In previous posts I have already presented the formula that relates the length of a throw

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

The main reason that the length of the throw is not exactly inversely proportional to the mass is the fact that the inertia of the thrower’s arm does enter into play. Were it not for the inertia of the arm, i.e. when

Applying the formula to athletics is far from easy because of lack of data. It is very difficult to find data on athletes who have thrown, in the same year, let alone in the same competition, implements of different weights. In an article of mine which appeared in New Studies in Athletics I managed to get hold of data from a competition in Poland where the athletes competed in shot put with the normal implement of 7.257 kg and afterwards with a 5 kg shot. The arm inertia

On the other hand the same analysis performed on the performances of three top junior athletes, who have registered throws with 7.25, 6 and 5 kg shots gave values in the 5-7 kg range, something much more reasonable. So, I expect that, for athletes who master the throwing style of light shots, the arm inertia will take values close to 5 kg. We can also extrapolate this value to the case of female throwers and wonder what would be the range of their throw with a men’s implement. Taking as an example V. Adams and her 21.34 m personal record assuming an arm inertia in the 4-5 kg range, and moreover that she would be able to accelerate equally fast with the 7.25 kg shot, we obtain a result of the order of 15 m, not bad a performance at all.

One could try to extend this result to the case of the other throws. After all the principle, that of the arm inertia is the same. However the precise value will certainly vary since the throwing style is quite different. On an intuitive basis we expect values smaller than the 5-6 kg of the shot put for all the other throws with values in the range of 2-3 kg for discus and hammer throw and less than 1 kg for javelin throw. Finding data in order to support this estimate is quite hard. However from time to time one gets lucky. While perusing the blog “Throwholics” I came across the junior performances of a portuguese hammer thrower, A. Silva. He has a personal record of less than 68 m but while a junior in 2005 he threw 76.07 m with a 5 kg hammer, 67.93 m with a 6 kg one and 59.05 m with the men’s implement of 7.25 kg. Using the values for the two heavier implements we find the value

Concerning the discus I was able to find some 2013 results by the junior world champion F. Dacres who threw 62.92 m with the 1.75 kg discus and 59.30 with the men’s, 2 kg, implement. The computed value of

In the case of javelin throw things become more muddy a) because of the very small weight difference between men and junior’s implement (just 100 gr) and b) because the light weight of the javelin requires further style adjustments which forces top quality junior throwers to avoid attempts with the heavier javelin. However as luck will have it there exists a most interesting result by the world record holder B. Spotakova. After having thrown 61.75 m in a local competition she participated at the men’s event throwing 51.97 m (with just two tries). These performances give a value of

Finally, no one has being as prolific in his performances with implements of different weights as the world junior champion J. Gill of New Zealand. In 2011 he threw 20.38 m with the men’s, 7.25 kg implement, 22.31 m with the junior, 6 kg, one and 24.45 m with the youth 5 kg shot. Moreover he had a 18.20 m performance with a 8 kg implement and an astonishing 37 m throw with a 1 kg shot.

Using the basic formula and fixing the value of

*L*to the mass*m*of the implement:$\text{L}=\frac{\text{a}}{\text{m}+\text{f}}$

The main reason that the length of the throw is not exactly inversely proportional to the mass is the fact that the inertia of the thrower’s arm does enter into play. Were it not for the inertia of the arm, i.e. when

*f*=0 in the formula above, the range of a very light projectile would tend to infinity when its mass tends to zero, something that is intuitively absurd.Applying the formula to athletics is far from easy because of lack of data. It is very difficult to find data on athletes who have thrown, in the same year, let alone in the same competition, implements of different weights. In an article of mine which appeared in New Studies in Athletics I managed to get hold of data from a competition in Poland where the athletes competed in shot put with the normal implement of 7.257 kg and afterwards with a 5 kg shot. The arm inertia

