David and Goliath: Strength and Power in Sport

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Top athletes seem to get bigger and bigger. How does size affect performance? Why do some sports have weight categories while others don’t? What types of lever are employed in sports events like gymnastics and wrestling and how much force does a karate blow need to exert to break a brick? These are some of the questions that we will answer by using simple maths.

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13 December 2011

David and Goliath: Strength and Power in Sport

Professor John D Barrow


Welcome to this second lecture in the series about maths and sport. 

The first topic I am going to talk about involves gravity in some way. I should get something straight from the very beginning, because, in everyday life, people who talk about weight are not usually talking about weight. They may be talking about mass, and sometimes, when they think they are talking about mass, they are actually talking about weight. 


What is the definition of mass and how is it distinguished from weight? There are two definitions you could adopt for mass. You could follow the spirit of Newton’s Second Law: if you apply a force to a body, if it is mass (M), then the acceleration that it experiences is given force, is mass times acceleration. This type of mass, which conventionally was always called inertial mass, could be defined as the ratio of the applied force to the object, divided by the acceleration that results. Let us, for the moment, just call that little m. So that is the definition of mass.


Now, weight is something different. Weight is a force, and it is the fourth force with which the Earth attracts that mass. Technically, that might be a different thing to the mass defined in this way, and so we use a capital M. The force with which an object is attracted toward the centre of the Earth would be given by the formula: W = Mg. Sometimes this is called the gravitational mass, and it might, in principle, be a different thing.


So, if we combine these two things, we now have a simple formula that tells us that the acceleration an object would experience if you let it go in the Earth’s gravitational field, A = M/m x g. If these were different - if M was, say, twice m, and for another object it was a half m, then when you drop them in the Earth’s gravitational field, they would fall with different accelerations. If these two types of mass are identical, the acceleration with which something falls in the Earth’s gravitational field is always the same.


This is an interesting feature because it means that you could carry out a high precision experiment to test whether these two sorts of mass are really the same. That has been done for some time, I think first in the mid-1960s. This was generally the most accurate experiment in physics that there was around. This is a measure of the accuracy. If you were to measure the difference in these two sorts of mass, divided by the sum, then this relative difference is less than 10 to the minus 12, one part in a trillion. A trillion is a thousand billion, or a million million.


Now, in Einstein’s Theory of General Relativity, which I am not going to talk about today, it is necessarily the case that these two sorts of mass are identical; his Theory of Gravitation exploits this apparent similarity and makes it exactly the same.


If you had two objects where the gravitational mass was the same and you put them on a balance, gravity pulls them toward the Earth and they are perfectly in balance. But if their inertial masses were different, when you accelerated this balance upwards, the scales would tilt because one of them would respond differently to acceleration.


That is just a little introduction to weight. Weight is a force, the force with which a mass, a quantity of matter, is attracted towards the centre of the Earth - or a small object, locally, towards the ground.


Let us have a look at gravity and your position on the Earth’s surface. If you hold a spring balance up, a hook with a spring with a mass on the bottom, then that object will measure the force with which the mass is attracted towards the centre of the Earth. But in practice, there is another force at work here. The Earth is spinning once on its axis every day; if you are located at this latitude close to the UK, what do you feel? You feel two forces: the force of gravity, which is the pull of all the mass inside you, roughly towards the centre of the Earth; but because the Earth is spinning, you are going round in a circle, and a force has to be supplied to make you move in that circle, but if you do move in a circle, you will feel the reaction of that force pushing you outwards. Like when a bus goes round a corner, you all fall towards the side, away from the centre of the arc that the bus is moving in.


What is the radius of the circle? You can see that, if you were down at the Equator, the radius would be equal to the radius of the Earth; but if you were to walk up to the North Pole, then you are not going round in a circle anymore. 


So, when you are located at some latitude here, you feel the resultant of these two forces: the force of gravity toward the centre of the Earth; and this circular or centrifugal force, pointing at right angles to the line through the North and South Poles. That force upon you is equal to your mass, times the radius of the circle you are moving in, times the square of the angular velocity, one rotation per day, of the Earth.


