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Mathematics and physics have many applications and in particular are very useful in sport. All sports, such such as athletics, sailing, cycling and skiing, are regulated by laws of physics. For each discipline, study mathematical models can be created with formulas to represent the sporting gestures. To make these models you must first study the laws of physics which apply to the athlete and the environment. From these models we can extrapolate optimum race strategy and technique of our sports. But not only! These models allow us to make "Predictions": and once the initial data is known it can play the action and deduce the final result.

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Now let us see some examples, we will look at some sports and show the relationship between maths and sports. We will look at: Skiing Pole-jumping Javelin Triple jump Sailing Football

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Skiing Looking at this sport, the athlete's system /environment in this case, skier/slope. Any body put on an inclined plane will slide along the line of maximum slope due to the weight force that is given by the mass to the force of gravity. Gravity is the force that pulls us down and therefore without it, it would not be possible to ski. There are also other forces that are to be considered by the force of gravity: the reaction to bind the soil that allows us to keep on going and also to make the curves, and friction.

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The friction is a force that allow us to have control of the skis, because it makes us slow down and without this friction we would continue to accelerate forever! The friction is caused by air resistance and contact with the snow. A competitive athlete looking to go down a slope as fast as possible can reduce friction to a minimum wearing an aerodynamic suit and wax under the skis. Usually a skier goes down a slope in a series of curves. The skier-to-corner / track can be studied from two points of view: that of the skier and the observer.

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In the non-inertial system comes into play the centrifugal force that tends to go straight to the skier. To counter-balance this force the skier leans towards the inside of the curve and thanks to the restraining reaction of the soil and the geometric conformation of the skis these are deformed and grip the ground causing the skier to turn. It is the ski blades that allow you to have the best reaction to bind the soil, so it is important to have them sharpened! In the inertial system instead there is the centripetal force that is equal in magnitude to the centrifugal force but has the opposite direction. This means that the vector points towards the inside of the curve. Now let's see in more detail what happens: For simplicity we begin by considering the skier who goes straight.

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We have said that any body put on an inclined plane sliding along the line of maximum slope due to the weight force P = mg, where m = mass of the body and g = gravity. P can be decomposed into two components, one parallel to the plane (Pt) to slide the body down the slope and one perpendicular to the plane (Pn) which pushes the body towards the floor. Pt and Pn are related to the weight P through the angle a which corresponds to the slope of the inclined plane and we have: Pt = PSEN (a) Pn = PCOS (a) (P) is applied in the center of gravity of the body.

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It is easily seen that if the plane is horizontal ie if = 0°Pt the force that accelerates the body is nothing, but if the plane is vertical, ie maximum Pt is Pt = P. A reasonable intermediate slope for a ski trail is such = 30. As sen30°that is 1/2 one has that Pt = P / 2 or the skier is pulled by a force equal to half of its weight. N is the reaction to bind the soil and prevents us from sinking because it is equal and opposite to Pn (Third Law). Friction is a force opposite to the movement exerted by two bodies in contact, and as we have seen the air resistance and friction force. The air resistance depends on the shape of the body, from the material with which it is made and the speed of the body.

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More precisely, we have that: Fr = CKrv2, where C = coefficient related to the shape and then in our case C depends on the position, Kr = friction coefficient which depends on the material of the contact surface that is material-air suit and v = speed skier. It may be noted that Fr varies as v2 which means that the air resistance increases as the speed increases. Moreover v = (Fr / CKR) 1/2 ie the speed increases, the smaller the coefficients C and Kr, which is why according to "egg" and the jumpsuit race you go faster! = 0°

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The friction force instead depends on the perpendicular component of the weight Pn and the coefficient of friction by contact-snow ski Ka. We have that F = PnKa. To minimize friction, reduce Ka then waxed ski is faster by having less friction. The equation of motion of the skier is going straight: but = P + N + Br + F as Fn =-N ma = Pt + Fr + Fa

