Influence of Environmental Factors on Shot Put and Hammer Throw Range

Journal of Biomechanics 35 (2002) 785–796

Inﬂuence of environmental factors on shot put and hammer throw range Ferenc Mizera, Gabor! Horvath*! Department of Biological Physics, Eotv. os. University, Pazm! any! set! any! 1, H-1117 Budapest, Hungary Accepted5 February 2002

Abstract

On the rotating Earth, in addition to the Newtonian gravitational force, two additional relevant inertial forces are induced, the centrifugal andCoriolis forces. Using computer modellingfor typical release heights andoptimal release angles, we compare the inﬂuence of Earth rotation on the range of the male hammer throw andshot put with that of air resistance, wind,air pressure and temperature, altitude and ground obliquity. Practical correction maps are presented, by which the ranges achieved at different latitudes and/or with different release directions can be corrected by a term involving the effect of Earth rotation. Our main conclusion andsuggestion is that the normal variations of certain environmental factors can be substantially larger than the smallest increases in the world records as acknowledged by the International Amateur Athletic Federation and, therefore, perhaps these should be accounted for in a normalization and adjustment of the world records to some reference conditions. Although this suggestion has certainly been made before, the comprehensiveness of our study makes it even more compelling. Our numerical calculations contribute to the comprehensive understanding and tabulation of these effects, which is largely lacking today. r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Hammer throw; Shot put; Range correction; Environmental factors; Earth rotation

1. Introduction estimation shows that under certain circumstances the change of the drag-free range of the two most inertially Nowadays, differences DLr of the throw distances Lr dominated throws, the shot put and hammer throw due of consecutive worldrecordsare only a few centimetres to variations in latitude can be greater than DLr: A in both the female andmale shot put, anda few further question is whether this change of the throw decimetres in the male hammer throw (Table 1). Thus distance is not overwhelmed by the effect of natural the question arises whether DLr have become compar- fluctuations of other environmental factors, such as able to the change DL of range L due to usual variations altitude, wind, or variation of the air density due to a of environmental factors. For the sake of fair compar- change in air pressure andtemperature, for instance. ison of sport performances, certain rules must be Although numerous excellent comprehensive and followedin international athletic competitions. For detailed analytical and computer studies exist in the example, a 2 m=s tail windspeedlimit for record literature on the mathematics andmechanics of the shot purposes is appliedby the International Amateur put andhammer throw (e.g. Garfoot, 1968; Tutevich, Athletic Federation (IAAF) to sprint races and hor- 1969; Lichtenberg andWills, 1978; Zatsiorsky et al., izontal jumps. Windalso appreciably influences the 1981; Hay, 1985; Hubbard, 1989; de Mestre, 1990; range of the throwing events. Zatsiorsky, 1990; Hubbardet al., 2001), detailed On the rotating Earth, due to the centrifugal and numerical computations of the effect of Earth rotation Coriolis (inertial) forces, the latitude and the release on range have never been reportedpreviously. Heiska- direction also influence the motion. A gross analytical nen (1955) gave a gross analytical estimation for the effect of the centrifugal force on the drag-free range of *Corresponding author. Tel.: +36-1-372-2765; fax: +36-1-372- the four throwing events. We do not know of any 2757. publishedcalculations of the effect of the Coriolis force E-mail address: [email protected] (G. Horvath).! on range.

Table 1 (a) Progression of the ten best consecutive worldrecords( Lr), their ω ω differences (DLr) andthe years in which they were achievedfor the male hammer throw andshot put (MegedeandHymans, 1991). The North Pole θ z present worldrecordshave not fallen since 1986 and1990