The interesting thing about this is that the effective force of gravity that you feel is affected by this rotation. If you are up at the North Pole, there is no effect of it at all because you are moving in a circle that has zero radius; but if you are down at the Equator, you will feel the biggest repulsive force away from the centre of the Earth that it is possible to feel. This means if you hold up your spring balance, which measures the resultant of those two forces on us at this latitude, you will feel the largest pull towards the centre of the Earth when you are at the North Pole or at the South Pole, and you will feel the smallest pull when you are at the Equator, when you are moving in the largest circle. So, the force due to gravity, if you like, the effective value of this little g that is about 9.8 metres per square second, acceleration due to gravity, varies with location on the Earth because of the Earth’s rotation. If you have a mass (M), its weight will be less at the Equator, measured by a spring balance, than it will be at the Poles. If you work out one rotation per day, use the radius of the Earth and so forth, you are looking at an effect of about 0.5% as you go between the Poles and the Equator, so about 500 grams in 100 kilogram mass.


If you are a weightlifter or a jumper, it becomes clear that there are better places than others in which to perform or to try and set records. Weightlifting is an obvious case: you want Mg to be as small as possible. But if you are a jumper or a thrower, then you might wonder about the maximum range that you can achieve if you launch yourself with some speed (V).  Anything that happens with gravity where you convert some motion at speed (V) into potential energy, distance up or distance along, this is always the telltale quantity – it is the quantity with dimensions of a length that you can make out of speed and acceleration due to gravity. 


So the maximum horizontal range for a long jump, for which you took off at 45 degrees, which no real long jumper is able to do at top speed – they are simply not strong enough – the range is V²/2g. The smaller the acceleration due to gravity, the further you will go.


If you are jumping upwards, half mV squared in kinetic energy, converted into MGH, then the vertical height that your centre of gravity will reach is V²/g. 


So, you can see, any type of jumping, throwing for distance, is always inversely proportional to g.


That is the first effect that is going to make you gravitate, as it were, to places where the acceleration due to gravity is smaller.


There is another effect that should not be ignored as well: the Earth is not perfectly spherical. So, in some places, Mexico City for example, you are at least a couple of thousand metres above sea level, a good deal further from the centre of the Earth than you are on the banks of the Dead Sea. If you are far from the centre of the Earth, then you will feel a lower force of gravity, a smaller acceleration due to gravity, than if you are closer to the centre.


So, if this is your weight, then the gravitational force is given by the mass of the Earth, times your mass, times Newton’s constant, divided by the square of the distance from the centre of the Earth: Mg = GMME/R2. So, as that distance goes up, your acceleration due to gravity is diminished. You can see there is a double-whammy here: if you could locate yourself at a place that is quite close to the equator, the rotation repulsively counters the pull of gravity, and you are also fairly far from the centre of the Earth, you are going to see an exceptionally smaller acceleration due to gravity, and it is going to be easier to lift heavy things. You will record a smaller weight when you stand on your bathroom scales and things like that.


Mexico City is an infamous venue of this sort. You are 2,240 metres above sea level and at relatively low latitude, so if you go there, the acceleration due to gravity is 9.779 metres per square second - your 100kg mass weighs 977.9 Newton’s. The worst-case scenario would be in areas such as Scandinavia, where you have acceleration due to gravity of 9.819. So this is the difference of about 4 Newton’s or so in what the weight is of the 100kg bar at those locations.


There are other effects that I am not going to go into today. If you are throwing things like the discus, you might even worry slightly about the rotation of the Earth.


The next type of problem involving power, weight and strength involves levers and leverage. You remember Archimedes boasted that, if someone would give him a long enough lever, he would be able to move the Earth. The most important points about a lever are the product of a force that you might apply and the distance from the point of application of that force to some fulcrum (balance point). The moment of this force here is the product of the force, times the distance to the fulcrum. What Archimedes was implying that if you made that distance great enough, the moment would be extremely large and you might move a load here that was 100 times the mass of one here by simply locating this one 100 times farther from the fulcrum.


This is a bit like financial leverage as well - the farther away you can get from the end point of the financial transaction and from the regulators, the better it is…


In mechanics and engineering, one is familiar with the possibility of having three types of lever. A lever has a balance point, a fulcrum, and there will be some load, which is applied. You will then have to make some effort here in order to balance the load. 


If one of your friends sits at the end of the seesaw, and if you get on the seesaw at the other end, if you have the same mass, and the same weight, and if this distance is the same as this distance, you will balance. This is called a class one lever, where the load and the balancing effort are on opposite sides of the fulcrum. This is a common balancing situation.


