Today’s approach to golf has changed a lot compared to the past. Many professional players use very complex techniques that involve a lot of physics and mathematics. Proof of this is that many golfers have degrees in physics and approach the game in a very analytical way. Is golf a sport or is it just a matter of physics? Golf is a delicate sport played on a very large open course. It consists in striking a solid ball along a special path, from the starting point (the tee) to an opening in the end area (green), by means of a series of strikes of steel bats different from each other. The player who completes the set of holes (usually 18) with the fewest number of strokes wins the match. Golf is one of the few sports that does not have a uniform course and every course is different. At many prestigious golf tournaments, players bring with them different types of clubs and balls. But some of them were seen with a calculator, compass and protractor, to better understand the direction of the shots and the slopes of the surfaces. The practice was soon banned from golf organizations but many champions are searching for the most scientifically effective way to put a golf ball into the hole. Golf Ball The ball is one of the most important tools in the game of golf. Its features have changed drastically over the years (see Figure 1). The first golf ball was made of solid wood (beech). Then the balls began to be made of leather and filled with feathers. It was surprisingly difficult and expensive. Later, the wood extracted from Gutta Percha, a very common tree in Malaysia, was used. The material is very hard but becomes soft and malleable at the temperature of boiling water. In 1932 a ball size and weight standard was approved according to the following characteristics: Diameter: 1.68 inches (42.67 mm); Weight: 1.62 ounces (45.93 grams). Golf balls have evolved and become faster. They can easily exceed a speed of 320 km / h, which leads to the accumulation of a lot of kinetic energy during their flight. Figure 1: Some golf balls used in different periods of history Today, golf balls are technically advanced. The most important types on the market include the following types: Rolled balls with a fluid core, for high rotation and low distance, suitable for professionals; Balls wrapped with solid core, good spin and good distance; Multilayer balls, consisting of a soft layer, a hard layer and a soft core; The two-piece balls, for good distance and long duration, are especially suitable for players with low technical level. Why does a golf ball have holes (dimples)? The golf ball does not have holes but it does have dimples and allows for better aerodynamics. When smooth balls were used, those that were a little worn out went even further. So scientists began to study the problem. If the ball is perfectly smooth, at high speeds, there will be too high pressure on its front to contrast with the air and a vacuum will be created in the back. A strong flow of air will cause circular movements around the ball, which will significantly slow it down. It cannot travel long distances due to the strong air resistance. On the other hand, dimples reduce vortices and increase lift. The more dimples a ball has, the greater its tendency to fly. But this is at the expense of distance (the higher the ball, the shorter the distance traveled). There are a perfect number of dimples in golf balls, which meet different kinds of playing needs. This number is almost always between 300 and 500 dimples per ball. The simulation in Figure 2 shows the behavior of two different spheres in a wind tunnel. One ball has dimples and the other ball is smooth. Both are not to scale but are drawn larger to be visible. The static and dynamic characteristics of the balls are as follows: Weight: 46 g; Diameter: 43mm; X initial velocity: 20 m/s; Y initial velocity: 20 m/s; Initial ball angle direction: 45°; The initial total velocity of the balls: 28.3 m/s; Initial angular velocity: 60 radians/sec; Vertical wind speed: 140 m/s (0.01 N/(m^2/s)). As can be seen from the simulation, the balls initially have a certain direction due to their initial velocity. Thus, strong headwinds deflect their direction. A whip ball is more aerodynamic than a smooth ball. The theoretical star-shaped ball will be faster. After a few seconds of travel, the whipped ball travels into a larger space. Figure 2: Simulating the trajectory of a smooth ball and one with dimples at the end, the dimples send the ball farther. This behavior is governed by the Magnus effect and Bernoulli’s theory. The Magnus Effect states that an object circulating in a liquid immediately pulls the liquid layer in contact with it, and the latter, in turn, pulls the adjacent layer with it. Numerous layers of fluid form around the rotating body, which rotates in concentric circles. If the movement is completely straight and there is no rotation, then the existing lines will be spaced evenly around the body. A smooth ball thrown at high speed will suffer severe deceleration caused by the impact on the air. High pressure is created in the front while an air vacuum is created in the back. On the other hand, the whipped ball produces less turbulence and can be thrown farther. The swing is the movement of golf made when holding a stick (excluding the putter) to raise the ball and direct it towards the goal, as shown in Figure 3. It is a very difficult movement, since, in a short time, a player must dose many kinds of angles and movements, Which is crucial in the final blow. There are two main components that go into the physics of golf: a good swing speed for the arms and shoulders; Release the wrists at the right moment so that they rotate freely while keeping the stick. Its