I am going to talk about sport today, and in line with some of the things that we did in the previous lectures, I am going to try and show you how rather simple mathematics can tell you things that you would not otherwise know about a variety of sporting activities, and if you are a sporting competitor or a coach, you might also learn something to your advantage. I am going to pick on some fairly diverse examples which will cover statistics, movement, balance, taking force moments and so forth, so I am going to jump around a bit between different sports.
Let us begin with something that we looked at briefly in one of the previous lectures and it is the issue of balance and balance sports. In gymnastics and many other events, you are faced with the problem of balancing yourself on a precariously narrow line or surface. This is a case of people trying to maintain their balance, a clear example of which can be found in tightrope walkers, some of whom often carry long poles. You do something similar to carrying a long pole if you want to sort of balance to walk across a narrow piece of wood or something - you put your arms out. Why do you do that, and why do people carry long poles?
If you stop the ordinary scientist in the street and ask them for an instance answer, they will probably mumble something about centre of gravity - that it is to somehow make your centre of gravity lower so you are more stable - but if you think twice about that, holding a pole actually raises your centre of gravity and so makes the situation worse on this question. True, you can shift it a bit from side to side and change where your centre of gravity lies, to make sure it is in line with the wire, but it does not seem to help in this way. So what is going on here is something a bit more subtle than just mere centre of gravity. It is what mathematicians and engineers call inertia. Besides your weight, what counts in determining how quickly you move and how difficult it is to stop you moving is how your weight or your mass is distributed, and that determines what mathematicians call inertia or moment of inertia. As the word suggests, your inertia is your tendency not to move, and, likewise, once you start moving, how difficult it is to stop you moving.
What matters with inertia is not just the mass of an object. Let us suppose we are looking at a sphere where, if the mass is distributed far from the centre, the inertia is higher than if the mass is evenly spread. So if you had two spherical balls, made of different material, so you could not tell just by looking at them which was which. They have the same mass, so you weigh them and they are the same. One is solid, so the mass is distributed through the whole of the volume; and the other is just a shell, so all the mass is all near the surface. In this case, it is the shell one which has a higher inertia - if you try and roll it, you will find it is harder to move this one than it is to move the one that is solid one. If you roll them down the same slope, you will find that they will take different times to roll to the bottom. So what the inertia looks like, typically, is the mass of the object, times the square of its radius, times some factor that determines how concentrated the mass distribution is. That concentration factor for the hollow shell is two-thirds, and for the solid sphere it is two-fifths, so the inertia for the shell is larger than for the solid sphere.
So this is the idea of inertia, and you can see that by moving lots of mass far away from your centre, you increase your inertia, and what that does is it means that you move more slowly. High inertia means that it is slow to move, low inertia means that it is quicker and easier to move. So when you hold the long pole, when you wobble, you wobble more slowly, and so you have got more time to move your feet and adjust and keep your balance. If you did not have the pole, you would wobble very quickly, and it would be much more likely that you would fall off. You can try this for yourself, if you have got a long ruler and a short ruler or a pen. If you try balancing the little pen on your finger, you will find that it is quite difficult because once it starts to wobble, it will fall off, but if you have a metre ruler or a long bamboo pole, it is much easier because you can adjust. So, the higher inertia gives you more time to adjust. You wobble more slowly, and the time it takes to wobble through a complete period is proportional to the square root of the inertia which we defined a little earlier.
So you can see why it pays to put your arms out if you want to stay still, but where else can we see this idea in sport?
If you look at Olympic cyclists, you will notice that their bikes are not like your bike. Their bicycle wheels are disks, unlike your bike wheel, which is probably a ring with spokes going to the centre. If you try to cycle down the road with disk wheels like in the Olympics on the front of your bike, you would pretty soon come to grief because, as soon as you turn slightly or the wind catches the wheel, it would swing round. The wind would not blow through the wheel, but would blow into it and turn it, and you would have an accident. On racing bikes there are no brake - you keep the bike perpendicular to the surface of the track, and you just go as fast as you can. So the wheel, if you are cycling correctly, is always facing directly in the direction you are going, so there is not problem with disk wheels in these conditions.
So why is a disk? The disk has different inertia. The inertia of the disk is a half times the mass times the radius squared, whereas for the ordinary type of wheel, the inertia is twice as big. So these disk wheels have smaller inertia, they respond more quickly to applying force on the pedals - you can change speed much more easily. So these are dynamically much more effective.