This is actually the situation that you would encounter in many sorts of sport. One example would be rowing, where you are sitting, in your sliding seat, you are applying a force here, there is a fulcrum which is the row-lock, and then you are applying your force to overcome the load, which is the drag on the water. You can probably think of many other examples of this sort.


The second type of lever has both the load and the effort on one side of the fulcrum. This example has the load closer to the fulcrum than the effort, meaning that you would only need to apply a smaller force as part of the effort because you gain from the large distance away. You probably do this type of class two leverage occasionally, if you go to the gym or keep fit at home, doing something like press-ups. This is a situation where the fulcrum is your feet, on the ground, the load is the weight of most of your body, and then, at the other end, with your arms, you apply the effort to push that part of your body up. If you raise your feet up and you put them on a stool, then you will need rather more effort to push your body up, feeling a bigger component. If you are really ambitious and you do a handstand against the wall, then you would have to push your whole body weight upwards against gravity, and it would not really be a lever anymore.


The third type swaps the position of these two, so you need rather more effort. It is not the sort of lever that you might choose to use to be efficient. You might choose it in order to make life hard for yourself because you want to get a better training effect, for example, if you were a strength athlete. In this case, the effort is closer to the fulcrum than the load. A good example would be if you are holding barbells or weights and curling them: the effort that you are applying is to overcome the load that you are holding in your hands; the effort is closer to the fulcrum at your shoulder.


Within sport, you will see that almost every activity involves leverage of some sort, whether it is running, cycling, throwing, jumping, and lifting things or gymnastics. You can spend many happy hours figuring out which of these varieties of lever is involved (maybe more than one). Even in running, you are using the ball of your foot rather like a fulcrum, driving through your foot to support your body weight and to propel yourself forward.


The classic and rather dramatic sport where all sorts of levers are on show at one time is wrestling. Now, Olympic wrestling is not really like the wrestling that you know and love, with Giant Haystacks and people in great costumes, so forget all that. Olympic wrestling, and particularly Greco-Roman wrestling, where there is a restriction on what sort of moves you are allowed to use, is a much slower affair and appears to require much more strength and a good deal more suppleness and mobility. This is the most famous American wrestler, Cael Sanderson, who was Olympic champion in 2004. I think his record in collegiate wrestling was something like 169-0, so he had never been defeated in his career in the US.


In a sport like wrestling, those three varieties of lever are by no means all equally good or equally sought after. The best type of lever, as I said earlier, is the class two, where you are going to be able to get away with a lot more applied force on your opponent by applying a lesser force at a distance which is greater than the impact on your opponent. So the class two lever is the best type of wrestling hold, the class three is the next best, and the boring old class one, like the seesaw, is really the least effective and the least sought after.


If you are watching a sport like judo or wrestling, you can keep an eye out for these different types of lever and how the fighters try to manipulate their opponent into a position where they can use their own body weight against them to create one of these leverages.


Another amusing place where leverage is rather dramatic is in rugby. The fellow who is going to jump for the ball thrown in from the lineout is generally propelled into the air by two of his gigantic colleagues – he is usually pretty gigantic as well. You can see that there is a small force problem here. These two people are applying a force that is probably a little bit bigger than their weight upwards at this angle (F cos this angle), and this fellow here (F of cos of this angle), and what is opposing those two pushes is just the weight of this gentleman. What is striking about this is the height to which you can expect the jumping forward to go. So, these people with their arms outstretched, this at least two metres, so you might have people here who are over two metres tall, even before they start stretching their arms out, and you are pushing this fellow up at least half his body height, and then he has got an outstretched hand… In total, you are looking at four and a half to five metres at the top of the jump. So, the person throwing in the ball has got to time this rather precisely. The ball has got to come in at the peak of its trajectory at just the right moment to reach the outstretched hand. This is a pretty alarming height to be pushed up to.


Just for comparison, this is the world high jump record, 2.45 meters. This is the world pole vault record of Sergei Bubka, 6.14 metres. We will probably see somebody win the Olympic Games with six metres and a bit next year. The rugby player is not very far off that. It is a long way down.


I shall now tell you something about karate. Karate, unfortunately, is not an Olympic sport. It has tried to become one, and sometimes it reaches the last stages of the IOC elections, but it never seems to get enough votes to become a full Olympic sport. In a way, this is sad because there are a lot of exponents of the sport the world over. 


