Abstract

The purpose of this project is to explore and investigate the theory behind a conversion.

In the modern game kicking has become an increasingly important feature. With the defence of the teams often being very evenly matched, or when the weather is so bad that attacking with ‘ball in hand’ is at a minimum, kicking becomes a very good way of gaining points. If a team has a good kicker it will put their opponents under pressure not to infringe the laws of the game (a penalty) in their own half as they know this will be punished by 3 points to the other team. This will then mean the other team have a bigger advantage in the breakdowns etc.

Therefore it is very important for teams to take full advantage of kickable opportunities. This paper looks at how factors affect the placement of the kick with concentration on wind, so that a kicker has a maximised chance of success. It gives practical solutions that can be easily used on the field of play by beginners and professionals alike.

It uses a powerful computer package called maple to help with any heavy lifting, and to plot graphs accurately. Excel is also used to produce graphs and tables of values.

The conclusions are what every kicker from beginner right through to the top level should know about (the top level players are probably making adjustments to their kick placement without really knowing why) to maximise their chance of success.

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Acknowledgements

Many thanks must go to my project tutor Dr Michael McCabe, who has been extremely patient throughout the project. Also helping me to design and build my models. Without him it would have been an extremely uphill task from the outset. Special mention must go to Mr Colin White, who with no reward to himself, allowed me to pester him about various problems when stuck, also for providing help and support in numerous ways.

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Contents

1.0. Introduction…………………………………………………………………………………………..……..………….....41.1. Rugby……………………………………………………………………………………………………..……………..41.4. Problem………………………………………………………………………………………………..……………....7

2.0. Rule of thumb with modification………………………………………………………………..….…….………82.1. Chapter conclusion………………………………………………………………………….……….…………..12

3.0. Kicking with and without drag……………………………………………………………………..……..………133.1 Drag less Case………………………………………………………………………………………..…….……….133.2 Simple Drag…………………………………………………………………………………………..………………163.3 Chapter conclusion………………………………………………………………………..……………..……….17

4.0. Wind models………………………………………………………………………………….………………..………….184.1 Headwind……………………………………………………………………………………….……………..………184.2 Tailwind……………………………………………………………………………………..………………..………..234.3 Crosswind from left to right…………………………………………………………….…………..…………264.4Crosswind from right to Left…………………………………………………………………….……………..294.5 Collection of Models………………………………………………………………………………….…………..314.6 Model for any wind direction………………………………………………….………………….………….324.7 Chapter conclusion…………………………………………………………………………………..…………….33

5.0. Practical Introduction……………………………………………………………………..…………….……………..345.1 Before……………………………………………………………………………………….………….………………..345.2 After……………………………………………………………………………………………………………………….35

6.0. Conclusion and limitations………………………………………………………………..….……………………..37

7.0. Further work………………………………………………………………………………….…………….……………….38

8.0 Bibliography………………………………………………………………………………………………………………….40.9.0 Appendix……………………………………………………………………………………………………………………….41

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1.0 Introduction

Notation:

Throughout the project I will try to keep the notation the same.

S – width of the posts (rugby union I will take this as 5.6m) h – height of the posts to the cross bar (this will be a constant 3m) w – the distance from the posts to the place where the try is scored, the ‘offset’ distance D – Distance from the ‘try line’ to the placement of the kick. V0- initial velocity g – this is gravitational pull and can be assumed to be 9.81 θ0 – launch angle t – flight time (seconds) P – placement of kick Ϛ – wind (speed of which will be noted when used)

Purpose:

The purpose of this project is to create a model for the ‘perfect conversion’. We will try and build a model which has a practical use, and that should influence the success of conversions/penalties in a game of rugby. The model should be able to be used for standard kicks on a still day and also on windy days. The main factors of the model will include; launch angle, launch velocity and ball placement for any given ‘offset’ distance. It will also include a factor to take into account various wind speeds. After the model is complete then we will test it by means of a practical to go some way in seeing if the model was a success.

1.1. Rugby

A try is scored in rugby when a player places the ball in the box created by the ‘try line’ the dead-ball line and the two side-lines, this can be done anywhere in this box (this is worth 5 points). After this, the team that have scored have the chance to ‘convert’ the try for an extra 2 points; here they attempt to kick the ball through the goal posts. They can do this anywhere parallel to where the try is scored, i.e. at a right angle to the try line; I will refer to this throughout the project as the ‘conversion line’ Another chance for points comes from a penalty, this occurs when there is an infringement of the rules, this is worth 3 points. The player must take the kick from anywhere back from the point where the penalty is given; again in a parallel way. The player can only move the ball back, so it may not be possible to kick from the optimum point every time as this may be further forward.There is point on this line which maximises the angle subtended by the goal posts, the larger this angle, the bigger the size of the ‘rectangle’ is for them to aim at. This angle is different for different offset values. Also the further back the ball is taken, the power needed for the trajectory increases - and so the error will increase. The project is all about the perfect place for these two things to be optimized.