Male hammer throw Male shot put Rp R

Lr (cm) DLr (cm) Year Lr (cm) DLr (cm) Year ϕ rotating Earth equator with geoid shape 8038 6 1980 2200 15 1976 Re 8046 8 1980 2215 15 1978 8064 18 1980 2222 7 1983 8166 102 1980 2262 40 1985 8180 14 1980 2264 2 1986 South Pole 8389 209 1982 2272 8 1987 8414 25 1983 2284 12 1987 8634 220 1984 2291 7 1987 8666 32 1986 2306 15 1988 (b) throwing ground 8674 8 1986 2312 6 1990 North release x z velocity release v0 angle α distance (range) To fill this gap, we performedan in-depth computer release height h L throw modelling (Mizera, 1999) including, in addition to the West East y release direction effect of Earth rotation, all other relevant environmental β (azimuth angle) factors affecting the range of the male shot put and South hammer throw. We computedandcomparedthe effect of variation in air resistance (drag), wind speed, air Fig. 1. (A) Geometry andvariable definitionsof the rotating Earth pressure and temperature, altitude, latitude, release geoid. (B) Definition of computer simulation variables. Terminology directionandgroundobliquity on throw distancefor andgeometry of the release ( v0; h; a; b) andimpact ( L) parameters. typical release (initial) conditions. Practical correction maps were computed, by which the ranges achieved at different latitudes and=or with different release directions can be correctly comparedwith each other in such 2. Computer modelling of the flight trajectory of the shot a way, that the throw distances are corrected by a term and hammer involving the influence of Earth rotation. We show that the magnitude of change in range due to changes in Consider a Cartesian system of coordinates x–y–z latitude is comparable to or even larger than those of fixedto the Earth surface (Fig. 1B). The x- and y-axis other environmental factors. Analysing the evolution of point northwardandwestward,respectively, andthe z- the worldrecordsof shot put andhammer throw, we axis points vertically upwards (perpendicular to the demonstrate that the time has come to take into account geoid surface, Fig. 1A). According to Landau and the effect of certain environmental variables on range. Lifschitz (1974), in this rotating reference system the In this work the word‘‘environmental’’ is usedto motion equation of a thrown shot or hammer is include all factors examined, even Coriolis and centrifugal force effects which depend abstractly on latitude d2r dr and direction in much the same way as air density does m % ¼ mgðj; HÞþ2m % Â o 2 % % on altitude. To our knowledge, this is the first precise dt dt analysis of the effect of Earth rotation on throw distance kArðp; TÞ dr 2 À % À W ; ð1Þ in comparison with the influences of other environ- 2 dt % mental factors. In this work we deal only with the male shot put and hammer throw. The corresponding female throwing where Â represents the vector cross product. The events wouldrequire a separate study,because the variables andformulae usedin the computer simulation masses anddimensions of the women’s implements are are the following (Fig. 1): different. In the discus and javelin throws, due to (i) r ¼ðx; y; zÞ: position vector of the implement. % atmospheric fluctuations (windandchange of air (ii) j: latitude (positive or negative on the northern or density), the effect of aerodynamics on range may southern hemisphere, respectively). overwhelm the influence of all other environmental (iii) H: altitude determining the gravitational factors. acceleration g andthe air pressure p in the following F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 787 way: the drag coefficient of the shot (k ¼ 0:47) andhammer s (kh ¼ 0:70). The exact value of kh varies along the 2H À0:000123 mÀ1 H gðHÞ¼g0 1 À ; p ¼ p0 e ; trajectory and depends on the attitude of the wire and R handle relative to the flight path. Furthermore, the 2 6 g0 ¼ 9:81 m=s ; R ¼ 6:368 Â 10 m; handle moves erratically around the head during flight, p0 ¼ 101 325 Pa; ð2Þ the consequence of which is that its drag varies too. k ¼ 0:70 is an averagedeffective value (Hubbard, where R is the