This raises an interesting question: what sports should be in the Olympic Games? I have a simple criterion for whether sports should be admitted or not, and that is simply to ask the question: is winning the Olympic Games the pinnacle of achievement in your sport? If the answer is no, then you should not be in the Olympic Games. That is quite a good criterion because it removes football, golf and tennis – all the sports that you might want to remove. It would also remove Formula 1, which I am sure will be there soon!


This is the party piece of karate black belts: having a stack of planks, for example, or a brick or bricks, and breaking them.


When you or I attempt this, we do not succeed because, when we apply our blow, we chicken out at the last moment and decelerate because we think we are going to hurt our hand. Of course, we decelerate at exactly the point where we should be accelerating. You think you are applying a hard blow, but at the last moment, you act as if you are swotting a fly. Whereas, the true exponent is really aiming the blow at a point that is lower than the impact point, and that ensures that the impact point is hit at maximum acceleration.


What would you need to split a block of wood? If you look carefully at what goes on in these demonstrations, you will notice that it is not one big block of wood, but lots of planks; you are not breaking a very thick block of wood but sequentially breaking a lot of thinner planks. That is much easier.


To break a plank like this, it does not matter too much what the area is - say 20 x 30 x one cm. What you have got to do is break through a slice of atomic bonds. If you make the plank thicker, there are going to be more atomic bonds in that slice to break and you will need more force. So, for something that is made of wood, one centimetre thick, you are looking at 3100 Newton’s. For comparison, if you are a typical person, weighing 70kg, your weight is 686 Newton’s. So, you will need the equivalent of five body weights to break that.


If you had brick of the same area, but four times thicker, you will need a little bit more. Brick is a bit more brittle than the wood, with its fibrous links.


A karate black belt will be able to bring their hand in at least seven metres per second to hit the top plank. What is the mass of the arm that is moving at that speed? Something like 3.4-3.5kg, so it is not the whole body mass that is coming in at that speed - it is just the arm. You can work out the momentum that is being transferred – it is the mass times the arm speed, and that is about 24-kilogram metre per second. From that, we can work out the acceleration and, therefore, the force that is being applied.

Well, acceleration is really just speed divided by time, so it goes from this speed V to zero, when it hits the top, in a very short space of time, so the contact time with the wood at the top is just a few milliseconds…okay, five milliseconds, you will see on film, typically.  So you can estimate what’s the acceleration that’s involved with striking the top plank.  It is mV divided by this time…five milliseconds…4800 Newton’s…  So this is easily enough to break this block or to break the brick, so it is not really even a challenge for a top karate exponent.  He can put quite a number of planks here and follow through and break each one after the other. The key is this high speed at impact and this very short time interval where the speed changes and that is the acceleration. If you slow your arm as you come in to hit, it does not matter what speed you started at, you will be hitting it at almost zero and there will not be any acceleration at all and you will just hurt your hand. Needless to say, do not try this at home!
Well, I want to look at a different sort of force now that is created by rotation in gymnastics. There is a gymnastics exercise that is called the giant swing and it is performed by men, for example, on the high bar here. Women do not do the high bar. They have asymmetric bars, they have a slightly different exercise, but they can do swings of this sort on the asymmetric bars, and do. But I could not help mentioning, I mean, the term “the giant swing” means two things in the world: one is as a gymnastic exercise; and the other, historically, is this fantastic object. 


People who have been to Bangkok will have seen this – it is called the Giant Swing. This is 30 metres high. It is an object that was used for a ceremony that goes back several hundred years.  It is a Hindu ceremony, which was to mark the success of the rice harvest. What would happen would be people would attach themselves, at the top here, on long ropes, and they would swing up as high as they could, and people would hold money and coins on very, very long bamboo poles, higher and higher, some distance away, that they were supposed to grab. So this continued right to the 1930s. It was then discontinued because of a number of fatalities. 


Here are some old pictures I found. So this is a picture in 1910, so you can see somewhere here who is just swinging through the base, and then he is going to swing up the other side and try to grab things.