basic stages are: landing sway finish. The backward motion is a very important motion as the body must be prepared to generate the maximum speed and energy imparted upon impact through the downward swing. It takes into account the player’s physical characteristics, motor skills and club movement. The purpose of this stage is to load the muscles like a spring, collecting the additional kinetic energy that will then allow the ball to be hit with maximum speed and force. A downtrend involves a reversal between ups and downs and the movements and positions assumed in the loading phase are reversed in the launch phase. If you try to hit (not cross) you are increasing the pressure of the hands and stiffening the arms excessively, eliminating the necessary fluidity and control. Finishing the golf shot is not just an aesthetic pose but serves to provide the player with a lot of data regarding the shot taken. This information is provided by the rotation of the feet, the position and rotation of the hips and shoulders and the general balance of the body. Figure 3: The four basic phases of the swing swing, the synergistic action of a large number of muscles is of particular importance, allowing the club to accelerate to more than 160 km / h in a very short time. To achieve the best results, the body should be stretched on itself as quickly as possible. Even if gestures seem easy and accessible to everyone, they are a very complex set of techniques that can only be acquired after years of practice and hard training. All muscles are connected to each other like many links in a chain. The main goal is to transmit the maximum energy of the club as accurately as possible. Pulse of force Now let’s examine the collision between the putter and the golf ball. Given the force that changes over time, affecting the body for a limited period of time: The thrust is a physical vector quantity I: The unit of measurement for the pulse is kg (m/s2) s = kg (m/s) . Applying Newton’s second law, we get: Therefore, the impulse caused by the forces acting on the body in time ∆t is equal to the change in the body’s momentum in period ∆t. In the event of collision, the two interacting bodies exert forces on each other that change rapidly over time (impulsive forces). It is very easy to calculate, for example, the coefficient of the average force acting on the ball in the interval ts, relative to the effect between the putter and the golf ball. We have the following starting data: ball mass (m) = 50 g (weight-force P = 0.5N); initial velocity Vi = 40 m / s; Duration of contact between the club and the ball ∆t = 1 ms. The firing range of this type can reach about 150 meters. The initial momentum of the ball is 0. And finally: in the interval – the motive force acting on the ball is very high and the other external forces present are negligible. Precisely for this reason, this is an impulsive approximation. The golf ball looks solid. But, during impact with a blow, it can deform by up to 2 centimeters. The ball is made of elastic material and follows, to a good approximation, Hooke’s law with an elastic constant of 760,000 N/m. Before computers and 1000 fps cameras, it was nearly impossible to calculate the exact swing speed. Using new measurement techniques, it was found that average swing speed varies greatly from man to woman and from novice to professional. A golf swing should have a balance of power and control, but it is very important to swing the putter as fast as you can and reach the highest speed possible with the head of the putter. Only in this way does the club enjoy enormous energy. The distance the ball travels is completely unrelated to the speed at which the bat swings. The best players have an average swing speed between 170 and 180 km/h. The strength index is a value between 0 and 100 that indicates how hard the golf ball is hit. The power indicator uses the putter’s mass, swing speed, and swing-down time to calculate the momentum (or kinetic energy) transferred to the ball upon impact. As can be seen from Figure 4, the physics of oscillation can be treated as a two-dimensional problem. The following factors come into play in it: “r” is the radius of the golfer’s swing; “P” is the position of the handle at which the golfer takes the club; “G” is the golf club’s center of gravity; “LG” is the distance from “P” to the center of mass “G” of the golf club; “θ” is the angle of swing formed by the arms of the golfer with the vertical position; “α” is the angle that the putter makes with the golfer’s arms. The position of the center of mass can be expressed by the following relations: And using Newton’s second law we can write the general equation for the force in the x direction: where: “sum Fx” is the sum of the forces in the x direction; “m” is the mass of the stick; “aGx” is the acceleration of the center of mass in the x direction with respect to the Earth. Figure 4: Oscillation is a two-dimensional problem. Conclusions Golf professionals combine their traditional training with full computer analysis. Complex calculations and mathematical illustrations are sure to help players throw the ball greater distances and with greater accuracy, thanks to the perfect coordination of movements. When the rotation of the pelvis, shoulders, and arms is correct, each part transfers energy to the next, even the head of the racquet. If you add the correct weight transfer during the swing, the kinetic energy acting on the ball will depend more on the mathematical gesture and not on the player’s pure physical strength. All these results are now possible thanks to the advanced scientific equipment available to the most important golf schools. .

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