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Zane Assen Aquinas Catholic College Menai How much does an angle change to launch a soccer ball in its parabola from increments of 10 metres? Written By: Zane Assen Age: 12 Years School: Aquinas Catholic College Menai Grade: 7 Key Words: Parabola, Projectile Motion, Speed, Distance, Time, Angle, Height, Newton's First Law of Motion, Air Resistance, Altitude, Gravity, Speed, Force, Directrix, Newton's Second Law of Motion. Table of Contents Introduction........................................................................................................................................1 Planning...............................................................................................................................................3 Literature Review..........................................................................................................3 Purpose & Aim..............................................................................................................4 Hypothesis.....................................................................................................................4 Materials & Methods.........................................................................................................................5 Materials.......................................................................................................................5 Using the JUGS Ball Machine......................................................................................5 Results..................................................................................................................................................6 10 Metres.....................................................................................................................6 20 Metres......................................................................................................................6 30 Metres......................................................................................................................6 40 Metres......................................................................................................................6 50 Metres......................................................................................................................6 Discussion............................................................................................................................................7 Conclusion.........................................................................................................................................10 Acknowledgement.............................................................................................................................11 References.........................................................................................................................................12 Appendices........................................................................................................................................14 Log Book............................................................................................................................................26 I Zane Assen Aquinas Catholic College Menai Introduction: Parabolas are special arcs which display the flight of a projectile with gravity acting against it. Parabolas have the same length from any given point on the arc to the focus, the point where the arc curves around, and the directrix, the straight line outside the arc with both the arc and focus in its boundaries. Parabolas are useful for viewing the way an object is projected in this arc and an insight to the path of any object's flight. The sport of soccer is subject to numerous forces and parabolic arcs, which show how projectiles move in accordance with gravity, wind resistance and the launching force. The main element of soccer exposed to the most forces is the ball. The prime force experienced by a ball is from a foot impacting on the ball at an angle, sending the ball in a parabolic arc of motion. The arc of the ball with the speeds and angles are seen through Newton's first law of motion, the law of inertia. This predicts the behavior of objects, in this case the ball, when all forces applied are balanced. This causes an object to stay at rest or continue moving with an acceleration of 0 m/s². This is also known as an equilibrium. It can be assumed that objects at equilibrium will not accelerate e.g. a ball at rest. Figure 1: Flow chart of Newton's First Law of Motion 1 Zane Assen Aquinas Catholic College Menai Examples of angles in soccer are demonstrated through Newton’s First Law of Motion. When in flight and decreasing in height, forces