*f*computed based on their result gave values in the 6-12 kg range. However one must take into account that these athletes did not have any experience with light implements and thus the large value of the computed inertia could be attributed, at least in part, to a sub-optimal style.On the other hand the same analysis performed on the performances of three top junior athletes, who have registered throws with 7.25, 6 and 5 kg shots gave values in the 5-7 kg range, something much more reasonable. So, I expect that, for athletes who master the throwing style of light shots, the arm inertia will take values close to 5 kg. We can also extrapolate this value to the case of female throwers and wonder what would be the range of their throw with a men’s implement. Taking as an example V. Adams and her 21.34 m personal record assuming an arm inertia in the 4-5 kg range, and moreover that she would be able to accelerate equally fast with the 7.25 kg shot, we obtain a result of the order of 15 m, not bad a performance at all.

World and Olympic champion Valerie Adams

One could try to extend this result to the case of the other throws. After all the principle, that of the arm inertia is the same. However the precise value will certainly vary since the throwing style is quite different. On an intuitive basis we expect values smaller than the 5-6 kg of the shot put for all the other throws with values in the range of 2-3 kg for discus and hammer throw and less than 1 kg for javelin throw. Finding data in order to support this estimate is quite hard. However from time to time one gets lucky. While perusing the blog “Throwholics” I came across the junior performances of a portuguese hammer thrower, A. Silva. He has a personal record of less than 68 m but while a junior in 2005 he threw 76.07 m with a 5 kg hammer, 67.93 m with a 6 kg one and 59.05 m with the men’s implement of 7.25 kg. Using the values for the two heavier implements we find the value

*f*=2.8 kg. Using this value and the performance with a 6 kg hammer we can now use the expression for*L*in order to predict the value of the performance for a 5 kg hammer. We find 76 m in perfect agreement with the real value.
AntÃ³nio Vital e Silva

Concerning the discus I was able to find some 2013 results by the junior world champion F. Dacres who threw 62.92 m with the 1.75 kg discus and 59.30 with the men’s, 2 kg, implement. The computed value of

*f*turns out to be 2.4 kg, well within our estimates. (Dacres went on to throw an impressive 66.75 m in his first senior year, but this is another story). Similarly using results of M. Nesterenko’s*annus mirabilis*2008 we find a value of*f*slightly smaller than 2 kg (but his incredible world junior record of 65.31 m with a men’s implement is a blatant foul as one can see in the video).
World junior champion Fedrick Dacres

In the case of javelin throw things become more muddy a) because of the very small weight difference between men and junior’s implement (just 100 gr) and b) because the light weight of the javelin requires further style adjustments which forces top quality junior throwers to avoid attempts with the heavier javelin. However as luck will have it there exists a most interesting result by the world record holder B. Spotakova. After having thrown 61.75 m in a local competition she participated at the men’s event throwing 51.97 m (with just two tries). These performances give a value of

*f*=0.5 kg. This means that, where she to throw the equivalent of her world record with a men’s javelin she would easily attain 60 m. Conversely, where she to throw with a 400 gr javelin (half the weight of the men’s one, just as in the discus and hammer case) one would expect a throw of over 85 m. Be it as it may, this record is still very far from the men’s one of 98 m. And then there is the question of whether an adult thrower can efficiently throw such light implement, normally reserved to master competitors (women over 60 and men over 80 years of age).
World and Olympic champion Barbora Spotakova

Finally, no one has being as prolific in his performances with implements of different weights as the world junior champion J. Gill of New Zealand. In 2011 he threw 20.38 m with the men’s, 7.25 kg implement, 22.31 m with the junior, 6 kg, one and 24.45 m with the youth 5 kg shot. Moreover he had a 18.20 m performance with a 8 kg implement and an astonishing 37 m throw with a 1 kg shot.

World junior champion Jacko Gill

Using the basic formula and fixing the value of

*f*at 6 kg we take*a*=268 m so as to reproduce the performance with a 6 kg shot. We obtain thus the following predictions for implements of weights 8, 7.25, 5 and 1 kg respectively: 19.1, 20.2, 24.4 and 38.3 m in fair agreement with the registered performances (and, quite surprisingly, all way down to 1 kg).
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