1.
Introduction

Football is a sport l am fascinated with and its played with
a total of 22 players on the pitch, each team has 11 players in total. It is
played on a rectangular field with a spherical ball. The field is split into
two halves with a goal at either end of the field. There is one goalkeeper per
team and usually four defenders, four midfielders and two strikers. The game is
generally played for a period of 90 minutes. There are 2 halves, 45 minutes
each, plus a few additional minutes if needed due to any delays within the 2
halves due to injury or decision delays. The game can be extended by 30 minutes
of extra time if the teams are drawn. The main objective is to score as many
goals as possible into the oppositions net. I always saw football as an
extremely simple sport that doesn’t really need mathematical calculations to
make you more successful but the older l got, l realised that wasn’t the case.

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Although penalties are one of the easiest ways of scoring a goal, they should
not be under-estimated.

The dimensions of a pitch varies but then there are limits
within which it must be. The length of the pitch must be between 100 yards (90
meters) and 130 yards (120 meters). And the width of must be between 50 yards
(45 meters) and 100 yards (90 meters). The dimensions vary as each team has the
pitch size that suits their style of football play. The distance between the
penalty spot and the goal is 12 yards (11 meters).  The size of the goal is 8 yards (7.32 meters)
apart making a total of 16 yards (14.64meters).

A diagram demonstrating the pitch is arranged can be seen
below.

Figure 1: Dimensions of a football
pitch

Figure 2 shows conversion rate based on the nine possible
areas a penalty taker can shoot with the percentage of scoring in a goal post. A
goal post is has 9 possible areas which a shot can be made being the top left,
top right, top middle, left, right, middle, bottom left, bottom right and
bottom middle. Figure 2 represents from Euro 2012 all penalties taken in during
the competition with the percentage of goals scored at each possible area in
goal. The 2 best places to shoot a penalty being the bottom left and right.

Figure 2: The
percentages and areas of the goal scored

The objective of my IA is to find the best angle and distance
to score a goal from a penalty with a shot placed on the ground. We will
consider 2 different parts of the goal to shoot the ball at being, first the
bottom right and then the bottom left. It is true that for all mathematical
models including this one, trying to consider all possible physical effects is
quite unrealistic. The modelling process will use trigonometry concepts mostly
related to triangles being trigonometry ratios, Pythagoras theorem and sine or
cosine rules.

2.
Rationale

In our modern day football, the penalty kick is considered
more than a golden opportunity for the penalty taker to increase his/her goal
tally. The penalty taker is unchallenged by any opposing player except the
goalkeeper who stands in between the goal posts. Thus, the penalty taker has an
because penalties in most instances, determine the final outcome of the game. My
Internal Assessment seeks to analyze some variables involved in a penalty kick
and attempt to devise the best angle to kick a penalty to insure a high success
rate. The fundamental variables of a penalty kick are the angle at which the
ball is kicked, distance the ball has to travel to enter the goal post and also
the velocity of the shot. Our analysis will be done using right angled
triangles and trigonometric ratios, also, the sides of the triangle will be
calculated using Pythagoras theorem and Sine Rule. The best angles will be
determined from shots placed in the area of goal at the bottom left and right
which will be difficult for the goalkeeper to reach. For this to be achieved,
we need make some assumptions as well. We will also consider a margin of error
to make our work more realistic. After the best angle and distance is
established, l will pick 5 players from my school to try it out to find a
margin of error.

3.
Methodology

When observing penalty kick takers taking penalties, we
notice that the possibility of them scoring is dependent on their shot being
placed within the framework of the goal line. Therefore it would be reasonable
to assume that the range to score is determined by the angle at which the ball
was kicked with respect to the horizontal and vertical axis of the ball’s path.

Table 1, identifies the physical constants we will encounter.

Table 1: Physical
constant

Physical
constant

Symbol

Value

Width of the goal

W

8 yards

Height of the goal

H

8 feet (ft)

Horizontal distance from center of the goal area to the
penalty spot

L

12 yards

Figure 3: The best
area to shoot the ball on ground

The total length of the goal is post from one end to the
other is 24feet. So from the center to the left or right is 12 feet each. The
best region to place the shot is the place that will be difficult for the
goalkeeper to reach which is the place more closely to the goal post itself. The
average height of a goalkeeper in the English Premier League is 6feet, 3inches.