So that is the first lesson about inertia: that whenever you see there is something rotating or moving, like a wheel, it pays to keep the inertia small.
If you are an engineer of course and you are building a building, you want to keep the inertia large. If you are putting a girder across the ceiling of this building, you want the inertia to be large, so that when forces and strains develop in the building, it is harder to move and buckle the girder. So you will notice that sometimes girders come in a sort of an H shape, like the letter H. A lot of mass in the girder has been moved away from the centre, and what that is doing is increasing the inertia, making it harder to shift and deform and move that girder, which is essential for the structure.
When you rotate this wheel about a centre, we are thinking about it rotating just in one direction, but complicated movements can occur where you rotate around any of the three of the axes that the object has. So if we have something like a tennis or squash racquet, we could rotate it by holding it vertically in the air by the handle and you could spin in around the axis of the body of the racket, spinning it in your hands as you see squash players do between points; or you could lay it flat on the floor and spin it around on that axis; or you could throw it in the air and throw it along the axis at 90 degrees to the previous one. For a solid object, there are three directions around which you could rotate it, and the distribution of mass is different in the different directions. What this tells us is that there are, in effect, three inertias for the rotation and movement about each of the three axes. There is a rather remarkable and interesting property of motion around these axes, and that is, one of the inertias will be the largest, one will be the smallest, and one will be in between - what we call the intermediate inertia. When you rotate an object about the axis which has the intermediate inertia, that motion is unstable.
So, if you are a gymnast and you are doing a somersault on the beam or on the floor exercises, you can change the moments of inertia of your body's mass distribution and you can make sure that, when you do a somersault on the beam, you also do a twist - you cannot avoid doing a twist. So, the intermediate moment of inertia instability is a very interesting and sort of crucial ingredient, if you are a gymnast or someone who is doing high-board diving or something like that. You can manipulate these moments of inertia of your body so as to either have the twist or do not have the twist.
For example, the way you manipulate the inertia would be, if you are a diver, you get into the tuck position and then when you do a somersault when you fall, there will be no twist at all, because that is not the intermediate axis of inertia for your body. Similarly, if you stretch your body completely outwards, you can do a full turn without the instability.
Another important place where we see inertia playing a role was in the recent Winter Olympic Games: ice skating. Here, again, we are looking at rotation. Your inertia determines how easy it is to spin, but when you do spin, there is a quantity which is conserved - it is not energy, one conserved quantity, unless there is friction and so forth - but it is the moment of the angular rotation or the angular moment. What that means, in practice, is that what is conserved is your rate of rotation times the square of your size.
So if you are an ice skater and you start with your arms outwards, the square of your size is large, because it is like holding the long pole. But then, as you start to spin, as you draw your arms in, the square of your size decreases rather dramatically, and so your spin rate will increase. So the rapid pirouetting of an ice skater drawing her arms in is a consequence of reducing the skater's inertia and the conservation of angular momentum.
Divers can do it as well. If you start to rotate after you leave the high board and then you draw your arms in, into a tuck position, you will rotate faster; and, conversely, if you bring your arms out, you will start to rotate more slowly.
Let us move on from inertia now to look at a different sport and a different type of consideration. What we are going to look at is rowing or rigging rowing eights. This is not fixing the result of the boat race but rigging is the term for how the oarsmen are sort of laid out. What is known as the standard rig is where the oarsmen are on alternating sides, right/left, or starboard/port. The question is: is this the best way to organise your oarsmen?
The rules say you can put your oarsmen pretty much where you like, so you could change them around, if you wanted to. This was a problem I became interested in just over a year ago, I started to think about it for some other purpose and I ended up writing a paper about it, which attracted all sorts of attention from coaches and so forth, and there was an article, in the end, in the last Rowing Biomechanics Journal or something, developing it in further detail. But, regardless of that, let me tell you that it was just concerning the recognition, first of all, that the standard rig is not in fact a very good idea. It is not the optimal way to set up the oarsmen. Why is that?
If we consider a four, will begin by making the assumption, as physicists generally do at the outset, that all the oarsmen have the same strength, they are all identical and they all exert the same force. What we are going to be interested in is, if we stand at one end of the boat, and take moments of the forces which they exert, is the overall moment the turning force on the boat zero, and we are going to see it will not be. With this configuration, what happens is that, with each half-stroke, the boat wiggles, left to right, left to right.