are unbalanced when following the parabolic arc. When speed, the initial kicking force and the normal force, the forced that counteracts gravity decreases. This is when gravity overpowers the other forces and completes the parabolic arc when forces remain as unbalanced. When this occurs, the arc ends and the initial force isn’t present, causing the ball to stop. When increasing the kicking angle by kicking the ball higher, the ball will have more time in the air to have larger unbalanced forces, thus having a shorter distance compared to a smaller angled kick. In this experiment, the aim is to acquire added knowledge on how much of an angle you would need to score a goal above the goalkeeper when needed, and to understand the uses and characteristics of parabolas. 2 Zane Assen Aquinas Catholic College Menai Planning: Literature Review: Parabolas are the arc in which a projectile moves. Parabolas contain both a vertical and horizontal velocity. These components are dependent on the speed and direction the ball is kicked to maintain its parabolic arc. When there is an acceleration on the ball, the velocities are unbalanced as forces, reflecting Newton's Second Law of Motion. This is due to the increasing of one force whilst the other remains constant or slowly declines. For example, on Science360, the vertical and horizontal velocities are changing due to gravity's effects on the trajectory of the ball. When the ball begins to follow the path of the parabola, the vertical velocity overpowers the horizontal, causing the ball to go higher rather than farther. When gravity acts on the ball it reaches the apex of the parabola. This means that there is no vertical velocity. When this occurs the speed of the horizontal velocity is at its highest as there is no other velocity counteracting it. Following this, vertical velocity increases under the effect of gravity, pulling the ball back towards the ground. The water parabola experiment by Physikanten shows that the parabolas forces are varied like Newton's second law with acceleration due to unbalanced forces. This is from the force being applied to the water to form the parabola. When this occurs, the water separates as the pull of both forces causes it to form small droplets. This will assist my investigation in demonstrating parabolas and how height and distance can affect the flight of the projectile. This means that the ball won’t form smaller items as it is a whole solid compared to water as a liquid. It also will assist the investigation in the understanding of parabolas and the angle, height and length of a parabola to launch the projectile perfectly to reach the specific target. 3 Zane Assen Aquinas Catholic College Menai Purpose and Aim: The purpose of this experiment and investigation is to understand the angles used when playing soccer and shooting to score a goal above the goalkeepers head. Parabolas, angles, height, speed and distance for a clear shot at goal will become understandable towards soccer players. This will further help strikers to score goals from any distance back from the goal line. The aim of this experiment is to find the angle change when shooting towards the middle at the top of the goal. Hypothesis: This experiment will show the shot angle will be dependent on the distance, height and specific speed to reach the highest point in the goal. After moving back in an increment of 10 metres, it can be estimated that the shot angle will increase approximately less than 10 degrees each time. This means a distance increase would only be found with an increase in the size of the angle. To estimate these listed below, the short distances (10-30m) used trigonometry tables1 to find the altitude or tangent of the ball in flight. 1 See appendix 3 for Trigonometry Tables 4 Zane Assen Aquinas Catholic College Menai Materials and Methods: Materials: - WORKZONE Architect Protractor - Ball Machine/Air Cannon - Pencil - Soccer Oval - Size 5 Certified Soccer Ball - Soccer Goal Using The JUGS Ball Machine: The Ball Machine has to be in a specific condition to use. - Plug the machine into an electrical source - Align the machine with your target - Dry the spinning wheels and ball with towel if wet. - Adjust angle with knob under the machine for correct height, angle, parabola etc. - Push the ball in between the two rotating wheels and release the ball. - Record time and angle with a WORKZONE architect’s protractor and any stopwatch. 