Importance of kicking:

Kicking in the modern game has become vital for the team to do well, there are numerous examples where a team has had all the territory and possession, but a few infringements of the laws within

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kickable distance has allowed the other team to win. A good kicker will put pressure on the other team, ensuring that they are careful around the breakdowns etc. so that they don’t give away unnecessary points. This will put them on the back foot and cede further advantage to their opponents. Statistics support this argument, in the 1999 Rugby union world cup there were a total of 20 countries competing, and they played 41 matches, during which 572 place kicks were attempted. Of these 334 were penalties with the remainder being conversion attempts following a try. Out of the points scored, 62% were a result of a kick. (Jackson, 2003, p. 805)

Post/Pitch size:

The goal posts are in the centre of the pitch, the pitch size must be as close as possible to 100m in length and 70m in width. The goal posts themselves are 5.6 meters wide and the horizontal bar is 3 meters off the ground. There are no limitations to the total height of the vertical bars, but the International height for the height of the goal post is 17m. A pitch with all the relevant markings is shown below.

Ball:

It is played with an oval-shaped ball known as a ‘prolate spheroid’ (Lipscombe, 2009) with a maximum diameter of 190mm (12 in) and length of 300mm (24 in).

The Chase:

As soon as the ‘kicker’ starts their run up, the players on the opposing team are allowed to attempt a charge down, they can run from anywhere on the try-line the aim being to intercept or to disrupt the kicker. An average kick run-up is thought to take around 5. It is thought that professional rugby players take around 2 seconds to do the first ten meters and then can run up to 10m/s thereafter. My trajectories will have to clear the obstacle that these ‘chasers’ form.

The kick:

There are many different kicks in rugby each with their own specific uses, including drop-kicks, spiral/torpedo kicks, punts, grubbers and the place kick. I am interested here in the place kick. It is done off a ‘T’ ranging from 20mm to 100mm high. Formerly the rugby ball was placed upright; now however they are leant forward more and more, with the impact being onto the ‘point’ of the ball. This technique makes it very similar to the ‘punt kick’ which is helpful as it will broaden range of studies that I can draw knowledge from. Arguably this method results in greater accuracy as the ball will rotate/tumble in a backspin manner resulting in it cutting through the air like a disc and thus holding it onto the projected line. It is also known to increase the range, with the distance of kicks in the modern game getting longer and longer. This is due to the fact that the ball is harder on the end, and so the deformation that happens on impact, returns to shape quicker than when the ball is kicked using the

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old method of kicking (on the side). The transfer of velocity is greater, with less being absorbed by the ball. (Lipscombe, 2009)

Figure 1.1.1. A Birds-eye view of a rugby pitch.

Longest ever conversion:

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The record for the longest conversion ever is held by Francois Steyn, the ball travelled 68 meters horizontally and is the highest ever kick from a conversion. The initial velocity was 43m/s, and he imparted a massive 1455N on the ball. (This is thought to be around 400N bigger than the average kick).

Drag:

As the rugby balls length is far longer than its width, this complicates the computation of drag, on most conversions – as said above, the ball spins end over end in a backspin manner. This results in the area that’s pushing through the air changing many times during flight. Various studies have shown that the range is decreased by between 24- 34% (Hartschuh, 2002), while others suggest that as soon as the ball drops below 15 m/s the drag increases dramatically – due to friction build up; when the ball is travelling fast the particles around it get swept past, but as it slows down they ‘clog’ up the surfaces thus creating more drag.It is interesting to note here that other types of kicks in rugby such as the ‘torpedo’ or ‘screw’ kicks result in the drag being reduced up to 5 times less than the tumbling kick (Murakami, Kobayashi, & Seo, 2006) this is due to the conical part of the rugby ball screwing through the air like a torpedo, it can also result in interesting swerves which is handy when ‘kicking for touch’

1.2. Problem:

So now my problem becomes clear, I need to;1. Maximise angle of the ball in regards to the try line2. Minimise power and launch angle needed to clear the goal posts3. Include a small discussion on drag4. Incorporate wind into the model.

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2.0. Rule of thumb with modification

Chapter Overview: Calculating the value of ‘D’ to maximise angle for different offset values and so looking at the rule of thumb and the refinement of this model.

A general rule of thumb when taking conversions is to take the ball back the same distance (D) along the conversion line as the offset distance (W). Here we are going to try and refine this model for the use of more experience/accurate kickers. We can create a model to maximise the angle of opportunity, i.e. calculate the size of D which results in the biggest rectangle to aim at for. The problem is similar to a golfing putt. We will ignore the cross bar and drag and just optimise the angle.

Birds-eye ViewThis diagram shows a birds-eye view of the first problem. I need to find the value of ‘D’ which will maximise β. As the triangle is right angled these calculations are relatively simple.

Figure 2.1. Birds-eye view of conversion geometry

Applying simple trigonometry;

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tan γ=¿ WD

∧so γ=tan−1 WD

¿

tan α=¿ W +SD

with α=tan−1 W +SD

¿

And so, as β = α – γ, we have;

β=tan−1 W +SD

−tan−1 WD

It is now a simple problem to write a maple program to calculate this value for various values of W. Appendix 2 shows the code. The graph below is when the offset distance is fixed at 4m, we can see from this that the optimum distance from which to take the conversion is between 6 and 8m back from the try-line. The angle of opportunity drops off very steeply the closer to the try-line you get, in comparison to the gradient on the left which is gentler.

Practical tip: take the ball back slightly further if you’re not sure of the distance.