So how would we demonstrate that? Well, think of an individual oarsman. You have got your oar going out from the boat, the oar sits in the row-lock, and when you pull the oar, a force is exerted on the row-lock and therefore on the boat. Like any force, you can split it into two components: one in the direction in which you are going, and another force at 90 degrees to it. The thing is that, in the first half of the stroke, when you are leant forward and the oar is behind your back, that other component of the force is towards the boat. In the second half of the stroke, when you are leant back and the oar is in front of your body, it reverses and is in the opposite direction. So what is happening is, in the first half of the stroke, there is an overall force from the oarsmen on this side and the oarsmen on that side pointing into the boat, and in the second half of the stroke it goes the other way.
What is interesting about forces are things called moments. Remember what Archimedes taught us: if you want to figure out where the balance point is, you hang a weight on either end, then you get the balance by multiplying that distance by the weight on one end and getting it equal to the distance times the weight on the other end. So this is called the moment of that force, and the moment of the weight on one end tends to pull it in one direction, whereas the one on the other end pulls it in the opposite direction, but they have to be equal for there to be equilibrium.
Suppose we work out the moments of the forces on the boat. Well, we have got some distance here from the stern for each of the rowers, who row on alternate sides. Since we are only concerned with the moment, we only have to worry about the components of the forces that act at 90 degrees to the boat. These will be acting in different ways according to which side of the boat you are on, so we can call one side positive and one side minus. We can work out the moment for the rower nearest the rear of the boat as the distance from the stern, S, multiplied by the force, N, which would be negative or positive according to which side the rower was on. If the oarsmen are distance R apart, the moment for the next person, whose oar is on the other side, is N times his distance, S, plus R. The moment for the next person, on the other side, is S plus 2R. For the last person in a four-person boat, it is S plus 3R. I am going to call the forces +N in the first half-stroke and -N in the second half-stroke, indicating the direction in which they act. So the oarsmen on either side will have opposing N's in that one will be positive and one negative for each half of the stroke. From this we can see that the sum of the moments about this point being exerted on the board is:
So in the first half-stroke, it is -NS + N(s+r) - N (s+2r) + N(s+3r). That is the sum of the moments on the boat in the first half-stroke. The first thing you notice is that all the things with the Ss all just cancel out (-, +, -, +), so S does not matter, because it never appears. But for the others, we have got Nr - 2Nr + 3Nr, which leaves a + 2Nr moment on the boat pushing in one direction, perpendicular direction of motion in the first half-stroke. It is 2N, the force, times the distance between the rowers. In the second half-stroke, N turns to -N, and the moment on the boat is -2Nr. This means that what is going to happen is the boat is going to wiggle as it goes along, and if there is no cox, with a four, you are going to have to correct this yourself. If there is a cox, he will do a lot of the correcting, but by moving his rudder, you will do more work going through the water. So the question then arises: is it possible to organise the rowers so that this moment on the boat is zero, so there's no wiggle?
Now, in the early 1960s, there was an interesting story in Italy, on Lake Como, with the top Italian cox-less four who were associated with the Moto Guzzi Club. This is the famous engineering company that, in effect, invented motorboats, makes motorcycles and so forth. One day, the four were to go down to Lake Como to do their trial ahead of the European Championships. For one reason or other, and it is not clear whether this was some sort of joke on the coach or just an accident, the crew rigged up the boat in a different way: they had the front and rear rowers on the same side, and the two middle rowers on the opposite side. When the coach arrived and saw this, he was rather annoyed, and told them they were going to have to row the trial, he had a target time and they were going to have to row it in this configuration, and if they missed the target time, they were just going to have to row the trial all over again around 15 minutes later.
So they rowed the trial, but they broke their course record! So the Moto Guzzi engineers had a look at what was going on here, and one of them deduced that what was going on is that this is a very efficient situation - because there is no moment on the boat. We can see this if we are to feed it back into our earlier formula:
We can see from this that there is no net moment force on the boat in this configuration. They went on to win the European Championships that year, and I think, two years later, won the Olympic Games as well.
The next question I asked is: what are the solutions for Eights? This is a slightly more elaborate question. If you look back at the sort of calculations that we are doing here, you notice the distance of the stroke from the end of the boat does not seem to matter - it always cancels out. So you do not really need an S, so let us just call it 1, rather than have this symbol S sitting around. Similarly, for the others, if you are wanting to show that there is a zero wiggle, it does not really matter what R is, so we could set R equal to 1. What that means is that what these zero moment configurations amount to, in the case of an Eight, is saying: can you take the first eight numbers and sprinkle down four plus signs and four minus signs in between them so that the sum is zero?