5 Zane Assen Aquinas Catholic College Menai Results2: The size 5 certified soccer ball weighed 0.42 kg. This ball was launched from 10 metres (m) away from the goal was projected at 15˚. The average velocity was 14.29 meters per second (m/s) towards the goal. The ball took an average time of 0.7 seconds (s) to reach its target. The size 5 certified soccer ball weighed 0.42 kg. This ball was launched from 20m away from the goal was projected at 20˚. The average velocity was 13.25 m/s towards the goal. The ball took an average time of 1.51 s to reach its target. The size 5 certified soccer ball weighed 0.42 kg. This ball was launched from 30m away from the goal was projected at 25˚. The average velocity was 13.51 m/s towards the goal. The ball took an average time of 2.22 s to reach its target. The size 5 certified soccer ball weighed 0.42 kg. This ball was launched from 40m away from the goal was projected at 30˚. The average velocity was 10.44 m/s towards the goal. The ball took an average time of 3.83 s to reach its target. The size 5 certified soccer ball weighed 0.42 kg. This ball was launched from 50m away from the goal was projected at 40˚. The average velocity was 10.35 m/s towards the goal. The ball took an average time of 4.83 s to reach its target. 2 Refer to Appendix 4 for Results in Tables 6 Zane Assen Aquinas Catholic College Menai Discussion: Isaac Newton’s First Law of Motion states "An object will remain at rest or in uniform motion in a straight line unless acted on by an external force." In these cases, the net of the goal, air resistance and gravity are the external forces of the experiment. Once reaching the apex of the parabola, gravity acts on the ball decreasing most of the initial velocity generated from the machine. Parabolas are the arc displaying the flight path of an object without any other forces than the initial force e.g. kicking the ball and the force of gravity. This arc is only valid when gravity is present and the initial force applied carries the ball or item off the ground. Once this occurs, the ball will continue in its upwards motion until air resistance and gravity can redirect the ball downwards resulting in a parabolic arc. To calculate shot angles that result in the ball following parabolic motion, the tangent triangle can be utilised. The height and length to the specific target when parabolas aren't effective as an external force, the goal's net, and acts on the ball before the ball can slow down and gravity can overcome the other forces. To do so, you need to divide the length of the shot distance by the height. Figure 1: Tangent Triangle for calculations 7 Zane Assen Aquinas Catholic College Menai This displays that the angle being calculated would be the result of the height divided by the distance, then referring to trigonometry tables3 to find the hidden angle. The equation on the previous page is very accurate in comparison to the results. The accuracy for the 10m test for example was 1off the actual measurement. It was clearly observed that when the ball was launched from far distances, there was a smaller velocity as there was more air time. This caused gravity and wind resistance to have a greater effect on the ball and its motion from the ball machine. The ball as expected slowed when bouncing in the 40 and 50m tests as well. In terms of accuracy of the tests, the short distances were easier to maintain accuracy than the larger distances. This was due to parabolas only being found and lasted for the longest time in the larger distance as they were in the air long enough for gravity and wind resistance to take control and lead the ball to a state in which makes it at equilibrium, hence at rest. When bouncing occurred throughout the 40m and 50m tests, this reduced the height of the apex of the parabolas after bouncing, making the parabola smaller after each bounce. Forces that are acting upon the ball are slowed down by the pull of gravity. Smaller sized parabolic arcs are formed from this as the bounce on the ground reduces the balls kinetic energy, thus reducing the height and length of the ball’s flight. When firing the machine, the 40 – 50m tests didn’t reach the target without a bounce. The machine had limits and was unable to launch the ball 40 and 50m to hit the target without the ball bouncing. 3 See Appendix 3 For Trigonometry Tables 8 Zane Assen Aquinas Catholic College Menai It is also essential to notice that speed indicators were incorrect on the machine as when launching the 50m test, we achieved a speed of 10.35 m/s or 37.27 km/h but the machine was suggesting we were launching the ball at 85 km/hr. This was due to differentiating speed indicators of m/s for my calculations and results, with the machine with a more sensitive speed. This was found when placing the results into a table in the appendices. False speed indicators were not displaying the actual speed, but still proved reliable as it stayed at the same speed as the other shots in the tests, from varying distances. When testing, it must be as accurate and valid as possible. For a fair and validated test, repeating the test 3 times is best to find averages and for consistency. 