Billiard Ball Research Articles

Freezing in space-time: A functional equation linked with a PDE system

We analyze the functional equation F(x+F(x))=−F(x) and reveal its relationship with a system of partial differential equations arising as the hydrodynamic limit of a system of pinned billiard balls on the line. The system of balls must freeze at some time, i.e., no velocity may change after the freezing time. The terminal velocity and the freezing time profiles play the role of boundary conditions for the PDEs (qua terminal conditions, despite being “initial conditions” from the wave equation perspective pursued here). Solutions to the functional equation provide the link between the freezing time and terminal velocity profiles on the one hand, and the solution to the PDE in the entirety of the space-time domain on the other.

On pinned billiard balls and foldings

On pinned billiard balls and foldings

Magnetically Driven Manipulation of Nonmagnetic Liquid Marbles: Billiards with Liquid Marbles.

Liquid marbles are droplets encapsulated by a layer of hydrophobic nanoparticles and have been extensively employed in digital microfluidics and lab-on-a-chip systems in recent years. In this study, magnetic liquid marbles were used to manipulate nonmagnetic liquid marbles. To achieve this purpose, a ferrofluid liquid marble (FLM) was employed and attracted toward an electromagnet, resulting in an impulse to a water liquid marble (WLM) on its way to the electromagnet. It was observed that the manipulation of the WLM by the FLM was similar to the collision of billiard balls except that the liquid marbles exhibited an inelastic collision. Taking the FLM as the projectile ball and the WLM as the other target balls, one can adjust the displacement and direction of the WLM precisely, similar to an expert billiard player. Firstly, the WLM displacement can be adjusted by altering the liquid marble volumes, the initial distances from the electromagnet, and the coil current. Secondly, the WLM direction can be adjusted by changing the position of the WLM relative to the connecting line between the FLM center and the electromagnet. Results show that when the FLM or WLM volume increases by five times, the WLM shooting distance approximately increases by 200% and decreases by 75%, respectively.

What would have happened? Counterfactuals, hypotheticals and causal judgements.

How do people make causal judgements? In this paper, I show that counterfactual simulations are necessary for explaining causal judgements about events, and that hypotheticals do not suffice. In two experiments, participants viewed video clips of dynamic interactions between billiard balls. In Experiment 1, participants either made hypothetical judgements about whether ball B would go through the gate if ball A were not present in the scene, or counterfactual judgements about whether ball B would have gone through the gate if ball A had not been present. Because the clips featured a block in front of the gate that sometimes moved and sometimes stayed put, hypothetical and counterfactual judgements came apart. A computational model that evaluates hypotheticals and counterfactuals by running noisy physical simulations accurately captured participants' judgements. In Experiment 2, participants judged whether ball A caused ball B to go through the gate. The results showed a tight fit between counterfactual and causal judgements, whereas hypotheticals did not predict causal judgements. I discuss the implications of this work for theories of causality, and for studying the development of counterfactual thinking in children. This article is part of the theme issue 'Thinking about possibilities: mechanisms, ontogeny, functions and phylogeny'.

‘N1+把+N2+V1+在/到+장소(위치)’ 문형과 ‘N1+把+N2+〔V1+R(결과보어)〕’문형의 유사성과 ‘타동성’의 정도성 차이

This paper intends to study the difference in the degree of ‘transitivity’ in the two phrase types through a comparative study of X type and Y type of the ‘把’ construction. First, I would like to discuss the difference between the subjectivization of the ‘Transitivity’ action chain events that appear in the billiard ball chain movement diagrams of the ‘passivity event’ and ‘Transitive event’ events of the X type and Y type. Second, by comparing the degree of ‘Transitivity’ of ‘Agency’, ‘Affectedness’, ‘Kinesis’ of X type and Ytype, it was explained which sentence type showed higher degree of ‘Transitivity’

Introduction to the Special Issue: Pluriversal relationality

Both relationality and separateness are aspects of our everyday lives. How we engage these phenomena hinges on the particular existential assumptions that we take for granted. Within the discipline of International Relations (IR), both relationality and separateness have informed how global politics is studied and practiced. How states and their relations are conceived has, for instance, varied by the distinct degrees of privilege given to separation and interconnection: from notions of completely autonomous units like billiard balls to always emergent phenomena co-constituted through relations. The plurality of trajectories that inform this Special Issue illustrate how much broader the spectrum of relational engagement can be when we are cognisant of the impact of these existential assumptions on forms of life, knowing, and knowledge production in International Relations. By highlighting a spectrum of relational engagement, we raise important questions about the way the various knowledge frames in IR are acknowledged, legitimised, limited, and reproduced.

Collisions of three billiard balls

Newton’s cradle is commonly used to show that a collision of three or more identical balls in a row is one that conserves momentum and kinetic energy. An experiment is described where one billiard ball collides at an angle with another two at rest. The collision also conserves momentum and kinetic energy and has several surprisingly simple outcomes where two of the balls emerge at right angles to each other.

A single collision between three objects cannot be described using only two conservation equations

A collision of one object with two or more objects is relatively complicated in general, but a simple example is provided by Newton’s cradle since all the objects are identical and in line. In the present paper, an experiment is described where a heavy mallet collides head-on with two billiard balls. The two conservation equations indicate that many different outcomes are possible. The experimental results are surprising but can be explained in terms of the relative mass and stiffness of the mallet and the balls.

Motions of a Billiard ball After a cue Stroke

We study the collision between the cue and the ball in the game of billiards. After studying the collision process in detail, we write the (rotational) velocities of the ball and the cue after the collision. We also find the squirt angle of the ball for an oblique collision which represents the deviation of the ball from the intended direction.