The standard rig, where the rowers are on alternate sides, has the following sum for the moment:
The mathematical question is: how many solutions, how many ways are there that you can organise the sums and differences of those first eight digits, with four pluses and four minuses, to give the sum of zero?
What I noticed in this is that it is a simple example of a rather sophisticated number theoretic problem called the Zero Subset Sum Problem. People in that integer sequence field were rather amazed and excited that someone had actually found an application for some of the things that they do, and offered to publish a paper on this in their rather high-brow journal.
Once you look into this, you can prove that there are only four possible solutions to this problem. This first one is rather predictable. The Italians found this, long ago, in the early -50s - you just take two of the possible formulations we found for the fours which have 0 moment and put one behind the other. This means that is it effectively the sum of two zeros, which came out of the rowing formations for the fours. There will be a zero wiggle for this Eight, and this is sometimes called the Italian Rig or the Spaghetti Rig:
Rower: 1 2 3 4 5 6 7 8
Side: L R R L L R R L
The other commonly used type of rig of this sort is a German Rig:
Rower: 1 2 3 4 5 6 7 8
Side: L R L R R L R L
These are the two known solutions of this problem, which are used periodically, even today, in competition. But there are also two new solutions to this problem:
Rower: 1 2 3 4 5 6 7 8
Side: L R R L R L L R
Rower: 1 2 3 4 5 6 7 8
Side: L L R R R R L L
You can see that the first one is like an Italian Four, but followed by another one that has been turned upside down. The second one is more unusual still. You have got four all on one side, and then with twos at either end. These are all the possible solutions of this little finite integer sum problem.
I have actually been contacted by some people in the US who were going to try out these configurations in some national intercollegiate championship. It will be interesting to see what happens.
There are other concerns with positioning rowers. You do not want people to be putting their oar into rough water created by the person in front, or banging their oar into the person who is sitting in front of them, so a coach will take into account other things, and also the fact that rowers are not all the same. So you will carry out an analysis of what the force curve is with time of each of your oarsmen's pull on the oar, organise your crew to balance the stronger against the weaker, and so on, but the same principles apply. You will have slightly different solutions, but the same basic idea will help you.
Mathematically, you can go on and analyse all sorts of unusual things once boats get rather larger. There is a little theorem that if you have a boat that is balanced, so you have got the same number of oarsmen on one side as you have on the other, you can only have a zero wiggle configuration if the number of oarsmen is divisible by four. So you can do it with a four, with an eight, and a twelve, but not with a six. With a larger number like these, the number of possible configurations of the rowers grows dramatically. This is one of these problems where a small increase in the complexity of the problem produces a huge number of new computations and calculations, so it is really quite formidable, even with a crew of twenty oarsmen, to work out the zero moment configurations. To give you a clue about how to do this, if you pick the first and the last, the next one and the next but last, and so forth, you will notice there is something of a pattern, so you can start to generate solutions fairly straightforwardly. But I will leave it to you to work these out for yourselves.
Moving on from rowing, we can look at the shot put. The current record-holder is Randy Barnes, who has thrown the shot 23.12 meters. His career ended rather abruptly when he failed a drugs test and was banned. If you look at the top ten list of performances in men's and women's, there is not a single performance in the men's top ten that has occurred since 2004, and in the women's top ten, there is not a single performance since 1990, which is a reflection of a new stringency in drug testing and monitoring of what is going on in that event.
I am going to tell you something about throwing a shot, if this is something that you do in your spare time, or throwing anything really. There are two surprises. The first thing to recognise is that we are looking at projectiles, and one of the things you may remember from school about projectiles, or just have a feeling for intuitively, is that, if you throw something at some angle, or you have a cannon here on the ground and you fire a projectile, and you want to maximise the range (let us forget about air resistance and so forth here), then the launch angle you want to use to maximise the range is 45 degrees. The range, or the distance of flight, if you fire something from the ground, follows a parabolic trajectory, as Galileo first discovered, and the range is given by the square of the speed divided by twice the acceleration due to gravity, times the sine of twice the angle at which it is launched.
The maximum value of sine is one, and it occurs when the angle whose sine you are taking is 90 degrees. So if you pick this angle Θ, theta, to be 45, two theta will be ninety, and this will be one, and the maximum range will be the launch speed squared over twice the acceleration due to gravity. So, if you go to high altitude, where the acceleration due to gravity is less, g will be smaller, everything else is the same, you will throw things further, but if you are training to throw something, you can see that if you can do something physiologically, technically, to increase the speed with which you can launch it, there is a really big pay-off, because it comes in as the square.