average of the Earth radius. h 1989). In the case of the typical average velocities of (iv) o: Earth angular velocity vector, with amplitude % 15 and30 m =s of the shot andhammer duringflight, o ¼ 7:27 Â 10À5 sÀ1; anddirection parallel to the axis of respectively (Hubbard, 1989), the air resistance is rotation. Its Cartesian coordinates would be o ¼ x proportional to the square of the relative velocity vector o cos j; o ¼ 0; o ¼ o sin j if the Earth possesseda y z dr=dt À W: spherical shape. However, the Earth has a flattened % % Motion equation (1) involves all relevant environ- geoidshape, approximately an ellipsoidwith rotational mental (physical andmeteorological) factors determin- symmetry (Fig. 1A). Due to this flattenedshape, ing the trajectory of a thrown shot or hammer. We apart from the equator andthe poles, at latitude j the solved(1) by means of Runge–Kutta numerical integra- angle between o andthe Earth surface is yðajÞ: If R % e tion of fourth order. and Rp are the Earth equatorial andpolar radii, respectively, y andthe coordinates of o can be expressed % as (Fig. 1A) 3. Results R2 tan y ¼ e tan j; o ¼ o cos y; o ¼ 0; R2 x y p In the case of the male world-record hammer throw oz ¼ o sin y; ð3Þ (achievedin 1986) the initial conditions of the computer simulation were (Fig. 1B): h ¼ 1:8m; a ¼ 441 (ballisti- where R ¼ 6:378 Â 106 m; R ¼ 6:357 Â 106 m: e p cally optimal release angle) and v ¼ 29:28 m=s corre- (v) mg: gravitational force, where the ‘‘apparent’’ 0 % sponding to a throw with L ¼ 86:74 m: For the gravitational acceleration (Caputo, 1967) male world-record shot put (achieved in 1990) the g ¼ðg ; g ; g Þ¼½0; 0; ÀgðjÞ; 1 % x y z release conditions were: h ¼ 2:25 m; a ¼ 37 and v0 ¼ 14:5m=s corresponding to a throw with L ¼ 23:12 m: In gðjÞ¼9:78049 ð1 þ 0:0052884 sin2 j simulations studying the influence of Earth rotation on À 0:0000059 sin2 2jÞ m=s2 ð4Þ range, the standard values of environmental parameters were: air temperature T0 ¼ 293 K ¼ 201C; air pressure involves both the geoid Earth shape (latitude dependent p0 ¼ 101 325 Pa; altitude H ¼ 0m; groundobliquity Earth radius) and the Earth rotation (latitude dependent o ¼ 0: As reference city, to which all records are vertical component of the centrifugal acceleration). normalized, we chose Athens. This was an obvious (vi) W: windvelocity vector. % choice, because both the next (in 2004) andfirst Olympic (vii) r ¼ 1:23 kg=m3: air density under normal con- Games will be/were heldthere, andbecause it is at a ditions (temperature T0 ¼ 293 K andpressure reasonably central latitude. Thus, as reference latitude p0 ¼ 101 325 Pa). It depends on p and T as follows: we chose the latitude of Athens j ¼ 381; andas p reference release direction we used the northern direc- rðp; TÞ¼ ; Q ¼ 287:05 J=kg K: ð5Þ QT tion b ¼ 01 (Fig. 1B). (viii) b: release direction (or azimuth angle) measured counter-clockwise from North. 3.1. Range versus latitude (ix) a: release angle measuredfrom the horizontal. (x) h: release height measuredfrom the ground. Fig. 2 shows the throw distance of the world-record Although it is thrower-specific, we assumed hh ¼ 1:8m hammer throw as a function of latitude for four for the hammer throw and hs ¼ 2:25 m for the shot put release directions. At a given latitude, due to the (Hubbard, 1989). Coriolis force, the difference of the ranges for different (xi) v0: release velocity. release directions is maximal at the equator and (xii) m ¼ 7:26 kg: mass of the male shot andhammer. gradually decreases to zero toward the poles. At a (xiii) t: time measuredfrom the moment of release. given release direction the range gradually increases dr % (xiv) 2 m dt Â o: Coriolis force. from the poles towardthe equator because of the krA dr % 2 (xv) ð % À WÞ : drag force due to air resistance, centrifugal force andincreasing Earth radius, which 2 dt % where A is the projectedarea of the shot have much greater influences on range than the Coriolis 2 2 (As ¼ 0:0095 m ) andhammer ( Ah ¼ 0:0138 m ), k is force. 788 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796

