state of the conventionalP′-trajectory. The negative residue att=0 confirms the repulsive contribution of a vacuum pole component withJ=0 to the generalizeds-reaction potential, as postulated by Chew (1965). TheP- andP′-poles can build the non resonating background.

A dynamical scheme for leading trajectory and theory of fireballs

The quantum-field theory based on the Bethe-Salpeter equation is applied to investigate properties of the leading vacuum trajectory for the elastic amplitude and to describe high-energy inelastic processes. It is shown that it naturally explains the production of fireballs at very high energies. Parameters of the leading trajectory are related to characteristics of inelastic processes.

Total cross-section forn-particle production in a multi-Regge model

The total cross-section σn(s) forn-particle production in a multi-Regge model is analysed in detail as a function of both multiplicity and total energy, using special techniques for the phase-space integration. It is shown that repeated exchange of the vacuum trajectory, even with finite slope, would lead to a cross-section which increased without bound ass→∞. The application of the results to analysis of experimental data is discussed.