So the standard picture tells us that 45 degrees is the best angle to launch your projectile at. Therefore, the first surprise, which may be obvious if you are thinking about it carefully, is that, for putting the shot, 45 degrees is not the optimal angle. Why would that be? Well, because you do not put or launch the shot from ground level. You launch it from higher than shoulder height. So it is a slightly different problem. We are launching from a distance higher, h, above ground level. If we launch from this distance, h, above the ground, the range is increased by a factor that can be seen here.
d flight = v2 sin 2Θ [ 1 + (1 + 2gh )1/2 ]
What you now want to ask is that, if you know what your launch height is, what is the optimal angle of launch going to be for you? This a slightly more complicated mathematical problem of calculus. However, if you work it out, you will find that as you launch at greater and greater speed the angle of lauch changes very slightly. The angle at which the maximum speed makes for the maximum distance comes out to be less than 45 degrees. The angles vary according to your ability, but they work out to be as follows:
21.3 metres (= 70 ft) optimal angle is 42 deg
15.24 metres (=50 ft) optimal angle is 41 deg
10.7 metres (= 35 ft) optimal angle is 39 deg
So, if you are a world class, potential Olympic champion throwing over 70 feet, 21.3m, then the best angle for you, from this analysis, is 42 degrees, not 45. If you are a rather more routine competitor, at club national level, throwing at around 50 feet or 15m, then your best angle will be even lower, at 41 degrees. If you are just a weekend shot putter, throwing about 10m, then you want an even lower angle of projection.
However, there is a second surprise because, if you actually look at detailed pictures of world class shot putters, they do not launch the shot at 42 or 43 degrees. They launch at an angle that is actually closer to about 37 or 38. So why is this?
The answer lies in the fact that we have been assuming, in the previous analysis, is that there is no correlation between the angle at which you launch the shot and the speed with which you can throw it but, in practice, there is. We find it much easier to move weights in some directions than in others. So if you lay on a weightlifting bench to carry out a bench press straight up, you will lift much more than if you try and bench press it at an angle of 45 degrees. So there is actually a very significant effect, that if you study shot putters throwing projectiles and what speed they manage to achieve after launch against the projection angle, it is by no means the same for all angles. So, the larger the angle, the lower the speed that you manage to achieve.
So if you fold this constraint of this graph into the previous analysis, you are now looking at an optimisation problem which is constrained by this data. If you look at the data, you will see that the actual best angle for a world class shot putter is somewhere around 34, 38 degrees, depending on their size and so on. In doing this, we see that by studying these sorts of events rather closely and taking into account all the constraints that exist, you can understand what happens, and, for an individual athlete, you could optimise what they should be trying to achieve in their training programme.
Well, that is enough of throwing things. We will now have a look at judging things. There are all sorts of judging systems in operation in sport. Some are automatic and quantitative - in football, for instance, you just add up the goals. But in some sports there is subjective judging, so you get a mark for performing some routine and you get another mark for how well somebody thinks you have done it. You have to be very careful with some of these judging systems that they do not introduce subtle paradoxes which mathematicians would outlaw from a voting or judging system right at the outset.
Traditionally, weird judging means ice skating, and it became so weird at the Salt Lake City Winter Olympics that it was subsequently all changed. There were two reasons it was weird: one was simply bias by certain judges, who were colluding; but the other was a mathematical deficiency of the judging system. I am going to show you what that deficiency was.
The situation before the last event by the last skater in the Ladies Figure Skating at that Olympics is summarised here:
Skater Score #1 Score#2 Total
Kwan 0.5 2.0 2.5
Hughes 2.0 1.0 3.0
Cohen 1.5 3.0 4.5
Slutskaya 1.0 ? ?
The way figure skating used to work at this time is that you had two programmes, a short programme and a long programme, and you were given some score for each of them and the skaters were ranked according to their scores. These rankings were then added together and the person who had the lowest final score was the winner. Because of this, the scores were really only important in creating the placing, so if you were first and first, you would have a lower score than if you were sixth and sixth.
The first odd thing about it is that, in all this excitement about getting 6s and 5.9s and so on in the judging, none of that counts for anything, because all it does it determine your position, but then all those scores are completely forgotten - they are not used in any other way.
