Math Behind Penalty Kicks In Soccer

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David Tarry IB Math SL 11/2/2014 Math Behind Penalty Kicks In soccer, the penalty kick is a golden opportunity for the shooter to attain a goal. The kicker is unchallenged by any opposing player except the goalkeeper who stands on the goal-line 12 yards away. This one on one situation allows the player shooting to have a great advantage over the goalkeeper. Using the opportunity to the team’s advantage is of real importance due to the fact that in most instances they determine the outcome of games. This paper analyzes the variables involved in a penalty kick and maps out the best methods to use to ensure a very high success rate. The two fundamental components of a penalty shot are the angle at which the shot is kicked and the velocity of …show more content…

Best region to allow for error on the ground. At this point we have only considered the best shot on the ground. Considering the best shot in the air, ideally the ball would be placed right under the cross bar. FIGURE 3. Best region for a Penalty in the air with error margin. However, when deciding to kick the ball in the air you have to take in count another type of error. When kicking the ball on the ground you only have to worry about horizontal error, but when you add height on the ball vertical error comes in to play. The vertical component tells us that the height of the ball must be at least greater than 6.5 feet to avoid the goal keeper and the horizontal similar to the error on the ground. These two will combine to create a region very difficult if not impossible for a goalkeeper to cover during a penalty shot. Figure 4. Best shot on ground and in the air. sin⁡(θ)=opp/hyp 11ft/√((〖36〗^2+〖11〗^2)) = sin θ → sin θ = 16.99° 16.99° is the best angle on the ground si n⁡(θ)=7/√((〖37.64〗^2+7^2)) → sin θ = …show more content…

The effect of air resistance on the ball would only be minor, but would need a lot of calculations which is why it was left out. We are also going to assume that there is no “curl”, or “bend” on the ball. Using the technique of bending the ball changes the straight line path of the ball making it hard to calculate. Also, we assume that the preparation leading up to the kick is perfect, and that there will be no errors in the initial shooting velocity. There is no error in the initial shooting velocity. All of these diagrams are shown for a right footed player (majority of players are right footed), however the diagrams can just be flipped to find the same numbers for a left footed

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