Here are two people on a frame, sort of a gondola, just getting going, on a swing, and people have erected some of these poles here with things on for them to grab. So this looks pretty hair-raising… So you think, this is 90 feet high, 30 metres or more…

Well, let us get back to safer pursuits in the gymnasium. This is the sort of exercise that a male gymnast would perform on this, and it is a sort of benchmark of a good gymnast, I think, if you can do this exercise. I remember, as a student, going on a sort of athletics training course to Loughborough, and one of the people there was somebody called Angus McKenzie, who was a very remarkable young athlete. He was national long jump champion, hurdles champion, decathlon champion, as a junior. He had never been on a piece of gymnastic apparatus before, and the sort of coach there, I think it was George Gandhi, had asked somebody if they wanted to have a sort of feel of this bar. McKenzie went on it and did a giant swing immediately, in the first time that he had ever touched the apparatus, which rather staggered people who were in the gymnasium.


So what I want to ask about this is: what is the force on you as you go round? Is this a dangerous exercise? Is it safe for young gymnasts to do this sort of exercise?


Well, you can see that there are two forces on you: it is the old story that we had of rotation and gravity when we looked at the motion of the Earth. But in this case, when you are at the bottom, say, you have got your weight acting downwards, but if you spin around, you are going around in a circle where this is the centre, and the centre of gravity of your body will be about there, go round in a circle, and you will feel a force pushing you outwards. So, when you are at the top, you have got that force going upwards, minus the force of gravity going down, but when you are at the bottom, you have got that force of gravity going on plus the centrifugal force going outwards, so the biggest force upon you is when you are at the bottom. It is like being on a rollercoaster, exactly the same - in fact; the mathematics is very, very similar. Your head, fortunately, is moving in a smaller radius circle. It does not feel such a big stress as the rest of your body does.


Well, let us have a look at a set-up. Let’s even work out, with some simple mathematics, what the force is.


So, suppose you start like this, at the top, or I picture you instantaneously when you are at the top, there is your centre of gravity, some distance, h, from your fingertips, and let us suppose you are spinning at an angular speed little w when you are at the top, and then you drop down to the bottom, and you will be spinning faster at the bottom because you gain this potential energy, and your angular speed at the bottom is big W. So what we want to work out is: what is the change in the energy from the top to the bottom?


Here are some formulae. So here is the same picture. Remember, little w at the top, big W at the bottom, and the rotational energy at the bottom is a half x the moment of inertia, x the angular speed squared. The energy at the top is a half iw squared, and you have got potential energy change because you go from this location to this location, your centre of gravity falls to h below the bar from h above, and so this is the change in the potential energy, 2Mgh. The maximum force, as we saw, is obviously when the gymnast is at the bottom – feeling the weight plus the centrifugal force. So, that is going to be the weight, plus the rotational force at the bottom. And we just turn these around, and you have got this formula here.


The moment of inertia is a quantity that looks roughly like your mass x the square of the distance from your centre, so your size squared. If you are a solid sphere, it is two-fifths Mh squared. If you are a hollow shell, it is two-thirds Mh squared, and so on. So, this is the force that you feel at the bottom: it is your weight plus two factors, determined by your inertia and your rotation. If, at the top, you started from a handstand, so you were not actually moving, so you just dropped, that would be the smallest angle of velocity that you could end up with at the bottom – you would just cross off this term.



If you look at film, like we saw of this fellow here, you can actually count the number of times they go round, and so you can see what the angular speed is, and it is about two-thirds revolutions per second, is the typical speed at the top, and the body size, about 1.3 metres, so what is the force on the gymnast? Well, it looks like the weight, okay, is a one. Here is the inertial factor, sort of four, and typically, falling from the top, about 1.26 Mg, so it is like experiencing six g at your centre, so this is quite a serious force. 


You would not allow a rollercoaster to be built which had the riders experiencing a force of that magnitude, and that is of course why rollercoasters are not semi-circular – they have a slightly different teardrop shape to reduce the force on you at the bottom. 


If you started from a handstand, so you lose this term here, you would only lose this term here, and you would be down to about five g.


The force on your head is a little smaller because your head is a much smaller distance away from your arms, so you could reduce this h factor here. Suppose you halved it – that would be rather dramatic, so you would reduce this maybe to two or three. So those forces are, not so seriously affecting your head, and therefore the blood supply to your brain. 


But this is pretty much the same as the story for a rollercoaster. Of course, the velocities involved with the rollercoaster are much greater, because you are falling a very great distance to build up speed.


Okay, let’s look at weightlifting, so this sort of little gentleman, who is pretty much the strongest, pound for pound, person in the world, and he lifts about three times his body weight. I think he is barely about five foot tall, okay, but pound for pound, he is the strongest person in the world – certainly the strongest weightlifter, with three Olympic gold medals. He would have had four, but the fourth Olympiad he would have competed at, he had changed nationality and was not eligible to compete again. 