Figure 2.2. Line graph showing angle subtended by posts for various values of D for a fixed offset distance (4m)

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Figure 2.3. Graph showing angles of opportunity for different values of ‘D’ at various ‘W’ values (with the W values being illustrated on the graph).

The points where these lines are at a maximum are the solutions to the problem. We can now find the points of optimum opportunity by finding the maxima for each value of W. I do this by differentiating β with respect to D, this becomes;

dβdD

= −W +5.6

D2[1+(W +5.6 )2

D2 ]+ W

D2[1+W 2

D2 ]If I then equate this to zero and use maple to solve for D, it becomes;

D=0.2√25∗W 2+140∗W

Using the Binomial Theorem;

D=√W 2+285

W

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¿W √1+ 285 W

¿W [1+ 285W ]

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Which, when W is large;

≈ W [1+

12∗28

5 W ]¿W +2.8

So from this our refined rule of thumb is D=W+2.8, however because W is the distance to the nearest post, as in figure 2.1, we need to add half S to get the correct distance, and with S=5.6, our rule is;

D = W + 5.6

So if a try was scored at an offset distance of 10 meters from the centre of the pitch, the optimum distance along the conversion line to take the kick would be 15.6m.

To check this rule, and to find the rule of thumb for smaller values of we can plot the graph of;

D=0.2√25∗W 2+140∗W

Against the W=D line (the original rule of thumb), the maple code is in appendix 3.

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Figure 2.4. The lines: D=0.2√25∗W 2+140∗W and W=D

And now plotting the differences between the two lines:

Figure 2.5. The difference between my plotted line and the line x=y

We can see that the difference between the two lines starts to tend to 2.5 after the offset value goes beyond 5m so our rule of thumb seems correct. In reality the kicker won’t take a kick closer than 5m to the posts (as he has to clear the crossbar too), so we can ignore these smaller offset distances.

2.1 Chapter conclusion Found optimum distances to maximise angles for values of offset. Found refined rule of thumb

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3.0. Kicking with and without drag

Chapter overview: Discussion of drag, investigating the differences between a drag less and a simple drag case.

3.1. Drag-less case

If we assume a drag less case, and consider a kick from straight in front of the posts, we can then find the minimum launch angles and initial velocities to clear the cross bar.

Figure 3.1.1. Side view of problem

We need two equations here which will generate the optimum velocities and launch angles for a given distance, in this case the landing height will be 3m above the launch height (the height of the cross bar)

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In (White, 2011) these equations are defined to be:

Minimum angle of projection;

θm=ψ+12(90−ψ )

With:

ψ=tan−1 hD

Minimum velocity:vm=√sim

g¿¿

Using Pythagoras to find distance from ‘P’ to the crossbarsim

=√D2+h2

With: g=9.81 θm = minimum angle of projection h = height of crossbar = 3m D = distance from posts vm = minimum velocity

Appendix 4 contains the maple code finding the minimum launch angles and initial velocities for a sequence of distances from posts. (I have used ‘d’ instead of D to be distance from try line as maple doesn’t allow a sequence to be assigned to D).

D Minimum Angle

Minimum Velocity

5 60.48 9.3010 53.35 11.4815 50.65 13.420 49.27 15.0925 48.42 16.6330 47.86 18.0335 47.14 19.3440 46.9 20.5745 46.91 21.7250 46.72 22.82

Figure 3.1.2: Table of minimum launch angles and initial velocities.

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5 10 15 20 25 30 35 40 45 500

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20

30

40

50

60

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Optimum angle to clear cross bar (no drag)

Distance from posts (m)

Optimum angle (degrees)

Figure 3.1.3: Optimum angle to clear the cross bar

The minimum angle is very steep close to the try-line, this is to be as expected as the ball has to clear the bar; and from this close it will have to climb steeply to do so. When D>20m the angle of launch doesn’t seem to change much, it looks like its tending towards 45 degree.

5 10 15 20 25 30 35 40 45 500

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10

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20

25

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Minimum Velocity to clear cross bar (no drag)

Distance from posts (m)

Velocity needed (m/s)

Figure 3.1.4: Minimum velocities for various distances.

The relationship between velocity and the distance the ball travels is much steeper than that of launch angle. If we look at the graph, it requires an initial velocity of 10m/s to go 5m, however if we double the initial velocity, the ball will travel almost 8 times as far. When we include drag, we would expect this gradient to change.

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3.2. Simple Drag - 30% ruleIn (Hartschuh, 2002) it stated that a simple measure of drag is that a ball kicked with the same velocity and launch angle will have a reduction in range of 30%. To model this we can change ‘D’ from the previous equations of θm and vm to 1.3*D. The maple code is in appendix 5. Below are the resulting optimum velocities and angles.

DMinimum Angle

Minimum Velocity

5 65.3 1010 56.6 1315 53 1520 51 1725 49.9 1930 49 2035 48.5 2240 48.1 2345 47.7 2550 47.4 26

Figure 3.2.1: Table of optimum velocities and angles for simple drag.

5 10 15 20 25 30 35 40 45 500

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40

50

60

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Optimum angle to clear goal posts (simple drag)

Distance from posts (m)

Optimum angle (degrees)

Figure 3.2.2: Optimum angles for a range of distances.Again the gradient (and therefore the launch angle) is steeper as we get closer to the posts. However its roughly 5 degrees higher in figure 3.2.2 than in 3.1.3, and it takes longer to tend to 45, this makes sense as the drag has the effect of pulling the ball back down to earth sooner, so it needs the extra elevation to get it the required distance.