Billiard ball vs rubber ball collisions

When two billiard balls collide head-on on a smooth, horizontal surface, momentum and kinetic energy are both conserved to a good approximation. If the experiment is repeated with two rubber balls, then kinetic energy is not conserved and neither is momentum. The latter result can be explained in terms of the external friction forces involved.

An Improved Upper Bound on the Number of Billiard Ball Collisions

We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\log K_+ \leq c(d) n \log n$.

Logical and Physical Reversibility of Conservative Skyrmion Logic

Magnetic skyrmions are nanoscale whirls of magnetism that can be propagated with electrical currents. The repulsion between skyrmions inspires their use for reversible computing based on the elastic billiard ball collisions proposed for conservative logic in 1982. Here we evaluate the logical and physical reversibility of this skyrmion logic paradigm, as well as the limitations that must be addressed before dissipation-free computation can be realized.

Recent Articles

Recent Articles

Collision of two coins on a horizontal surface

Oblique angle collisions of two penny coins on a smooth, horizontal surface were filmed with a video camera to investigate the physics of the collision process. If one of the coins is initially at rest, then the two coins emerge approximately at right angles, as commonly observed in billiard ball collisions and in puck collisions on an air table. The coins actually emerged at an angle less than 90 degrees due to friction between the coins, which also resulted in both coins rotating after the collision. At glancing angles, the friction force was due to sliding friction. At other angles of incidence the coins gripped each other and the friction force was then due to static friction.

Topological deviation of membrane waves as a model of gravitational lens effect

Elastic membranes are often used as didactic demonstration of gravitation from the general relativity perspective. Indeed, trajectories of rolling spheres such as billiard balls influence each other through the deformation their mass print within the membrane tissue as would the space-time curvature of gravity. The analogy is pushed here using membrane waves. Indeed, rules of topology apply to masses and waves. We show through experiments and curved manifold simulations that wave propagation due to topological deviation of a two-dimensional flat fabric membrane, justifies classical analogy with gravitational lenses. This allows revisiting through membrane waves, the famous 1919 Eddington experiment that demonstrated light deviations of stars in the vicinity of the sun. It is our hope that the experiment described here will provide new insight for the testing of geometry effects on wave propagation. Also, inspired by acoustic metamaterials, it is our hope to use a more correctly constructed membrane to observe quantified exchanges of geometric waves with their elementary mesh.

Oblique impact of a hard ball on a soft surface

The impact of a hard ball on a soft surface involves deformation of the surface, with consequent changes in the coefficient of restitution (COR) as the angle of incidence is varied. Experimental results are presented for oblique impacts of a golf ball with grass and carpet surfaces, and the impact of a billiard ball with clay and foam surfaces. It was found that the COR depends on two main factors. One is the usual variation of the COR with impact speed in a direction normal to the surface. The other involves deflection of the ball arising from indentation of the surface. The indentation forms a ramp that acts to project the ball at a steeper angle.

Bounce of a biassed ball

If a smooth ball is dropped vertically without spin on a smooth horizontal surface then one might expect the ball to bounce vertically without spin. If it does not then the centre of mass of the ball does not coincide with its geometric centre. An experiment is described where a billiard ball and a superball are deliberately biassed by adding a small steel ball on one side. The results agree well with theoretical calculations.

Theoretical Chemistry and the Calculation of the Atmospheric State

Theoretical chemists have been actively engaged for some time in processes such as ozone photodissociation, overtone photodissociation in nitric acid, pernitric acid, sulphuric acid, clusters and in small organic acids. The last of these have shown very different behaviours in the gas phase, liquid phase and importantly at the air–water interface in aqueous aerosols. The founder of molecular dynamics, B J Alder, pointed out long ago that hydrodynamic behaviour emerged when the symmetry of a random, thermalised population of hard spheres—billiard balls—was broken by a flux of energetic molecules. Despite this, efforts over two centuries to solve turbulence by finding top-down solutions to the Navier–Stokes equation have failed. It is time for theoretical chemistry to try a bottom-up solution. Gibbs free energy that drives the circulation arises from the entropy difference between the incoming low-entropy beam of visible and ultraviolet photons and the outgoing higher-entropy flux of infrared photons over the whole 4π solid angle. The role of the most energetic molecules with the highest velocities will affect the rovibrational line shapes of water, carbon dioxide and ozone in the far wings, where there is the largest effect on radiative transfer and hence on calculations of atmospheric temperature. The atmospheric state is determined by the interaction of radiation, chemistry and fluid dynamics on the microscopic scale, with propagation through the mesoscale to the macroscale. It will take theoretical chemistry to simulate that accurately. A challenging programme of research for theoretical chemistry is proposed, involving ab initio simulation by molecular dynamics of an air volume, starting in the upper stratosphere. The aim is to obtain scaling exponents for turbulence, providing a physical method for upscaling in numerical models. Turbulence affects chemistry, radiation and fluid dynamics at a fundamental, molecular level and is thus of basic concern to theoretical chemistry as it applies to the atmosphere, which consists of molecules in motion.

Visualizing Fluid Flow Around a Baseball Using Water Instead of Air

The flow of air around a baseball and over the seam acts to slow the ball and to deflect it sideways. Turbulent flow can be visualized, and sideways deflection of the ball can be observed clearly if the ball is dropped in a glass fish tank and filmed with a high-speed camera. Results are presented for a baseball and also for a billiard ball with a metal ring attached to simulate the effect of the baseball seam.

Time-traveling billiard-ball clocks: A quantum model

Closed timelike curves, along which an object can travel to its past, are interesting because they are predicted by general relativity and yet need to be reconciled with quantum mechanics. The authors consider a quantum model of a paradox where a billiard ball can travel along two trajectories: one unperturbed and the other where it can collide with its past.