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

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

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).

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

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

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

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

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.