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5 10 15 20 25 30 35 40 45 500

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10

15

20

25

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Minimum velocity to clear crossbar (with simple drag)

Distance from posts (m)

Velocity needed (m/s)

Figure 3.2.3: Minimum velocities for a range of distances.The gradient here is steeper, for example, if we kick the ball at 10m/s it will go 5 meters, however if we double this then it will only 6 times as far (instead of 8 when there is no drag). The relationship is not totally linear.

3.3 Chapter conclusion Found minimum angles and velocities to clear posts Found minimum angles and velocities for simple drag Commented on difference

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4.0. Wind Models

Chapter overview: In a game of rugby there is rarely no wind at all, the kickers need to know how different types of wind will affect their kick. If the wind is at their back (tailwind) in which direction should they move the ball to ensure maximum angle of opportunity? We will consider crosswind, tailwind and headwind. If possible give practical advice for kickers, also a general model for any angle of wind.

We will assume that there is no drag and no spin. This will mean that the model won’t be accurate in a mathematical manner, but hopefully it will be sufficient for practical use.

Below is the wind speed explained so we can put the results into perspective:

0.3 – 1.5 m/s = Light air = some drifts gently 1.6 -3.4m/s = Light breeze = leaves rustling 3.5 -5.4m/s = Gentle breeze = leaves moving constantly 5.5 -7.9m/s = Moderate breeze = small branches move 8 -10.7m/s = Fresh breeze = small trees start moving 10.8 – 13.8m/s = Strong breeze = umbrellas difficult to hold, big branches moving

The highlighted condition is the one for which data has been calculated.

4.1. HeadwindThe wind will swerve the ball in a manner as shown below, so we want to find the ‘angle of opportunity’ i.e. how much we have to aim for. We can find this by extending a line parallel to the try-line from our point P (as shown in the figure 4.1.1( and then working out the two angles ϑ1 and ϑ2.

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Figure 4.1.1: Birds-eye view of headwind problem.

If the initial ball velocity of the ball is v0 then the two components as shown above are:

vw=v0 cos (θ0)

vD=v0 sin(θ0)

However as we have the headwind working against the ‘D’ component we have:

vD=v0 sin(θ0)−u

The distance the ball has travelled along the ‘D’ direction is given by: D=(v0 sin (θ0 )−u )∗t

And in the ‘W’ direction:W =v0 cos (θ0 )∗t

Solving these for ‘t’ we have:

t= D(v0sin (θ0 )−u )

∧t= Wv0 cos (θ0 )

Now we can equate them:W

v0 cos (θ0 )= D

(v0 sin (θ0 )−u )

Cross multiplying, this becomes: W (v0sin (ϑ )−u )=D ( v0cos (ϑ ))

Simplified:

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Wsin(ϑ )−Dcos(ϑ )=v0 W

u

If we then choose a point on the pitch where a theoretical try has been scored, to the right of the posts, with W= 10, and D=10; this will allow us to solve for the two values of ϑ. ϑ1 is the angle of projection that will hit the near post, ϑ2 is the angle that will hit the far post, it stands to reason then, that if we do ϑ1 - ϑ2 the angle left over will be any projections that go through the posts. This may be possible by hand, but we can easily write a code to get maple to do the heavy lifting for us. The maple code is in appendix 7.

With ‘H’ denoting headwind we have:

NearpostH : Wvo cosϑ

= Dvo sinϑ−u

For the far post we have the same as the near post, but now the distance that the ball is moving sideways, is W+ (the width of the goal post), which is W+S (from figure 4.1.1)

FarpostH : W +Svo cosϑ

= Dvo sinϑ−u

So angle of opportunity is:

ϑ=NearpostH−FarpostH

So we now have a model with various parameters. We can therefore change one of the parameters while keeping the other ones the same to see the effects of each one separately. We start with head wind and then do the same with tailwind.

The parameters that can be changed within the model are:1. D – which is the distance back along the ‘conversion line’(m)2. u – which is the wind speed (m/s)3. vo – which is initial velocity of kick (m/s) (NB we will need to keep v0 >u, or else the ball wont

move) 4. W – the ‘offset distance’ (m)

We will keep all parameters fixed as follows (unless it is the one being changed): Vo = 20 m/s, D = 10m, W= 10m, u = 10m/s. All the axis will have the same scale to ensure comparing is made easier.

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2 4 6 8 10 12 14 16 18 20 22 24 260

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Angle of opportunity for fixed velocity, offset and headwind

Distance from try-line (m)

Angle (degrees)

Figure 4.1.2 Model created by varying distance from try-line (D)The distance along the conversion line that optimises the angle of opportunity is between 8 and 12m, this is roughly 4.5 meters closer to the try-line than suggested in the rule of thumb (where there is no wind of course) we can see that the angle drops off more steeply the closer to the try-line you get.

Practical Advice: It’s best to err on the side of caution and increase D, as long as you can exert enough power to clear the posts from this spot.