86.80 Table 2 Male world-record hammer and shot range L as a function of altitude 86.75 East H for Athens’ latitude j ¼ 381; o ¼ 0; W ¼ 0m=s; T ¼ 201C: The North, South 86.70 West dependence is approximately linear 86.65 Altitude H 0 200 400 600 800 1000 86.60 (m) 86.55 Hammer 86.74 86.86 86.97 87.08 87.19 87.29 throw L (m) 86.50 Shot put L 23.120 23.125 23.129 23.133 23.137 23.142

throw distance L [m] 86.45 (m) 86.40 86.35 Table 3 86. 30 0o 45o 90o The effect of vertical level difference Dz along throw distance on the equator latitude ϕ pole male world-record hammer and shot range L for Athens’ latitude j ¼ 381; T ¼ 201C; p ¼ 101 325 Pa; H ¼ 0m; W ¼ 0m=s: The depen- Fig. 2. Male world-record hammer throw distance L as a function of dence is approximately linear latitude j for four release directions (East, West, North, South) when T ¼ 201C; p ¼ 101 325 Pa; H ¼ 0m; o ¼ 0; W ¼ 0m=s: Dz (cm) À20 À10 0 10 20 Hammer throw L (m) 86.93 86.84 86.74 86.65 86.55 Shot put L (m) 23.33 23.22 23.12 23.02 22.91 hammer throw shot put 90 23.25 shot 89 put 23.20 88 23.15 world record lives at altitudes o1000 m; andevery Summer Olympic 87 world record shot put 23.10 Games venue in the last 100 years has occurredat hammer throw 86 23.05 altitude o100 m with only three exceptions (Munich, hammer H ¼ 320 m; Atlanta, H ¼ 450 m; andMexico City, 85 23.00 throw H ¼ 2200 m). The increase of range with altitude is due throw distance L [m] 84 22.95 to the decrease of the gravitational acceleration g and 83 22.90 the air density r; the latter as a result of the decrease in air pressure p: 82 22.85 -10 -5 0510 head wind tail wind 3.4. Inﬂuence of ground obliquity wind velocity W [m/s] parallel to the release direction

Fig. 3. Male world-record hammer throw distance L as a function of Table 3 shows the influence of groundobliquity on windvelocity W parallel to the release direction for Athens’ latitude range of the hammer throw andshot put. If the vertical j ¼ 381; T ¼ 201C; p ¼ 101 325 Pa; H ¼ 0m; o ¼ 0: level difference of the throwing ground is Dz along the throw distance L; groundobliquity is definedas o ¼ Dz=L: The greater the level difference, the more the 3.2. Influence of wind range differs on an oblique ground from that on a horizontal ground. The maximal obliquity permitted by Fig. 3 plots the influence of wind(parallel to the the IAAF along the range is omax ¼ 70:001: The release direction) on hammer throw and shot put range maximal permittedvertical level difference of the at the reference latitude of Athens j ¼ 381; as a function groundis about 78:7and72:3 cm for the hammer of the component of the windvelocity vector parallel to throw (L ¼ 86:74 m) andshot put ( L ¼ 23:12 m), the release direction. The effect of the wind speed on respectively. range is non-linear. The increase in range from a tail windis smaller than the decrease from a headwindof 3.5. Influence of air pressure and temperature the same speed. Tables 4 and5 show the influence of air pressure p 3.3. Range versus altitude andtemperature T on the hammer throw andshot put distance. We used Dp ¼ 72 kPa and DT ¼ 7201C for Table 2 shows how the throw distance of the hammer the maximal size of reasonable air pressure and throw andshot put increases with altitude H ranging temperature perturbations, respectively, because gener- between 0 and1000 m : We chose this altitude range, ally, larger changes do not occur under normal because the majority of the world’s human population meteorological conditions. F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 789

Table 4 +1.5 Male world-record hammer and shot range L as a function of air 1 1 pressure p for Athens’ latitude j ¼ 38 ; T ¼ 20 C; H ¼ 0m; o ¼ 0; +1.0 W ¼ 0m=s: The dependence is approximately linear

Air pressure p (kPa) p0 À 2 p0 À 1 p0 ¼ 101 325 p0 þ 1 p0 þ 2 +0.5 Hammer throw L (m) 86.83 86.78 86.74 86.70 86.66 Shot put L (m) 23.122 23.121 23.120 23.119 23.118 0.0

