Download How much does an angle change to launch a soccer ball in its

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
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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
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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.
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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.
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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
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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.
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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
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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
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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 1off 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
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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.
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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.
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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.
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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].
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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].
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Aquinas Catholic College Menai
Appendices:
Appendix 1: Images Of Methods and Testing
Figure 5: Equipment used for testing.
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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
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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
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Aquinas Catholic College Menai
Figure 2: 20m Calculation for Test
Figure 3: 30m Calculation for Test
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Aquinas Catholic College Menai
Figure 4: 40m Calculation for Test
Figure 5: 50m Calculation for Test
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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
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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
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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
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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
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Aquinas Catholic College Menai
Appendix 4: Figures
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Aquinas Catholic College Menai
Log Book:
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