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Angle of opportunity for fixed velocity , offset and distance

Wind speed (m/s)

Angle (degrees)

Figure 4.1.3 Model created by varying wind speed (u)There is a negative correlation between wind speed and angle of opportunity, this is to be expected, as the bigger the headwind the more the path of the ball will deviate over its flight path and so the angle when it approaches the posts will get smaller, and therefore the ‘angle of opportunity’

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5 10 15 20 25 30 35 400

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Angle of opportunity for fixed offset, wind speed and distance

Velocity (m/s)

Angle (degrees)

Figure 4.1.4 Model created by varying Velocity (Vo )We don’t have any data for V0 = 5 or 10, as these are lower than the wind speed. The angle of opportunity does increase in proportion to velocity.

Practical Advice: Kick it as hard as possible without sacrificing accuracy.

0 5 10 15 20 25 30 350

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10

15

20

25

30

35

40

45

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Angle of opportunity for fixed velocity, wind speed and distance

Offset distance (m)

Angle (degrees)

Figure 4.1.5 Model created by varying Offset Distance (W)As expected this graph shows a negative correlation, the angle of opportunity gets smaller as the offset distance grows. It’s not quite a linear relationship, with the difference between 15m and 30m offset being 7 degrees smaller than the 0 to 15 range. It would be interesting to see here if the angle

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would tend towards anything if we were to carry on increasing the offset distance – which there was no point doing here as these values have already hit the touchline. 4.2: Tailwind:

Figure 4.2.1 Birds-eye view of tailwind problem.As we can see the paths of the ball are now bent in the opposite way to what they were in figure 4.4.1.

We can easily change equations NearpostH, and FarpostH by changing it to sin(ϑ) + u (as now we are adding on the wind factor because it’s behind us)

Resulting in:

NearpostT : Wvo cosϑ

= Dvo sin ϑ+u

FarpostT : W +Svo cosϑ

= Dvo sin ϑ+u

ϑ=NearpostT−FarpostT

Where ‘T’ denotes tailwind.

The initial conditions remain the same; Vo = 20 m/s, D = 10m, W= 10m, u = 10m/s

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2 4 6 8 10 12 14 16 18 20 22 24 260

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Angle of opportunity for fixed velocity,offset and tailwind

Distance from try-line (m)

Angle (degrees)

Figure 4.2.2. Model created by varying distance (D)Not only is the optimum distance now higher than in figure 4.1.2 (14-16m instead of 8-12m) it’s also over a meter larger than the rule of thumb. As we are taking the theoretical kick from the right of the posts the angle of opportunity is also much larger – with a maximum of around 17 degrees rather than 8. The gradient of the line - and therefore the angle of opportunity - doesn’t drop off so steeply as we get closer to the try-line meaning there is more room for error.

Practical advice: Take the kick back a bit, especially as the distance of the kick won’t be a problem with the wind behind you.

1 2 3 4 5 6 7 8 9 10 11 120

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Angle of opportunity for fixed velocity, offset and distance

Axis Title

Axis Title

Figure 4.2.3 Model created by varying Changing Wind (u)Unlike figure 4.1.3, there is a positive linear correlation, the angle increases with wind speed. This can be explained as the stronger the wind the more it will blow the trajectory of the ball round so that it approaches the posts at an angle close to the ‘normal line’.

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5 10 15 20 25 30 35 4013

13.5

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14.5

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15.5

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Angle of oppourtunity for fixed offset, wind speed and distance

Velocity (m/s)

Angle (degrees)

Figure 4.2.4 Model created by varying kicking velocity (Vo) Here the lower values of V0 were still calculated as the wind was blowing with the ball rather than against it. This did result in some unusual results as above. The graph suggests that a kicker will have more success if they just ‘guide’ the ball towards the posts and let the wind carry it over. (Incidentally this is what a lot of kicking coaches will tell you to do all of the time to stop you ‘snatching’ at the ball)

0 5 10 15 20 25 30 350

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Angle of opportunity for fixed velocity, wind speed and distance

Offset distance (m)

Angle (degrees)

Figure 4.2.5 Model created by varying offset distance (W) The angle of opportunity goes down as W increases (which is to be expected), it does however start a lot bigger than in figure 4.1.5, and only dips below 10 degrees after W=12.5, whereas it goes below 10 degrees after just 7.5 meters in figure 4.1.5

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4.3. Cross windNow we have found a model for headwind and tailwind, it can be adjusted to model crosswind. The wind is now affecting the ‘W’ value, as it will be blowing across the pitch.

4.3 left to right

Figure 4.3.1 Birds eye view of crosswind problem with wind blowing left to right.

Here the kick is being taken from the left of the posts.

So now our equations become:

NearpostL WV 0∗cos (ϑ )+u

= dV 0∗sin (ϑ )

FarpostL W +SV 0∗cos (ϑ )+u

= dV 0∗sin (ϑ )

With ‘L’ denoting the side that the wind is coming from.

The same parameters that were changed in 4.1 and 4.2 apply here too and the conditions remain the same; Vo = 20 m/s, D = 10m, W= 10m, u = 10m/s

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2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 320

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Angle of opportunity for fixed wind speed, velocity and offset distance

Distance from tryline (D)m

Angle (degrees)

Figure 4.3.2 Model created by changing distance (D)Here the angle of opportunity is always very low (in fact these kicks are one of the hardest to succeed in getting), the ball is blowing the ball away from the posts as soon as the ball leaves the ‘t’. The best distance to kick from is between 14 and 18m but even these values leave you a very hard kick.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

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Angle of opportunity for fixed distance, velocity and offset distance

Wind speed (u) m/s

Angle (degrees)

Figure 4.3.3 Model created by changing wind speed (u)As we expected the angle of opportunity gets smaller as the wind speed increases.