A goalkeeper with stretched arms will be around 7 feet 2 inches making for more
than half the length from center of goal to the goal post. Therefore the best
distance out of reach from the goalkeeper will be 3feet from the goal post
towards the center line. Therefore making it almost impossible for the goalie
to reach 3feet. Therefore 3feet will be the ideal area to place the shot.

Before we start we should establish our assumptions;

1.
We
ignore grass resistance as a shot on the ground mostly encounter this.

2.
We
ignore a curl on the ball

PASTE DIAGRAM HERE

Figure 4: Angle
and Distance to score

Our diagrams are designed to favour a right kick taker as it
is an established fact that such a player controls and kicks better to the
right. The angle and lines showed are symmetric and therefore can be reflected
about the 36ft line, to represent left footed penalty taker. The distance from
the spot kick to goal is 36feet (12
yards). A right angle
triangle is formed and we can use trigonometry ratios to find Angle ?, which is
the perfect angle to place a penalty on the ground. To get angle ?, we need to
find the angles ? and ?. Then we subtract angle ? from angle ?. Please note all
angles is in degrees.

PASTE DIAGRAMS HERE

Figure 5, figure 6

? = Tan -1
12

36

? = 18.4° (3
significant figures)

= Tan -1 9

36

?
= 14.0° (3 significant figures)

Hence Angle
? = Angle ? – Angle ?

= 18.4°- 14.0°

= 4.4°

The
best distance to score a penalty kick will be found by the angle ? from figure
6. We will then double ? and use Sine rule to find the distance. Figure 7 below
shows the triangle that will help us achieve this.

Total
angles in a triangle is 180°. Thus from figure
6

180° = ? + ?
+ 90°

180° = 18.4° +90° + ?

180° = 108.4° + ?

71.6° = ?

Figure 7

PASTE IMAGE HERE

Sine Rule

Sin36.8 x b = 18ft x sin 71.6

b

b = 28.5feet (3 significant figures)

The distance to necessary to score a penalty kick is
28.5feet.

a.
Comment on result

The findings above informs us
that to take a penalty with a shot on the ground then the best angle within 14.4o at a distance of
28.5feet.

b.
Comparing to real life data

A recent analysis on penalty kicks taken in previous world
cups from 1982 – 1998 shows the percentages of them being scored with the shot
place at the bottom left or right. In most cases the advancement of most teams
during these years especially has been decided on penalty shootout. A greater
part of these penalty kicks were shot to the ground either to the left or
right. The table below shows by year the percentages of penalty shootout scored
with shot placed on the bottom left or right.

Table … Recent analysis on penalty
kicks from previous worldcups.

1982

1986

1990

1994

1996

Average

77%

77%

73%

75%

80%

76%

I got some friends l do football practice with to try the
angle out. The goalkeeper was 6feet and the other was 6feet 5nches along with 2
penalty takers. Each player took 10 penal shots with the aim of placing the
shot at measured 9 feet from the center. I observed that because the goalkeeper
was shorter than the average English Premier League goalie of 6feet 3inches
shots placed less than 9feet were still goals. Hence the height of the goalie inversely
affects the angle range to score. Thus to score a taller goalie the required
angle is lesser than 14.4 o lesser the angle and the shorter the
goalie the required angle is more than 14.4 o.

3.
Limitations

Every form of research has its limitations and mine is not
an exception. My limitation include the limited knowledge l have in physics, as
it is a basic requirement to find that of a penalty shot at the top corner of
the goal post. There are characteristics to consider which due to my current
knowledge in math l cannot analyze. They include the weather, psychology of the
penalty taker and also that of the goalkeeper and grass resistance.

4.
Conclusion

My IA shows in an ideal situation for a penalty taker who
lacks tactical discipline and ability can execute a well-placed penalty shot.

The skill needed is the ability to place the ball at an acute angle far from
the goalkeepers reach to the goal post. Once you get both right, then the shot is
a most certain goal.