Pool is a perfect sport that is tested with physics and mathematics. In many universities, professors use this game to explain some concepts about physics. The theory of bounces and spikes is the basis of the game, and sector professionals can perform amazing tricks, sometimes mysterious and magical, and they may not even be aware of the equations behind them, they rely only on experience. On a perfect table (with sides that respond perfectly and have a very smooth surface), friction is almost completely eliminated. The balls must also be perfect and well calibrated. Newton’s laws can be observed here, with theories of conservation of energy. In the event of a collision between balls (or between a ball and a bank), in fact, momentum is conserved, along with angular momentum, thus exchanging spin and creating unexpected and astounding consequences. The balls (as well as the signals) must be well calibrated to be able to perform key strokes. Let’s see the basic physics behind this cool green board game. Balls As can be seen from Figure 1, billiard balls are balls, that is, geometric solids whose distance between all points on the surface from the center of the ball is equal to a constant value (the radius of the sphere). The main feature of spherical bodies is that they are three-dimensional shapes with a minimum surface-volume ratio. This relationship explains why many things in nature, such as water droplets, tend to have a spherical shape. Figure 1: A billiard ball is a perfect ball The second principle of dynamics states that when an object is subjected to a force, its acceleration and velocity vary. If this body meets another, momentum comes into play. For a good approximation, the balls are solid bodies, and the system is linear. Momentum is a property of an object with a given mass moving at a constant velocity and is the product of an object’s velocity multiplied by its mass, according to the equation: According to its very definition, momentum is a constant measure of an object moving at a constant velocity. In other words, an object retains its momentum if no external force intervenes in the system (conservation of momentum), parallel to the kinetic energy of an object moving at a certain speed v1 and having a certain mass m1. The equation is: here, the most decisive value is velocity. Recoil As with other sports, studies and motion detection analysis are nowadays carried out using special high-frequency cameras (up to 10,000 fps). Rebound analysis is performed by observing the behavior of a ball hitting the side of a general pool table, made up of a specific type of elastic rubber with a triangular shape covered with cloth (see Figure 2). Figure 2: The side of the pool table radically changes the bounce of the ball In principle, by pushing the ball along any primary diagonal, after hitting the first bank, a secondary trajectory is determined, which depends on the primary. Upon impact, it is possible to observe the deformation of the bank that accommodates a whole part of the ball. In this way, the ball rotates around this contact surface to be later reflected back on the playing surface. This dynamic shows that the ball is not rejected at a completely reflexive angle to the angle of incidence (see Figure 3). Furthermore, in the moments immediately following the bounce, the ball follows some vertical spins gained along the way that involve a trajectory deflection. In the event of a collision between balls, the rotation is always transmitted from one ball to another. Then, when the latter hits a bank at a certain angle, it exits the bank at a different angle than the angle of incidence, precisely because it has an effect. The Euclidean character (which states that the angle of incidence equals the angle of reflection) is violated by the theory of conservation of angular momentum. Therefore, the purely geometric path is always different from the actual path. The sides of the pool table are flexible and reflect, in principle, the radii of incidence with the paths of reflection, just as a mirror reflects a ray of light. If the ball was, in fact, made of an ideal material and had no rotation or friction, it would determine an exit path exactly identical to the entry path. Fortunately, the reality is quite different and for accurate shooting, there are many factors that come into play: this makes the game even more amazing. The balls on the table can jump, slide, spin or go straight. The ball is constantly changing its rotation and state of motion, also due to the conditions of the environment and the playing surface. There is, in practice, an infallible standard for calculating pool strokes, but there is always a margin for error. Depending on the type and condition of the bank, the trajectory of the ball is subject to a significant deflection after impact. Geometry simply determines the path of the mirror while physics determines the direction, state of motion and rotation. By combining these three components, each pool track is created. Normally, the trajectory does not have a straight but perfectly curved trajectory, the concavity of which changes according to the angle of incidence, the force of the shot, and the rotation. The bank contrasts