What I would like to see is: can we understand the trends in the world weightlifting records? So weightlifting is categorised by weight. The weightlifting world is very statistics-orientated. It tries to have very detailed formulae by which it can normalise the performances of lifters in all sorts of different weight categories, transfer them into points, so that it can compare weightlifters in different categories to say who is the best, who is the strongest, pound for pound.


Well, strength is something that is not quite the same as size, it is not the same as mass, and it is not the same as volume. We have had a little hint of this already. You remember when I talked about the karate black-belt breaking the plank, what did you have to do to do that? You had to sever or slice through a little cross-sectional area of atomic bonds. So what determines strength is an area. If I am holding a bamboo stick here, as it were, that is a mile in length, I will not find it any harder or easier to break than if it is just six feet in length, because I will have to achieve the same thing in both cases: I will have to split it and sever this little cross-section of atomic bonds.


So, in practice, strength is proportional to some area like quantity, proportional to size squared.


If you look at two cats, a nice example, what you notice is that the fully grown cat is not strong enough to support its tail – the tail always curves over, but the little kitten holds the tail bolt upright in a spine – it is strong enough to support its tail weight. So what has happened here is, cats have the same body plan: as they get bigger, they scale up in size. The volume of the cat has grown like R cubed, some size cubed, and the mass, and therefore the weight, grows in the same proportion, but the strength has grown more slowly, as R squared. So the strength per unit weight goes like R squared over R cubed, goes one over R, so the strength to weight ratio of these cats as they get bigger get smaller, and the big cat is not strong enough to support the weight of its tail.


You know this sort of thing probably from experience as well. When you were a child, you had no difficulty carrying another little child of the same mass as yourself on your back as a piggyback.  You probably even took part in piggyback races at the school sports. But if you try carrying another adult as a piggyback on your back, you will have much more difficulty. So, an ant can carry another ant on its back; a small dog, can just about carry another dog on its back without any problem; a horse cannot carry another horse on its back.


So, what we see here is, if you scale bodies up on the same plan, strength growing like size squared, weight growing like size cubed, therefore strength goes like weight to the two-thirds power.


If you try to make giants bigger and bigger and bigger and bigger on the same plan, eventually they would break because their weight would get so great that it would be sufficient to break the molecular bonds that hold their bones and their legs together.


It is the same with skyscrapers. If you tried to have a skyscraper that was ten miles high, the force on the base would be so great it would sever the molecular atomic bonds and it would collapse. It would just sink into the Earth’s surface by a [plastic flow].


Well, we are now in business to look at our weightlifting world records because the weight lifted is a measure of the strength, and the weight category that the lifter is in then tells us the weight.  They are quite narrowly defined.  So, we want to check this formula to see whether strength cubed is proportional to weight squared. So here is a log-log plot. So, three times the log of the strength would be a constant, plus two times the log of the weight, so here is the cube of the weight lifted in cubic kilograms, and here is the square of the weight, kilograms squared, of the weightlifter. Looking back a few years, these were the world records, a few years back, a little snapshot. The straight line is this rule. So you will see we are really pretty good in predicting and understanding what is going on, that the reason the world record trends grow as they do is just a reflection of this simple formula.


If you really wanted to take this seriously, it is not very good when you get up to the very, very high weights lifted and weights of lifter because, if the heaviest lifter really did lift what was required to be on this line, he is perilously close to breaking his cartilages, even his bones – you are putting enormous stress on yourself. So the very heaviest lifters tend to drop a bit below this line, and a rather more complicated formula is devised, so-called Simpson’s Rule, for weightlifters, which is used to normalise the performances to one scale. It is not based on physics or a rule like this at all; it is just based on statistics, and fitting them by a handy formula. So what you can also tell from this picture is that the lifter who is most above this line, to the left, is the strongest pound for pound lifter. The one who is most below it is, arguably, this one up here, the sort of world record, or even this one down here – these are the weakest.


Exactly the same principles would be at work if you were looking at how strength determines performance in an event like the hammer or the shot put, where strength is going to determine the launch speed of some projectile. So the same arguments work across different strength and power events.