9 Zane Assen Aquinas Catholic College Menai Conclusion: The hypothesis of this experiment stated that the angles will increase by approximately under 10˚ after each change in distance. The hypothesis also stated that this was due to parabolic arcs needing to change to suit height and distance to reach the target. This was proven partially correct by the results of the experiment because the angles mostly upscale by 5˚ between 10 to 30m, and 10 for 40 and 50m. In the 40 and 50m test, the angles upscale by 10˚ to unit so they can reach the far distances when they are not built to fire balls far distances at targets approximately 2.5m high. Fair testing was included in trials as it was clearly validated by experimenting the aim and was very reliable. This was because of consistency throughout testing the parabolas with the ball machine with the same target, angle, speed and distance. Each increment was tested three times to make it consistent. The issues with the testing were that the ball machine was unable to launch the ball 40 and 50m without bouncing to reach the target. Another issue faced was precipitation occurred when testing. This caused testing to be harder to complete in comparison to experimenting in good weather conditions. The testing was harder as the ball slipped out of the machine and was not launching properly due to less friction between the ball and tyres, and the added mass to the ball. Drying the ball and machine were the steps used to overcome this issue. This study can be clearly used by soccer players to learn how angles are important when shooting and that the parabolic arc will need to be at a larger height for more distance. When shooting at a larger height and angle, the ball has more distance than a smaller angle kick. A further and interesting future study might involve the curl of a ball, aiming in different areas, and with different elevations and weather. 10 Zane Assen Aquinas Catholic College Menai Acknowledgement: I would like to thank Nathan Denham from Nathan Denham Goalkeeping in Miranda, and Paul Smith from the Sutherland Sharks Football Club for providing the Jugs™ Soccer Ball Machine and permission for usage of the field. I also appreciate my parents for their transportation to the venue and for supplying sufficient tools for the experiment. I would also like to thank my brother, Jake Leite, for his help to understand parabolas and the structure of a scientific paper. 11 Zane Assen Aquinas Catholic College Menai References: Astro Navigation Demystified. 2016. Survival – Calculating altitude without an angle measuring instrument. | Astro Navigation Demystified. [ONLINE] Available at: https://astronavigationdemystified.com/survival-calculating-altitude-without-an-angle-measuringinstrument/. [Accessed 24 June 2016]. Definition of Parabola. 2016. Definition of Parabola. [ONLINE] Available at: http://www.mathsisfun.com/definitions/parabola.html. [Accessed 21 June 2016]. Fun Outdoor Trigonometry Experiments Part 1. 2016. Fun Outdoor Trigonometry Experiments Part 1. [ONLINE] Available at: https://education-teaching-careers.knoji.com/fun-outdoor-trigonometryexperiments-part-1/. [Accessed 18 June 2016]. Geometry Expressions - Katie Purdy. 2016. Mathematics On The Soccer Field. [ONLINE] Available at: http://www.geometryexpressions.com/downloads/Mathematics%20on%20the%20Soccer%20Field. pdf. [Accessed 3 August 2016]. Instructables.com. 2016. How to Make a Coaxial Air Cannon - All . [ONLINE] Available at: http://www.instructables.com/id/How-to-Make-a-Coaxial-Air-Cannon/. [Accessed 27 June 2016]. Newton's First Law . 2016. Newton's First Law . [ONLINE] Available at: http://www.physicsclassroom.com/class/newtlaws/Lesson-1/Newton-s-First-Law. [Accessed 23 June 2016]. Science360 - Video Library. 2016. Science of NFL Football: Projectile Motion & Parabolas Science360 - Video Library. [ONLINE] Available at: https://science360.gov/obj/tknvideo/fc729ef0-22ee-4f61-bb2a-b6c07685fb02/science-nfl-football-projectile-motion-parabolas. [Accessed 5 July 2016]. 12 Zane Assen Aquinas Catholic College Menai ShareefJackson.com. 2016. Sports Science: Basketball and Parabolas — ShareefJackson.com. [ONLINE] Available at: http://shareefjackson.com/blog/2012/11/8/sports-science-basketball-andparabolas. [Accessed 30 June 2016]. Soft Schools. 2016. Soft Schools - Distance, Speed and Time Triangle. [ONLINE] Available at: http://www.softschools.com/formulas/physics/distance_speed_time_formula/75/. [Accessed 29 June 2016]. TheScienceClassroom - Physics of Soccer. 2016. TheScienceClassroom - Physics of Soccer. [ONLINE] Available at: https://thescienceclassroom.wikispaces.com/Physics+of+Soccer. [Accessed 2 August 2016]. UCSB Science Line . 