-0.5 L [cm] of throw distances between ∆ Table 5 left and right sector border Male world-record hammer and shot range L as a function of air -1.0 temperature T for Athens’ latitude j ¼ 381; p ¼ 101 325 Pa; H ¼ 0m; difference o ¼ 0; W ¼ 0m=s: The dependence is approximately linear -1.5 0o 50o 100o 150o 200o 250o 300o 350o 1 Air temperature T ( C) 10 15 20 25 30 35 North West South East North sector centre line direction Hammer throw L (m) 86.57 86.66 86.74 86.83 86.91 87.00 Shot put L (m) 23.115 23.118 23.120 23.123 23.125 23.128 Fig. 5. The difference DL between the male world-record hammer throw distances calculated for the left and right border of 401 throwing sector as a function of the orientation of the sector centre line measuredcounter-clockwise from North for Athens’ latitude j ¼ 381; 86.80 latitude T ¼ 201C; p ¼ 101 325 Pa; H ¼ 0m; o ¼ 0; W ¼ 0m=s: 86.75 86.70 86.65 86.80 86.60 86.55 86.75 world record 86.50 86.45 86.70

throw distance L [m] 86.40 East 86.35 86.65 West 86.30 86.25 86.60 0o 45o 90o 135o 180o 225o 270o 315o 360o North, South North West South East North release direction β throw distance L [m] 86.55

Fig. 4. Male world-record hammer throw distance L as a function of 86.50 release direction b for Athens’ latitude j ¼ 381; T ¼ 201C; p ¼ ballistically optimal release angle = 44,2º 101 325 Pa; H ¼ 0m; o ¼ 0; W ¼ 0m=s: 86.45 42o 42,5o 43o 43,5o 44o 44,5o 45o 45,5o 46o release angle α