Well, let us look at one of those, putting the shot. This gentleman is the world record-holder. The world records in these events stem from a very, very long time ago – in women’s events, even more so. As you probably notice, if you follow women’s athletics at all, there are never any world records in women’s athletics unless they are new events. A very large proportion of the world records in women’s events were set in the 1980s. The major reason for that was that then rather stringent drug-testing came into play, for male growth hormone and other drugs, and so athletes today are really competing against standards in the past that were unfairly set. And there is therefore an argument that is fervently put forward by some people that one ought to start again, certainly women’s athletics, with new records. But men’s records in the strength events suffer a little from the same problem. The world record here, 23.12 metres, in the shot. We want to try and understand, in this event, how should you throw this shot in order to throw it as far as possible.


Well, from what you know about projectiles, from what we said earlier on, you would have thought that if you launch an object with some speed V, at an angle theta, then what your lecturers teach you in Mechanics at school and then at university is that the distance of flight, the range of this projectile, depends on the sine of twice the launch angle.  Well, a sine can never be bigger than one, and it is equal to one when two theta is equal to 90 degrees, and so launching at 45 degrees is what gives you the maximum range.


However, putting the shot is not quite as simple as that. There are two reasons why this is not the optimal angle to launch the shot at. 


The first, which you tend to see sometimes in articles on maths and sport, or in books about maths and sport, is that the 45-degree launch angle is the optimum angle, but only when you are launching from the ground and the shot then lands on the ground.  But shot-putters launch from about two metres above the ground, arm’s length from a very large person generally, and in that case, the maximum range formula is rather different. So, the range for a shot launched at a height H above the ground is given by the formula that we just saw, with another correction factor here that depends on that height. So, when the height is zero, this just looks like the previous formula, but with a height of two metres, you are looking at an optimum range that’s somewhere around 43-ish, 43-and-a-bit degrees, so it is a small difference. And you might have thought, well, that really solves that problem.


The unfortunate thing is that, if you look at shot-putters, like our friend here, in detail, and study how they throw things, they do not launch the shot at 45 degrees or even 42 or 43 degrees. They launch the shot closer to about 37 degrees. So you might wonder why on earth would that be. Why do they launch the shot at such a shallow angle?


The answer is that, in these formulae for throwing projectiles some distance, we think of the initial velocity and the launch angle, traditionally, as being two completely separate independent things that we are at liberty to set and achieve without worrying about how they are linked together. So you tend to think, well, our shot-putter can throw the shot at some speed V – how should he arrange that, you know, what should he make the take-off angle? Unfortunately, these are not independent.


If we got into the gymnasium and we study lots of shot-putters, and measure the angle at which they are launching the shot against the speed with which they can launch it, we will find that there is a definite trend. So this is the lab data…and you can understand intuitively what is going on here that as you increase the angle at which you try to launch the shot, the speed that you can achieve goes down. So as you try to apply a force with your arms, at a different angle, your strength really does vary as that angle alters, and that is what this is reflecting, and so, if we produce a simple formula to couple our V to theta, and then we go back to the problem of the range, we are looking to maximise the range, subject to this constraint that links the speed and the angle together, so it is a constrained optimisation problem.


If we look at some answers, there is not just one anymore. Here is the range, here is the angle that you project at, and here are the solutions. You can see that this, for example, here, is giving you an angle of about 38, 39 degrees, for these sorts of ranges up here.  The world record is way up here…world class performance above twenty metres… Again, you are looking at angles in the high-30 to low-40 degrees. So it is a different type of problem.


If you were thinking about long jump, the same type of principles would hold. People often imagine that, if you long jump and you take off at 45 degrees, this is going to give you the maximum distance you could possibly long jump. Nobody can run in at ten metres per second and launch into the air anywhere near 45 degrees, so the take-off angles are much, much lower – you know, maybe ten, eleven, twelve degrees or something like that. No one is strong enough to achieve a launch angle greater than that with those types of horizontal running speeds. 


So, in many of these events that involve throwing things or running and launching yourself, it is not a pure unconstrained projectile problem. There is a link between the launch speed and the launch angle that determines what the ultimate or optimal performance is.


Well, last example…we are going to look at things like this rather later on in the lecture series, about what happens when you apply power in an event and there are a number of people doing it.  The classic example is rowing, of course.  One can do this for canoeing as well, and it is probably more interesting there, but let us look at rowing, this example.


