On shot put and a remark of Willoughby

I am currently reading the book of  D. Willoughby “The Super-Athletes” published in 1970. The book has a long section of athletics which attracted my interest. While reading it, I fell upon a remark of Willoughby which made me react and I decided to publish my observations in the blog. 

Willoughby presents a table of performances in shot put obtained with implements of non-standard weights ranging from 3.6 to 25.4 kg. 

The performances are pretty old, registered over a period of 15 years from 1905 to 1919. Moreover they are not obtained by a single athlete but from several ones, which makes the comparison somewhat iffy. Still, the tendency is there and Willoughby manifests his astonishment as to the fact that there is no theoretical explanation for his observation that the length of the throw is related to the square root of the implement weight. The reason for this is very simple: the relation observed by Willoughby is a mere coincidence. There is no physical argument in favour of such a square root dependence.

Before proceeding further I would like to make a remark concerning Willoughby’s analysis, a remark which has a larger scope. It has to do with use of the square root. You cannot take the square root of say 16 kg. You can compute the square root only of a pure number. Of course, if you say that the length of the throw is (inversely) proportional to the square root of the mass of the implement, you let the proportionality factor take care of the dimensions. However this is not as simple as it sounds. Since one should work with pure numbers, one must divide the implement’s mass by some other quantity that has the dimensions of mass (and the same applies to the length). The problem is now what is the meaning of this quantity. This is something that can be answered only in the framework of a more or less elaborate model. Which brings us to the main point of this post.

The reason for the absence of a theoretical explanation of the square root is that the mathematical models of shot put do not predict such a dependence. I have published recently an article in New Studies in Athletics dealing precisely with the relation between an implement’s weight and the length of a throw in shot put. The model is based on  the separation of the throw into two phases. The first is the acceleration phase or “developing momentum in the run-up area” according to J. Silvester and the second is the throw itself i.e. “transmitting energy from the body to the implement”. An analysis with some simplifying hypotheses leads to a dependence of the length of the throw L to the implement’s mass m in the form
  

$\text{L}=\text{a}+\frac{\text{b}}{\text{m}}$

However as I explain in the same paper this model is too crude to provide an accurate description of the throwing process over a wide range of implements’ masses. A more elaborate model was thus proposed taking into account the slow-down of the athlete due to an excessively heavy implement and the inertia of the athlete’s body parts (essentially the arm) during the throw phase. In fact the latter turned out to be the main effect, leading to the following relation between L and m

$\text{L}=\frac{\text{d}}{\text{m}+\text{c}}$

Again no square root appears in the model. In order to appreciate the quality of the model we present a fit of Willoughby’s data with the two formulae given above. First we convert the data to the metric system (since working with imperial units is making me jittery).

Mass(kg)	Length (m)
3.63	20.60
5.44	17.55
7.25	15.62
8.16	14.09
9.53	12.87
10.89	11.97
12.50	11.26
15.00	10.63
16.66	9.72
19.05	8.63
25.40	7.91

Using these data we can now present a graphic with the best fit for both expressions.

For the first expression we obtain the dashed curve and parameters a=6.71 m and b=55.1 m kg. The second expression gives a much better fit with d=234 m kg and c=7.93 kg. I find that the data of Willoughby are in perfect agreement with my model. It would suffice to renormalise the parameters obtained so as to make the performance for a 7.25 shot agree with the current world record in order to establish upper limits for throws with lighter of heavier implements. 

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Revolutionary styles

In this post I will concentrate essentially on jumps and throws. This does not mean that there haven’t been revolutions in running events (the standardisation of stadia, the introduction of synthetic tracks, etc.) but they are not based on style. The ones that would qualify as style revolutions are related to hurdling (where what spurred the revolution was the introduction of L-shaped hurdles), with the straight lead leg style for high hurdles or the 13-stride style for low hurdles. However they have by now lost their revolutionary aura, being part of the standard technique.

I would like to start this post with a technique which I have already discussed in my post on flops and bends and which, while revolutionary 50 years ago, is now used by the totality of jumpers. (Jesús Dapena, of Indiana University, has some objections concerning the exclusive use of Fosbury flop, but I will not go into more details here. I suggest that the interested reader track down his work). What I would like to present, related to what is called now the Fosbury flop, is a photo from the 1906 Athens intercalated Olympics. 

I do not know the name of the jumper (he was a participant at the standing high jump event). This is definitely a “back layout” style, to use Fosbury’s own terminology. And that was more than a century ago. Most probably nobody noticed it at that time and if the athlete was not among the medalists the style was waved away as a crazy technique.

The style of triple jump was free till the beginning of the modern era. One could as well jump a step-step-jump or a hop-hop-jump. In fact the first winner of triple jump at the Athens 1896 Olympics used the latter style. With the standardisation of style to hop-step-jump there remained no possibility for major changes and only technical adjustments have been possible since. Long jump on the contrary has known a short-lived revolution in the 70s with the flip (somersault) style.

Alas, the IAAF banned this style on safety grounds and thus we will never know its real potential. I am not going to comment here on pole vault since I intend to devote a whole post to it. This discipline is undoubtedly the one where the style disruption has been radical. 

Concerning throws I must start with a mea culpa. In my post on throwing circles I had attributed the spin technique to Baryshnikov. That’s what I had always thought but while researching for this post I found out that I was wrong. In fact the spin technique is much older. The consensus concerning the inventor of this technique is that it was McGrath, winner of the 1965 USA championships, who converted from the glide to the spin technique with some success. 

His personal record, from 1966, with the O’Brien style, was a solid 19.59 m (and a 58.93 in discuss throw). In 1968 he started throwing with the rotatory style and managed to throw over 18 m. (O’Brien himself stated in an interview that he had experimented with the  spin style with “some success” as far back as 1962, but he never used that style in competition). The one who adopted the rotatory style and brought it to the highest level was Oldfield. In fact at a certain point this technique was called the Oldfield spin. This, highly colourful, athlete was the first to break a world record with his technique. Revolutionary though it may be, the spin style did not manage to displace the more classical glide. In fact Oldfield, now a coach, summarises the situation in a very clear way: “In very general terms, stronger athletes could be gliders. More dynamic athletes, those better at jumping and sprinting, could be spinners”.

Minor revolutions have been attempted in discus and hammer throws. Some people have tried a two-turn discus throw, the best performance to my knowledge being that of Sedjuk at more than 63 m. In the case of hammer throw, Piskunov has tried a 5-turn technique and managed to throw over 80 m. For me, these techniques are condemned from the start because of the inadequacy of the throwing circles. It suffices to have a look at a movie of a two-turn discus throw in order to convince oneself that only a 3 m circle would make this style really efficient. The same goes for the 5-turn hammer throw (but let us keep in mind that the current world record was obtained with a 3-turn technique).

Of course, the greatest revolution in throws was the invention of the spanish style in javelin throw. Unfortunately it was killed immediately by the over-conservative attitude of the international federation. This is a decision that I keep regretting. Throws of over 120 m would have been simply magical.