So, first of all, in the short programme, after all those marks come in, they decide who is in first place, and it was Kwan, and so she is given 0.5 mark. The second place, Slutskaya, is given 1.0 mark. Third place, Cohen, 1.5; Hughes, who was fourth, was given 2.0, and so on. So you see all the individual judging points of 6s and 5.9s count for nothing. It does not matter how far ahead of the second person you were, you cannot do any better than a 0.5 scoring mark.
Then, in the long programme, because it is long and there is more in it, the score you are going to get through winning it is not 0.5 but 1.0, and coming second, you will get a 2, and coming third, you get a three, and so we add those together. So the best you could do, if you won both, was to get a score of 1.5.
So what happened, after the first three had skated, Hughes was the best skater in the long programme, so had a mark of 1.0, Kwan was second best, had a mark of 2.0, and Cohen was third and had a mark of 3.0. So when we add them together, Kwan was in the lead with 2.5, Hughes was next with 3, and Cohen was third with 4.5. Slutskaya was yet to skate. What happens? Well, Slutskaya skates the second best long programme, so the marks are changed. Hughes, still has 1.0, Slutskaya has 2.0, and the others have 3.0 and 4.0. So, when you add the marks together, there is a tie of Hughes and Slutskaya, and then Kwan and Cohen. When there is a tie, the best long programme is taken to break the tie, so Hughes was the gold medallist.
Skater Score #1 Score#2 Total
Hughes 2.0 1.0 3.0
Slutskaya 1.0 2.0 3.0
Kwan 0.5 3.0 3.5
Cohen 1.5 4.0 5.5
But something very strange has happened here. You will notice, before Slutskaya skates, Kwan was ahead of Hughes. After Slutskaya skates, Hughes is ahead of Kwan. So how can the performance of Slutskaya change whether Hughes is better than Kwan? This is a situation which, when mathematicians develop voting systems - which is a very sophisticated area of mathematics - they try to rule out this possibility before they start. They usually introduce an axiom at the beginning, that your system must not allow this to occur.
There is an old joke about an American philosopher who was taken to lunch in a restaurant somewhere, and the waitress came around and said to everybody, "we've got either fish or we've got chicken - what do you want?" He said, "Well, I'll have fish." Then, after a little while, the waitress came back and said, "Oh, we've got another addition to the menu - there's pasta." "Ah," he said, "I'll have chicken then."
What has happened in our skating example is called the Independence of Irrelevant Alternatives. What has gone wrong here, what one knows about in voting systems, is that you must not add together preferences or orderings. You can add people's scores, but not the rankings. So if you had have kept all those 5.9s and 6.0s and so forth, and added them all up for the two programmes, there would have been a perfectly consistent and excellent judging system, which is rather like the one that they have since introduced because of that problem.
Why are adding preferences or rankings a problem? So, suppose that you have three competitors, A, B, and C, then we might think that a situation we would like is that if A beats B, and B beats C, then A must beat C. We know there are some things like that. So if A is taller than B, and B is taller than C, then A is going to be taller than C. But if A likes B, so if Angela likes Brian, and Brian likes Colin, that does not mean that Angela likes Colin. And if Arsenal beat Blackburn, and Blackburn beat Chelsea, that does not mean that Arsenal will beat Chelsea. So some relationships do not have that automatic transitive property, and preference voting is something that does not have this property.
Suppose you are voting for three political parties or individuals, and you ranked them. The first judge ranks them A, B, and C, so they like A most and C least. The next judge ranks them B, C, A, and the third judge ranks them C, A, B. Then what happens? Well, A beats B, B beats A, A beats B, so A beats B by 2:1. B beats C, B beats C, but C beats B, so B beats C 2:1. So A beats B and B beats C, so you would expect that A would beat C. But this is not so. A beats C, C beats A, C beats A - C beats A 2:1. You find sometimes in small committees, people decide that the right thing to do is to vote on the three candidates or three options that each of you prefer on the committee, and then combine those results. This, unfortunately, suffers from this type of fallacy, the so-called Nanson Paradox. So the golden rule is: do not add preferences or rankings. You can add scores, but not rankings.
Let us now look at another scoring problem, and this is about football. It is really to ask whether the English football Premier League is really a very competitive thing at all or is it just random. We can ask: how much does it differ from being just a random operation that we could replace by drawing, not just a few balls out of bags, like the Football Association do for different rounds of the cup, but, each week, drawing 38 pairs of balls out of a bag - would we get a very different type of league table distribution of points?