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5 10 15 20 25 30 35 400

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Angle of opportunity for fixed distance, wind speed and offset distance

Velocity (V) m/s

Angle (degrees)

Figure 4.3.4. Model created by changing kick velocity (v)We could include the low velocities here as the wind wasn’t blowing directly towards the kick, however in reality you would kick the ball harder than this to ensure it got to the posts. The angle of opportunity does get bigger if you kick it harder.

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

Angle of opportunity for fixed distance, wind speed and velocity

Offset distance (W)m

Angle (degrees)

Figure 4.3.5 Model created by changing the offset distance.Again as we would expect, the angle of opportunity gets smaller as W gets bigger.

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4.4 Right to leftChanging the equations is similar to what we did for headwind to tailwind, we just change the +u to a -u, as the wind is now going against the kick. This will result in the ball being blown so that it approaches the posts closer to the ‘normal line’.

NearpostR WV 0∗cos (ϑ )−u

= dV 0∗sin (ϑ )

FarpostR W +SV 0∗cos (ϑ )−u

= dV 0∗sin (ϑ )

With ‘R’ denoting side wind is coming from.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 320

2

4

6

8

10

12

14

16

18

20

Angle of opportunity for fixed wind speed,velocity and offset distance

Distance from tryline (D)m

Angle (degrees)

Figure 4.4.1. Model created by changing DThe angle of opportunity is very big here, the optimum distance being between 10 and 14m from the try-line. However there is not that much room for error here. Need to be as exact as possible in placement.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

5

10

15

20

25

Angle of opportunity for fixed distance, velocity and offset distance

Wind speed (u) m/s

Angle (degrees)

Figure 4.4.2. Model created by changing uThe harder the wind blows the more the ball is blown towards the ‘normal line’ resulting in a greater angle of opportunity.

5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Angle of opportunity for fixed distance, wind speed and offset distance

Velocity (V) m/s

Angle (degrees)

Figure 4.4.3 Model created changing VThis graph produces some unexpected results, the angle of opportunity is very high throughout, but it suggests that the best velocity to kick the ball is 18 degrees. Obviously this is not a practical speed to kick the ball as it probably won’t get to the posts, let alone over the crossbar. This is a good example of how this model fails.

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0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

Angle of opportunity for fixed distance, wind speed and velocity

Offset distance (W)m

Angle (degrees)

Figure 4.4.4 Model created by changing WThe angle of opportunity obviously is still going to decrease as W grows but it’s at a slower rate than figure 4.3.5.

4.5 Collection of Models

Headwind: W

vo cosϑ= D

vosin ϑ−u

Tailwind:W

vo cosϑ= D

vosin ϑ +u

Crosswind left to right:W

vo cosϑ−u= D

vo sin ϑ

Crosswind right to left.W

vo cosϑ+u= D

vo sin ϑ

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4.6 General model.

Figure 4.6.1 Diagram to show general model idea..We want to find a model that will allow us to put in any angle of ϕ in.

If we set ϕ it to 90, this means that we want to end up with the left to right model from above.

Wvo cosϑ−u

= Dvo sin ϑ

If we try D

vo sin ϑ+ucos (φ)And sub in ϕ = 90, we have

Dvo sin ϑ+u∗0

= Dvo sin ϑ

(As cos(90) =0)Now we need the other side, so we could try

Wvo cosϑ+usin (φ )

And again sub in ϕ = 90, we have W

vo cosϑ+u

We need ‘-u’ so this side becomes W

vo cosϑ−usin ( φ )= W

vo cos (ϑ )−u

(as sin(90) =1)

So our general model appears to be:

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Wvo cosϑ−usin ( φ )

= Dvo sin ϑ+ucos ( φ )

Where 0≤ ϕ ≤2*pi

If we test the model, with ϕ = 270, we should come out with crosswind from right to left.

Subbing this in becomes:W

vo cosϑ−usin (270 )= D

vosin ϑ+ucos (270 )

With sin(270)=-1, and cos(270)=0, we have W

vo cosϑ+u= D

vo sin ϑThis is correct.

4.7 Chapter Conclusion Headwind and tailwind model created, looked good for all parameters that were varied Crosswind from both sides created. Again looked good General model created which could be very useful in a further study, still needs further

testing.

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5.0. Testing the rule of thumb.

5.1. Practical 1Picking five points on the rugby pitch, with one directly between the posts and the rest of the pitch divided up as in figure 5.1 so that it replicated 5 theoretical tries being scored. After a warm up the participants (all aged between 18 and 21 and who kicked for their teams) individually took 10 kicks from each point. They did not know the nature of the study. The distance that they choose to place the ball from the try line was recorded, as was their success. The experiment was then repeated until all participants had completed 20 kicks. It was carried out on a still day, with the conditions being as similar as possible for each participant. The results are in the table below, with averages for distance each participant chose, the average success from each point as a participant, success as a group from each point, and then overall success as a group.