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Now we turn to the system and start from sciatore-che- curva/pista inertial system. Suppose that the skier falls along an arc of a curve of radius R at a constant speed v. The centrifugal force acting on the skier is: Fcf = mv2 / R In the non-inertial system the resultant of all forces acting on the skier is zero: Fcf + P + N + Br + F = 0 In the inertial component of the binding reaction opposite to the FCF and the centripetal acceleration acp. acp = v2 / R The equation of motion in this system is: macp = P + N + Br + F

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Pole-jumping Mathematics is a discipline that not only has implications for the theoretical but also the practical side of sport and we can find its use also in the pole vault. In fact, even if it is a purely sporting discipline, maths finds its highest expression along with physics. In fact, the discipline of the pole vault has three phases: 1)Jack of speed (run) 2)Achievement of the slider placed at a certain height from the ground using a flexible rod as a lever 3)Exceeded and fallen beyond the athlete of the slider.

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The mathematics in this case is present in the 2 and 3 phase, in fact in the 2 phase an important role of mathematics is represented by the right angle of inclination generated from the auction flexible with the ground to ensure the necessary thrust and elevation. While in last phase, to determine an analytical component of mathematics during the fall as the athlete performs a parabolic motion, forward, including through the use of physics, we can calculate the approximate speed at which the athlete reaches the mattress.

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Javelin The same principle applied to the previous sport, is also found in the javelin throw. In both sports the bodies move with a parabolic motion, the former is represented by the fall of the athlete, but in this case, is executed from the javelin equipped with a pushing force by the athlete. In fact just the use of parabolic motion causes a greater displacement of the javelin and the subsequent victory.

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Triple jump The triple jump or long jump, is another Olympic discipline in which mathematics is of great importance, because it is used to scan the rhythms of travel of the athlete, but also to determine the distance of fall following the parabolic motion.

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Sailing You can always write the physics of the sport (we'll even do it for the pole vault, in the second part of the article), but it is more difficult to talk about the mathematics of sport even if only because it is hidden inside the physics and equations. Mathematics, with physics finds its most concrete expression, otherwise it would be a simple game, an exercise in style, however intellectually challenging. Sailboats that fly over the water like Formula 1 racing cars... It is through the use of mathematics that these exist...

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In fact, thanks to mathematics and its properties used in many breeches of physics, it was possible to make structural changes to the sails allowing you get 100% maximum use of the sail by the biasing force of the wind, but also structural changes that allow the boat to glide smoothly getting on the water faster.

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Football The football can be described with mathematical language and mathematical methods applied with the world of football are as follows: 1.K-REGION 2.K-TEAM 3.K-PASS 4.K- EVENT

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Place of points that each player is able to achieve before all the others. K-Region represents a sort of "area occupied" by a given player at a given instant of the game (each color identifies a team). For a step to be successful, it is necessary that the player who is has the ball kicks the ball inside the K-Region of player receiving the ball. K-REGION

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K-TEAM System capable of processing characteristics of the players and calculate, for each, the performance indices, compatibility, use and gap compared to every event and every role. It is able to automatically calculate the best training holder.

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K-PASS Optimal transition (measures your understanding of the game). The figure shows a K-Pass. The red marker indicates the player who is positioned to receive the pass. The green line represents the optimal trajectory of the transition. The yellow line indicates the optimal movement of the player who needs to receive the ball.

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K- EVENT Sporting event described with the language of mathematics. The figure shows a simplified definition of 1v1 offensive. The blue marker is an attacker (A) holding the ball, the defender a red marker (D). The arrow indicates the direction of attack by A. So that you can talk about 1v1 offensive must satisfy two conditions: 1) A and D are located within a circle of radius r centered on the ball, within which there must be no other players. 2) D is located inside of the angle of X degrees which has vertex A and bisector passing through the center of the opponent's goal.

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Rescina Domenico V D

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Physics 111: Mechanics Lecture 5

Physics 111: Mechanics Lecture 5

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