So, if we have a boat with a number of oarsmen in it, then how does the speed which the boat can achieve depend on the number of rowers? You will see there is a sort of catch 22 here because, if you add more rowers, you get more power to move you along, but you are adding more weight to be moved. So what is the winning effect? If a boat moves through the water, the drag on the boat from the water is proportional to the speed squared and it is also proportional to the area of the boat’s surface that is in contact with the water, so it is a classic sort of resisted motion. Well, the surface area of the boat is going to be proportional to some length squared, the sort of radius of the curvature of the hull, and the volume of the boat will be proportional to the cube of that length. So it is our same issue…this is proportional to an area, this to a volume, and this volume will also be proportional to the number of members of the crew, because if you want to fit them in, okay, if you want to double the number of crew members, you have got to sort of double the length of the boat to fit them in. So the drag on the boat, proportional to V squared L squared, is proportional to V squared times N to the two-thirds. So, as you pack more people on the boat, this is how the drag force increases.


Well, what about the power? Here are our rowers. Well, if you have got N people sweating away, and each of them applies a power P – let us assume they are all the same in terms of strength – then the total power will be number of rowers, N, times the power that any one of them applies.  And that’s got to be the power, which overcomes the drag.


Well, if you want a power from a drag force, [by the] force, by the velocity, so it is V cubed times N to the two-thirds, so we’ve got now a relation between N, V and N to the two-thirds. V cubed is proportional to N to the third.  V is proportional to N to the nine. So the speed that the boat will go does indeed increase as the number of rowers increases, but not very fast…it goes as the ninth power of the number of oarsmen.


If you add a cox, an amusing little thing to do, if I assume my cox is a third of the weight of a rower, it is a slight variant because you are adding someone who you have got to fit in, he has got to be towed along, but he is not adding any power…just sort of verbal power maybe. So the formula just changes a little bit, with our third of a person there.


If we look at the 1980 Olympic races, you get a beautiful fit to that formula. So here is the single, [?], okay, and he has won, and here is the pair, the four, the eight…these are times in seconds, okay.  If you have a cox, you can see, when you have two, adding the cox really increases the time; adding the four increases the time… So this is almost exactly what our last formula predicts, so adding the cox slows you down, even though he may be helping you in certain ways – steer a true course or screaming at you to go faster. So if you want to go in a straight line, you are better without one. If you are in the Boat Race, it is essential because you are going round curves – you really need to steer. So, this simple scaling formula tells you how, over two kilometres, the winning time has this rather definite trend, which fits this N to the ninth rule rather well.


Last thought for the day. We have mentioned weight classes. One of the great paradoxes of sport is that we do indeed have weight classes in events like boxing and weightlifting and wrestling and judo, for obvious reasons, but there is a collection of other events that are equally reliant upon pure strength where there are not. So, there are no weight classes in shot put or hammer. There are none in Olympic rowing, although there are in other categories of rowing – so there is a lightweight crew competition, for example, there is a lightweight boat race. But there are no weight categories in these events, and so you can see what happens. If you are five foot tall, you do not become a shot-putter, and all shot-putters tend to become larger and larger and larger…it is a self-selecting bias. So, you could argue that there is a rather good case to have weight categories in shot put and other heavy strength events.


Also, if you are thinking along that line, I remember I gave a talk in Loughborough the other week, at the High Performance Centre in the Mathematics Department, and somebody came up to me afterwards, in the audience, and said he was a basketball player, and what could I recommend to improve his jumping and so forth. Well, the first thing that struck me about him was he was about the same height as me, so I recommended that he grow much taller! So, so it occurred to me that basketball might become a much more interesting sport, at least at the participatory level, if there were height classes for teams, so you would have a competition where nobody can be above seven feet tall, nobody can be above six foot six tall, and so on.


In the high jump, for example, you might wonder whether tallness really is an advantage – an interesting question. Most high jumpers, with just one or two exceptions - in men’s, Jacobs and Holme, are the two world-class high jumpers who are just of ordinary height…okay, 1.81 or something. But pretty much, they are very tall, moving towards two metres or so in height. 


So, for some reason, we have developed weight categories in some sports that reflect the fact that strength grows like that, two-thirds power of weight, and it grows like the square of size, but we have decided not to apply categories of weight or height into other events, for reasons that are not really obvious.


Okay, that is all for today. Thank you.





© Professor John D Barrow 2011

This event was on Tue, 13 Dec 2011


Professor John D Barrow FRS

Professor of Astronomy

Professor John D Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics.

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