Figure 5.1.1 Pitch divided into 5 pieces.

Point A Point B Point C Point D Point E

KickerSuccess D Success D Success D

Success D Success D Ave success

1 50 20 65 20 100 7.5 55 18 45 22 632 40 26 65 21 95 6 60 19 35 30 593 40 30 70 24 100 8 65 16 55 31 664 55 24 80 20 85 12 69 20 40 29 65.85 45 31 65 18 100 13 70 15 40 30 64

Averages 46 26.2 6920.

6 96 9.3 63.8 17.6 43 28.4

Ave success As Group 63.6Figure 5.1.2 Table of results ’before’ rule of thumb

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A B C D E0

10

20

30

40

50

60

70

80

90

100

Average success from each position

Average

Kicking Position

Success (%)

Figure 5.1.3 Average success from each position ’before’The graph and table shows some expected results, we expect people to be more accurate in front of the posts, and then to get worse as the offset distance grows. The data was also slightly positively skewed as the kickers were all right footed and so kick better from the left hand side.

5.2. Practical 2

In the second practical, we used the same pitch, the same day – so that the weather didn’t effect it too much, and the same kickers. They repeated the process, but now they were told where to place the ball, which was calculated through the rule of thumb. They didn’t know that this was the optimal place to kick from. Again they each did ten kicks form each place and then another ten after they had all kicked once. Results are recorded below.

Point A Point B Point C Point D Point A

Kicker Success Success SuccessSuccess Success Ave success

1 50 75 95 80 35 79.22 55 80 100 85 45 76.43 60 80 100 70 40 72.44 45 75 90 75 50 72.85 60 65 95 65 55 71.2

Ave success 54 75 96 75 45

Ave success as group 69Figure 5.2.1 Results table ‘after’rule of thumb

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1 2 3 4 50

10

20

30

40

50

60

70

80

90

100

Average success from each position

Series1

Kicking Position

Success (%)

Figure 5.2.2 Average of success from each position ‘after’

Again we can see some expected results, while the success in front of the posts is similar, as a whole the results are better (69% rather than 63%). The most improvement was on points B and D, this could have been because of chance, or the fact that they did benefit from placing the ball from our rule of thumb. however results on the outside were worse this could have been because the kickers wernt proffessionals. As whole the results were better.

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6.0 Conclusion

I have really enjoyed working on this project; it was challenging but also very rewarding to find reasons or formulae for things that I have done many times before without really knowing why. The most enjoyable part was applying what I have learnt and I think I could actually be a better kicker because of it (with some practise of course). There were some problems which had to be overcome, which prompted some out of the box thinking (a couple of times ideas came to me in bed which had to be jotted down).I enjoyed working with maple, which I didn’t realise was such a powerful tool (although at points it made me almost pull my hair out)

This investigation concludes with a rule of thumb, comparing and discussing drag, and a general model to use in any angle of wind. The rule of thumb can be used by the less experienced kickers while the wind model can be utilized by professionals.

I am most pleased with how the wind model turned out, although I did not have time to fully test it as thoroughly as id have liked to. I feel that I didn’t make many inroads into drag which is disappointing as it would have been nice to find out something new and novel. Next time I would spend more time doing research (although I did quite a bit this time anyway) to get a broader understanding of drag.

LimitationsThe testing of the rule of thumb yielded some positive results, however the reliability of them could be questioned. We really needed a wind free pitch (indoor) and professional kickers, with a lot more time than we had to complete this study. I would have also liked to test the wind models in a practical sense (wind machines etc.)For most of the models assumptions were made, it is impossible of course to do everything to a high degree of accuracy (and I don’t know if its need when considering such a large model as a conversion)Finally in the context of the real world, the model is only as good as the kicker. Many other factors which cannot be brought into the model, and cannot be measure could affect the kick. (Physiological factors, the occasion, gusts of wind etc. etc.)

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7.0 Further Work

1) A more complicated drag model could be found, an attempt I made at changing the variables of a golf ball trajectory from (White, 2011) to fit with a rugby ball is below.

Example kick: Below is a graph showing a kick with Vo=40, θo=45. Maple in appendix 6

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.000.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

Range inc. lift and drag

Range x (m)

Height y (m)

2) Would have liked to look at chasers, the problem in brief is as follows: As soon as the ‘kicker’ starts their run up, the players on the opposing team are allowed to attempt a charge down, they can run from anywhere on the try-line the aim being to intercept or to disrupt the kicker. Trajectories will have to clear the obstacle that these ‘chasers’ form. The obstacle will be approximately 3m in height.

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3) Having looked at the conversion, I would have liked to go on and look at other kicks, it wouldn’t take much more work to transfer the knowledge gained here onto the punt kick as it results in a very similar trajectory and spin. Also of interest is screw/spiral kicks, and seeing how far a ball could possibly be kicked when done in the ‘perfect’ manner. To look at the ‘up and under’ would have been nice, especially as it relates to me directly (being a fullback myself)

4) Another extension which would have intrigue me is the effect of spin, although various studies have been attempted, there hasn’t been one which has got to the bottom of it totally (with no disrespect to those that have tried) so it would have been nice to do something which was totally novel

5) I think an under developed part of the research is where on the ball is it most effective to kick it for different uses, namely accuracy and power. This would influence the placement of the ball on the ‘t’ which would then influence a lot of other things. I do however think that the way most people tend to kick it is most effective, or they wouldn’t have such large success rates.