2016. UCSB Science Line . [ONLINE] Available at: http://scienceline.ucsb.edu/getkey.php?key=2513. [Accessed 18 July 2016]. Water Parabola · Physikanten & Co. 2016. Experiments | Water Parabola · Physikanten & Co. [ONLINE] Available at: http://www.physikanten.de/experiments/water-parabola. [Accessed 26 June 2016]. Web Assign. 2016. Projectile Motion. [ONLINE] Available at: http://www.webassign.net/question_assets/buelemphys1/chapter04/section04dash6.pdf. [Accessed 25 June 2016]. Wiley. 2016. Wiley - Chapter 20 - Trigonometry. [ONLINE] Available at: http://www.wiley.com/legacy/Australia/PageProofs/MQ10_AC_VIC/c20TrigonometryII_WEB.pdf . [Accessed 16 July 2016]. 13 Zane Assen Aquinas Catholic College Menai Appendices: Appendix 1: Images Of Methods and Testing Figure 5: Equipment used for testing. 14 Zane Assen Aquinas Catholic College Menai Appendix 2: Calculations Tangent Equation: Pole =x Shadow x = Tangent Refer To Trigonometry Tables4 to find how many degrees to launch the projectile. Example - 10m Test 2.44/10 = 0.24 (2 Decimal Places) Refer to Trigonometry Tables 0.24 ≈ 14˚ Calculating Averages: (∑ All numbers in the data set) Number of results in the data set Example - Average Time for 10m Tests (0.75 secs + 0.63 secs + 0.73 secs) 3 4 = 0.7 (2 Decimal Places) Refer to Appendix 3 for Trigonometry Tables 15 Zane Assen Aquinas Catholic College Menai Speed Equation: Distance Time = Speed Example - Average Speed of 50m Test 50 4.83 = 10.35 m/s (2 Decimal Places) Parabolas: Figure 1: 10m Calculation for Test 16 Zane Assen Aquinas Catholic College Menai Figure 2: 20m Calculation for Test Figure 3: 30m Calculation for Test 17 Zane Assen Aquinas Catholic College Menai Figure 4: 40m Calculation for Test Figure 5: 50m Calculation for Test 18 Zane Assen Aquinas Catholic College Menai Appendix 3: Trigonometry Table Angle Sine Cosine Tangent 0° 0 1 0 1° 0.01745 0.99985 0.01746 2° 0.03490 0.99939 0.03492 3° 0.05234 0.99863 0.05241 4° 0.06976 0.99756 0.06993 5° 0.08716 0.99619 0.08749 6° 0.10453 0.99452 0.10510 7° 0.12187 0.99255 0.12278 8° 0.13917 0.99027 0.14054 9° 0.15643 0.98769 0.15838 10° 0.17365 0.98481 0.17633 11° 0.19081 0.98163 0.19438 12° 0.20791 0.97815 0.21256 13° 0.22495 0.97437 0.23087 14° 0.24192 0.97030 0.24933 15° 0.25882 0.96593 0.26795 16° 0.27564 0.96126 0.28675 17° 0.29237 0.95630 0.30573 18° 0.30902 0.95106 0.32492 19° 0.32557 0.94552 0.34433 20° 0.34202 0.93969 0.36397 21° 0.35837 0.93358 0.38386 22° 0.37461 0.92718 0.40403 19 Zane Assen Aquinas Catholic College Menai Angle Sine Cosine Tangent 23° 0.39073 0.92050 0.42447 24° 0.40674 0.91355 0.44523 25° 0.42262 0.90631 0.46631 26° 0.43837 0.89879 0.48773 27° 0.45399 0.89101 0.50953 28° 0.46947 0.88295 0.53171 29° 0.48481 0.87462 0.55431 30° 0.5 0.86603 0.57735 31° 0.51504 0.85717 0.60086 32° 0.52992 0.84805 0.62487 33° 0.54464 0.83867 0.64941 34° 0.55919 0.82904 0.67451 35° 0.57358 0.81915 0.70021 36° 0.58779 0.80902 0.72654 37° 0.60182 0.79864 0.75355 38° 0.61566 0.78801 0.78129 39° 0.62932 0.77715 0.80978 40° 0.64279 0.76604 0.83910 41° 0.65606 0.75471 0.86929 42° 0.66913 0.74314 0.90040 43° 0.68200 0.73135 0.93252 44° 0.69466 0.71934 0.96569 45° 0.70711 0.70711 1 46° 0.71934 0.69466 1.03553 20 Zane Assen Aquinas Catholic College Menai Angle Sine Cosine Tangent 47° 0.73135 0.68200 1.07237 48° 0.74314 0.66913 1.11061 49° 0.75471 0.65606 1.15037 50° 0.76604 0.64279 1.19175 51° 0.77715 0.62932 1.23490 52° 0.78801 0.61566 1.27994 53° 0.79864 0.60182 1.32704 54° 0.80902 0.58779 1.37638 55° 0.81915 0.57358 1.42815 56° 0.82904 0.55919 1.48256 57° 0.83867 0.54464 1.53986 58° 0.84805 0.52992 1.60033 59° 0.85717 0.51504 1.66428 60° 0.86603 0.5 1.73205 61° 0.87462 0.48481 1.80405 62° 0.88295 0.46947 1.88073 63° 0.89101 0.45399 1.96261 64° 0.89879 0.43837 2.05030 65° 0.90631 0.42262 2.14451 66° 0.91355 0.40674 2.24604 67° 0.92050 0.39073 2.35585 68° 0.92718 0.37461 2.47509 69° 0.93358 0.35837 2.60509 70° 0.93969 0.34202 2.74748 21 Zane Assen Aquinas Catholic College Menai Angle Sine Cosine Tangent 71° 0.94552 0.32557 2.90421 72° 0.95106 0.30902 3.07768 73° 0.95630 0.29237 3.27085 74° 0.96126 0.27564 3.48741 75° 0.96593 0.25882 3.73205 76° 0.97030 0.24192 4.01078 77° 0.97437 0.22495 4.33148 78° 0.97815 0.20791 4.70463 79° 0.98163 0.19081 5.14455 80° 0.98481 0.17365 5.67128 81° 0.98769 0.15643 6.31375 82° 0.99027 0.13917 7.11537 83° 0.99255 0.12187 8.14435 84° 0.99452 0.10453 9.51436 85° 0.99619 0.08716 11.43005 86° 0.99756 0.06976 14.30067 87° 0.99863 0.05234 19.08114 88° 0.99939 0.03490 28.63625 89° 0.99985 0.01745 57.28996 90° 1 0 Undefined 22 Zane Assen Aquinas Catholic College Menai Appendix 4: Figures 23 Zane Assen Aquinas Catholic College Menai 24 Zane Assen Aquinas Catholic College Menai Log Book: 25