The Premier League has got twenty teams. They each play the other 19 home and away, so they each play 38 games. If you look at the statistics of football matches, there is an interesting almost constant: there is a roughly one-in-four chance that a football match is a draw. Therefore, statistically, a quarter of the games in Premier League and the other major divisions are draws. So, let us assume that is the case, that the probability of a game's being a draw will be one-in-four, so there were will be a three-quarters probability that it will not be a draw. Let us assume half of those three-quarters, three-eights, will be a home win, and the other three-eights will be an away win. So if we had an eight-sided dice, we could write on two of the sides "Draw", on three of the sides "Home Win", and on the other three sides "Away Win", and we can play the whole of the Premier League. We used to do things like this when I was at school, with cricket and so forth, with more options. But what happens is rather interesting, as can be seen if we look at the comparison between the finishing position of the league from a few years ago to the position we get if we make it random:
What we have done here, after you have played all those games with your dice, you have labelled the teams, and the team that comes out at the top, I have just called team number one, and the team that comes at the bottom, which in that year was Wolves, is number twenty. The average finishing number of points is 52.5 points. Notice that the expectation value is not necessarily a value you can expect to get, because you cannot get 0.5 point. But you notice the pattern here. In contrast to this, you can see the actual points in the column further to the right. The first three teams are doing much better than random, they have a far bigger chance than three-eighths of winning at home and three-eighths of winning away, they are winning almost three-quarters of their games, home or away, without any trouble. But once you have taken out those top three teams, the rest of the league, the points distribution is very closely matched by the random outcome model, all the way down to the model. There is a bit of deviation towards the bottom, and I think you can understand that. It does not matter whether the good teams play home or away, they win over the poor teams all the time. For the teams like Wolves, it does not matter whether they play home or way - they lose all the time. But there are some teams in between, like Burnley, which tend to win at home but they generally lose away. So the model of having the same probability of winning home and away is not a good one, for some of the teams. But what this is showing you is that the top few teams - and if you looked at this year's league table, you would get the same basic idea - there may be four teams now rather than three, but they are far ahead of everyone else. But in the year we have the table for here, the fourth team, which I think was Liverpool, was closer to being relegated than to winning the league. So it is the most uncompetitive league you could ever imagine, and most of it is just statistically the same as tossing an elaborate set of coins in the air.
I have just two more sports I want to say something about, the first of which is weightlifting. The greatest weightlifter ever is a man named Naim Suleymanoglu. He won three Olympic Gold Medals and would have won another one had his country allegiance not swapped. He can lift more than three times his body weight, which is really quite extraordinary. What we want to know here is: how does the strength of someone like a weightlifter change with their body weight? There are weight categories in weightlifting, and the statistics are very carefully monitored as to what people's body weights are and the weight that they lift. It will not be unnoticed to you that there are other strength events, like putting the shot or throwing the hammer, where there are no weight categories. This seems bizarre, because strength clearly matters, and of course all the shot putters therefore are enormous - there are no little shot putters. You cannot make a living as a five foot tall shot putter!
We want to think about how strength changes with size, but let us first think about weight. Weight is just the force with which gravity attracts your mass to the centre of the Earth, so your weight is proportional to your mass. Mass is your density times your volume. We are all about as dense as each other because we are made of the same stuff. So, our mass then is proportional to our volume, and volumes tend to look like length cubed. So, if you are a large sphere, your volume will be four-thirds pir3, r is your radius. If you are a cube, it will just be r3. So in practice, your mass is proportion to your volume to some length cubed. If you were inflated, a copy made of you where all the lengths doubled, then your mass would be 2x2x2, 8 times bigger.
But the interesting thing is that, as your size gets bigger and your mass and your weight grow, your strength does not grow at the same pace, and that is because strength does not grow in proportion to your volume or your mass - it grows in proportion to an area. You see, if I was to break a badminton racquet, we would break it through a slice, a sort of cross-section, and so to break it, we just have to sever the atomic bonds across that cross-sectional area through the racquet. When you break your arm, what do you do? You just break through a slice, an area, of bone. So, your strength is proportional to area, which is a size squared, so as your size gets bigger, your strength does not grow as rapidly as your volume.
You can see that in practice if you look at big cats and little cats. The big cat is not as strong for its weight as the little cat - it cannot support, typically, its long tail. It has to bend it over and it is not strong enough to hold it upright. But the little kitten can support the tail upright, so its strength to weight ratio is different to the big cat.