6) The difference between right and left footed kickers is a problem in itself – with some teams actually having different kickers for different sides of the pitch- but is there an advantage to kicking from the left hand side if you’re right footed and vice versa.

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8.0 BibliographyBarritt, J. (2008). Place kicking in Rugby - The Continuum and Beyond. Rugby Football

Union.Croft, T., & Hart, D. (1998). Modelling with Pojectiles. Chichester: Ellis Horwood Limited.Haake, S., & Moritz, F. E. (2006). The engineering of Sport 6, Vol 1 and 2. Sheffield:

Springer.Hartschuh, R. D. (2002). Physics of Punting a Football. Wooster.Jackson, R. C. (2003). Pre-performance routine consistency:temporal analysis of goal kicking

in the Rugby Union World Cup. Journal of Sports Science, 803-814.Lipscombe, T. D. (2009). The Physics of Rugby. Nottingham : Nottingham University Press.Mestre, N. D. (1990). The Mathematics of Projectiles in Sport. Cambridge: Cambridge

University Press.Murakami, M., Kobayashi, O., & Seo, K. (2006). Flight dynamics of the screw kick in rugby.

Sports Engineering, 49-58.Ortega, E., Villarejo, D., & Palao, J. M. (2009). Difference in game statistics between

winning and losing rugby teams in the Six Nations Tournement. Journal of Sports Science and Medicine, 523-527.

Polster, B., & Ross, M. (2010). Mathematical Rugby. Victoria: Monash University.Vance, A. J., Buick, J. M., & Livesey, J. (n.d.). Aerodynamics of a Rugby Ball. Portsmouth:

University of Portsmouth.White, C. (2011). Projectile Dynamics in Sport. Abington: Routledge.

NB: Some of these have not been referenced directly as I just utilised them for background reading/knowledge.

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9.0 AppendixAppendix 1

Project Plan (PRO302 / U02849)

Date: 09/11/2012Student: Joshua MainSupervisor: Dr Michael McCabe

Provisional Title:

Projectiles in Rugby. Is there a perfect placement for a conversion?

The purpose of this project is to ascertain whether there is a certain place where a conversion should be taken from in order to maximise success. I am going to introduce the theory of projectiles. My project will include a short history of rugby, the basic necessary rules. I will discuss rugby union and rugby league as this study will be useful to both.

I will then go on to discuss kicks in the modern game, including types of kicks and statistics showing their importance. I will then introduce projectiles. I will then create a model for a conversion and then break it down link it to the projectile theory. There have been attempts to build both 2D and 3D models. I will attempt to check that these are correct and then extend them. I am going to add in the idea of a ‘chaser’. This will affect the placement of the kick as it now has to clear the chasers and the of course the goal posts.

At this point I will explain the use of any software that I am proposing to use.

After this I will build into my model other factors such as ‘chasers’, drag and the effects of using left and right kickers.. Another concept I might be interested in pursuing is spin.

Goals;

Main goal; is to build a model in which, when a try is scored you can put in the distance from the centre of the posts from which it was scored and the model will tell you how far back to place the ‘t’

Secondary goal; to comment on the effects of spin, drag and left/right foot kick takers.

Gather some data from different kickers to help show how successful/unsuccessful my study has been.

I will then conclude my study and decide whether it’s been a success.

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Chapter headings/content

Abstract

Introduction

o History of rugby

o Statistics of kicking

o End with my aim

Chapter 1, Projectile Theory

o Theory and introduction to projectiles

o How this will link to my project

o Previous models built (2D and 3D), how I can utilise them

Chapter 2, Modelling

o Building of the model

o Include ‘chasers’

o Limits and uses

Chapter 3, Model Extension

o Incorporate drag, spin, kickers off either foot

o Limits and uses

Results o Practical – if possible set up a test of my model with two kickers

Conclusion

o Explain each chapters findings

o Analyse practical- does it support my model?

o What I have learnt from the project

o What I would do differently

Plan/Schedule

9th November- hand in project plan, resource finding and background reading

23rd November- Draft of introduction, start using latex

30th November- Introduction complete

14th December- 1st Chapter draft

21st December/11th January- 1st Chapter complete

1st February- 2nd Chapter draft

8th February- 2nd Chapter complete

1st March- 3rd Chapter draft

8th March- 3rd Chapter complete

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15th March- Conclusion

15th-28th – Sort any problems

Resources:

Croft, T. & Hart. D. (1988). Modelling with Projectiles. Chichester: Ellis Horwood Limited.

DeMestre, N. (1990). The Mathematics of Projectiles in Sport. Cambridge: Cambridge University Press.

Haake, S., & Fozzy Moritz, E. (Eds.). (2006). The engineering of Sport 6 (Vol. 1.), New York: Springer.

Haake, S., & Fozzy Moritz, E. (Eds.). (2006). The engineering of Sport 6 (Vol. 2.), New York: Springer.

White, C. (2011). Projectile Dynamics in Sport, Principles and Applications, London: Routledge.

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Appendix 2

Appendix 3

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Appendix 4

Appendix 5

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Appendix 6

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Appendix 7

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