with the ball’s movement, allowing it to change direction. It features a triangular section and a portion of the ball’s energy is wasted and converted into heat. The bank cannot be considered a true engineering mirror, but rather an indefinable physical obstacle. Immediately after contact, the ball moves a short distance by sliding over the surface. Figure 3: Poolside is not a perfect “mirror” in pool, collision theory is perhaps most important (see Figure 4). The phenomenon of collision can be described as the collision of two bodies or two masses with each other. During this phenomenon, bodies exchange parts of their energy. Part of the kinetic energy is converted into heat and deformation, and the velocity vectors of both bodies change. When two billiard balls collide, the bump is nearly elastic. An elastic collision is one in which the kinetic energy of the system is conserved before and after the collision. Figure 4: Impact between two masses In the diagram, the vectors indicate the direction of motion of the bodies. The normal effect is the straight line of action on which the forces of influence act, passing through the point of contact of the two bodies and perpendicular to the tangent plane. Impacts As can be seen from Figure 5, an impact is the action that is transferred to the ball to initiate the process of lateral rotation (right or left). It is a phenomenon that occurs when the ball does not hit exactly at its center of mass (which corresponds to the geometric center). In most cases, you should not give the ball too much spin, as this can damage the shot due to excessive slip. The balls may have air bubbles inside, so the center of mass has shifted slightly from normal. The consequences of the effects are very complex. When the impact begins, the complex phenomena of friction, rotation, forces, slip, and angular velocities are included, and only a computer can simulate and solve them. Figure 5: Effects make the ball spin with different effects The Beautiful Spot Billiards physics predicts a point on a ball which when hit produces no frictional force between the ball and the pool table (see Figure 6). Knowing this allows for interesting shots to be performed. The following aspects are necessary if you want to know the height of the point: F: the force that the stick exerts on the ball; r: radius of the sphere; G: center of mass of the ball; g: gravitational acceleration on the planet, equal to 9.8 m / s 2; P: the point of contact between the ball and the plane; Fpx: the “x” component of the force acting on the billiard ball, at point P, which is a friction force; Fpy: the ‘y’ component of the force acting on the billiard ball, at point P. The height of hitting the ball so that friction does not develop at point P is calculated by the following simple equation: it is the result of very complex formulas, which include Newton’s second law, general equations of forces, general equations The torque of a rigid body about its center of mass. It doesn’t matter how hard you hit the ball in this mode. At point P, no frictional force will develop. If the stick hits above or below the “h” height, friction is generated as the ball glides across the table. Figure 6: Example of a sweet spot exercise We have a ball of mass M = 0.2 kg and radius R = 4 cm on the green table (see Figure 7). The player hits the ball with the stick in a horizontal direction at a height h of the table, giving it an impulse equal to: Question: Given h = R, determine the speed v0 at which the ball begins to slide on the table immediately after impact and its kinetic energy. Answer: The stick hits the ball on the center of mass (CM) line h = R, and thus does not provide any angular momentum. On the other hand, the ball undergoes a change in its momentum in the horizontal direction which is equal to the pulse of the force: knowing the impulse we can calculate the velocity v0 that occurs immediately after the collision, along with the kinetic energy: Question: Given h <> R, determine the value of h for which the ball begins a pure rolling motion immediately after impact. Answer: For a pure roll from the start, an initial angular velocity of the ball must be provided through the stick. This is only possible if h <> R. By integrating the equation of the ‘z’ component of angular momentum: and given that the initial angular moment is zero, the angular momentum is obtained after impact in the form: with: , the speed of rotation is: and with: you get On: Figure 7: Pool table for practice Conclusions The game of pool offers thousands of different ideas for studying the most important concepts of physics. The force of gravity always determines the static and dynamic behaviors of any action. Just try to imagine what a pool game would be like on another planet. There are many websites that describe fun environments to play pool, obviously 2D. There are also interesting simulators that allow you to monitor parameters such as kinetic energy, potential energy, total energy, and more. One of these is “Billiards”, click here to have a look. Anyway, we all agree that no computer can simulate the excitement that a real pool game can offer. .
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