So your strength is proportional to your size squared, which is your weight to the two-thirds power, so, if you like, your strength cubed is proportional to your weight squared. So if you tried to make people or buildings or trees that had the same design plan but just made everything bigger, eventually, they would break, because their strength would not keep pace with their size.
If the density of the material stays constant then strength α (weight)2/3
If we look at the world weightlifting records, the weight lifted is a measure of the strength of the weightlifter, but do the world weightlifting records, as you follow them across weights, follow this rather simple rule that we've found? As you might guess - or I would not be telling you! - they do, to quite a remarkable and impressive degree follow our formula with a great level of precision. You can see this for yourselves if you get the results for recent Olympics for all the weight categories and plot the cube of the weight lifted, which is effectively the cube of the strength, against the square of the weight of the weightlifter. You will see that the world records follow the straight line very closely, which shows that we are doing our maths correctly. The people who are above that line, would be the strongest pound-for-pound lifters, and the heavyweight lifter, who weighs about 100 stone and lifts goodness knows what, is relatively the weakest, because, if what he lifted really followed the predicted line, his bones would break - he could not maintained that force strength. In reality, weightlifting looks much more meticulously at these higher lifting problems, because of those constraints, and uses a more sophisticated formula to transform everybody's weights to the same scoring system, so that they can say who is the best ever weightlifter when you normalise everything for weights. But, the thought I will leave you with, on this, is that, clearly, if you did have hammer throwers and shot putters in every weight category, I suspect you would find a very similar trend and there would be an equally good argument to have weight categories for all those throwers.
Finally, I would like to finish on something rather unusual which I came across recently: it is perhaps the fastest, the most dangerous, the most lunatic sport that there is, and it is drag car racing. So drag car racing is really a matter of rockets on wheels, and they go a quarter of a mile, 400m, in a straight line. The world record is less than four seconds, from rest. If a Formula 1 racing car, at top speed, with Michael Schumacher behind the wheel, was to come past at full speed when the gun went, it would not beat a drag car to the finishing line, even though it was starting from rest.
When one looks at the numbers, you get a feel for why it is so crazy. These cars go from nought to 100 mph in 0.7 seconds, it takes them just under four seconds for a ¼ of a mile, and they have a top speed 330mph. But when you start to look at the forces, that is what is really alarming. The drivers are experiencing about six times their weight in force because of the acceleration, and then there is the deceleration at the end when parachutes are thrown out to stop the car. If you tried to ride as a passenger, you will be killed I think by this. People who do it a lot, what is going to happen to you, even if you have pressure suits on, is you are going to get detached retina.
The starting acceleration is actually considerably in excess of what is experienced by astronauts in the launch of space rockets or the space shuttle, so astronauts would hesitate to do this sport. And this is not mentioning the sound levels and the vibration levels that you have to experience as well - oh, and every so often, they crash as well!
But what is going on here, from a sort of Newtonian point of view, is rather interesting. It is a type of motion that you do not normally come across in textbooks or mechanics classes at school - it is motion at constant power. So, as we know, power is force times velocity, and force is mass times acceleration, dV/dt, time velocity, which is just a ½ d/dt of the square of the velocity.
Power, P = Force x Velocity = m x dV/dt x V = ½ d/dt(V2)
If the power is constant, d/dt of V2 is constant. If you start from rest, time zero, a place we will call X is zero, what happens is that the square of the speed is proportional to the time, because the power is constant. So after time t, you have gone a distance which is proportional to the 3/2 power of the time. And, if you change this around, what is amusing is that you find this little rule:
When I looked in books about this sport, this seems to be a rule of thumb - I do not know whether it was really worked out originally by an engineer in the same way - and it is called Huntingdon's Rule: that the velocity is proportional to the cube root of the power that you can generate times the distance you go, divided by the mass of the car.
Speed(Mph) = K X (Power in horsepower/mass in pounds)1/3
People in the business use horsepower for the power, and mass is in old-fashioned units like pounds, and the speed in miles per hour, and so on. And what you find is, when you calculate with this formula, you get an answer of about 270 for the proportionality factor, whereas what you observe in practice is somewhat smaller, about 225, and that is because this little model is not quite accurate. At the beginning, as we saw, if you remember, in the last lecture about rolling resistance in transport, this car has to roll a bit at first before the wheels really start to grip, so there is a little bit of rolling against friction at the beginning, and since this is a very short race, this is what is the added detail here. So, if somebody asks you, "What is the fastest sport?" or "What is the most dangerous sport?" I think the only possible answer is drag car racing.