3.6. Range versus release direction Fig. 6. Male world-record hammer throw distance L as a function of release angle a for four release directions (East, West, North, South) calculatedfor Athens’ latitude j ¼ 381; T ¼ 201C; p ¼ 101 325 Pa; Fig. 4 plots the hammer throw distance as a function H ¼ 0m; o ¼ 0; W ¼ 0m=s: The position of the maxima of the curves of release direction for ten latitudes. At a given latitude, belonging to the different release directions is practically the same and throw distance changes approximately sinusoidally with it is markedwith a vertical broken line corresponding to the value of release direction. the ballistically optimal release angle. In a stadium, the shot and hammer must hit the groundwithin a 40 1 sector. Fig. 5 demonstrates for 3.7. Range versus release angle reference latitude j ¼ 381 that, due to the Coriolis force, there is a small difference between the throw distances Fig. 6 plots the hammer throw distance for Athens for the left and right border of the sector. This difference (j ¼ 381) as a function of release angle for four release depends approximately sinusoidally on the direction of directions. Since the Coriolis force is relatively small, the the sector centre line. If the centre line points northward ballistically optimal release angle (when the range is or southward, the difference is maximal (about 1 cm for maximal) is practically independent of the release the hammer throw). Because the Coriolis force increases direction. We can see from Fig. 6 that the ballistically with decreasing latitude, this difference is somewhat optimal hammer release angle is about 44:21; which is greater at the equator (j ¼ 01) than in Athens optimal only if release velocity is independent of release (j ¼ 381), andcan reach a value of about 1 :5cm: In angle. Since no experiments have been carriedout the case of shot put this difference is o2mm; which can to determine the sensitivity of release velocity in be neglected. the hammer throw, this is probably a reasonable 790 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 assumption even though the true optimal release angle is the change of DL computedfor the male world-record likely o44:21: The ballistically optimal release angle is range of the hammer throw andshot put as a function of about 42:21 for the shot put. However, probably due to release direction b: Table 6 gives a quick overview of the ability of the thrower to generate higher release extrema of correction terms in the maps. speeds at lower angles, shot putters use a nominal In the ‘‘correction maps’’ of range L in Figs. 7–10A release angle of about 371 rather than ca. 421 (Hubbard, the calculatedvalues of DL are representedby different 1989). Recently, Hubbardet al. (2001) studiedthe grey shades within an annular area on the horizontal dependence of release velocity on other release variables throwing ground. In these angular coordinate systems in the shot put. the range is measuredradially,andthe angle measured from North is the release direction. As a demonstration 3.8. Correction map of the range versus latitude for using the correction maps in practice, consider the following example: In an athletic competition in Variations in latitude and release direction influence Sodankyla. a male hammer thrower reacheda range of hammer throw andshot put range. In orderto compare 70 m with an eastwardrelease direction.His trainer (or correctly the hammer throw or shot put ranges L the spectators, or officials) wouldlike to know how achieved with different release directions b at different great wouldbe his range in the reference stadium in latitudes j; the measuredthrow distances must be Athens. One first selects the corresponding correction modified in such a way that a term DL corresponding to map (Fig. 7A) then the eastwarddirection( b ¼ 2701) the influence of Earth rotation is added to them. This andmeasures a section, corresponding to the range L ¼ term, calledthe ‘‘correction term’’ of range, corrects L 70 m; from the centre of the map. Thereafter the value for the change of b and j on the rotating Earth. The of the correction term DL is readon the map. Finally, correction term DLðL; b; jÞ is positive or negative if the adding the correction term to the range L ¼ 70 m; the range increases or decreases due to Earth rotation, correctedrange is obtained. respectively. Measuredthrow distances modifiedby the The correction maps in Figs. 7–10B are asymmetric. correction term Lc ¼ L þ DLðL; b; jÞ can be considered One of the reasons for this is the non-linear effect of the as the ranges of imaginary hammer throws or shot puts wind. As we have seen above (Fig. 3), a head wind with performedwith the same reference release directionand a given speeddecreases the range to a greater extent at the same reference latitude. than the enhancement of the throw distance due to a tail To compute the extrema of the correction term DL windwith the same speed.The secondreason for the which can occur on the rotating Earth, as the extrema of asymmetry is the influence of the Coriolis force: the the latitude we chose the highest jmax ¼ 67:51 ranges are slightly greater eastwardthan westward. . (Sodankyla) andlowest jmin ¼À17:51 (Papeete) latitudes at which throwing competitions were ever organized. The northern direction b ¼ 01 was set as 4. Discussion the reference release direction at the reference latitude j ¼ 381 of Athens. Hence our imaginary ‘‘reference 4.1. Evolution of hammer throw and shot put world stadium’’ is placed in Athens, where the release direction records points northward. The numerical values of the correction term DLðL; b; jÞ were computedfor differentvalues The last 10 worldrecordsof the male hammer throw of LminpLpLmax and0 1pbp3601 for a stadium in andshot put are given in Table 1. On the basis of our Sodankyla(. j ¼ 67:51) andPapeete ( j ¼À17:51) in the results it is evident that the differences DLr of the ranges case of the hammer throw (Figs. 7 and8) andshot put of the consecutive worldrecordsare comparable to, and (Figs. 9 and10). in many cases smaller than, the values of the correction Figs. 7–10A and7–10B show the maps of the term DL of the world-record ranges (Table 6). Thus correction term DL of the hammer throw andshot put in our opinion, it wouldbe time to take into considera- range L computedfor calmness andan east-windwith a tion the influence of Earth rotation andother environ- speedof 2 m =s (permittedat tail windby the IAAF for mental factors (Table 7) on the range of these throwing recordpurposes to sprint races andhorizontal jumps). events. It wouldbe logical to accept the corrected Every point of the maps gives the numerical value of throw distances Lc as the real performances. These distance DL; by which the hammer or shot thrown in correctedranges can more fairly andphysically be Sodankyla. or Papeete wouldfly further or closer in comparedcorrectly with each other. Figs. 3, 7–10B, comparison with an imaginary hammer or shot thrown 7–10D andTable 6 demonstrate that a relatively small northwardin Athens on horizontal groundwith the windspeedof 2 m =s; for instance, has already so same initial (release) conditions and the same air great influence on the shot put andhammer throw temperature T ¼ 201C; air pressure p ¼ 101 325 Pa ranges that it shouldnot be permittedin the approval of andaltitude H ¼ 0m: Figs. 7–10C and7–10D show the worldrecords. F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 791

hammer throw in Sodankyla, reference stadium in Athens ∆ step of L = 2 cm +20.6 (a) (c) +20.4

+20.2

N +20.0 L [cm] of the

β ∆ +26 +12 +12 +24 calmness cm WE cm +19.8 S +19.6

+19.4 world-record hammer throw

calmness correction term +19.2

world record

+100 (b) (d) +80

+60

L [cm] of the +40 east-wind of 2 m/s

east-wind ∆ -64 -16 +40 +128 +20 cm cm cm cm 0 2m/s -20 world-record hammer throw

correction term -40

-60 step of ∆L=8cm NorthWest South East North release direction β

Fig. 7. (A, B) Map of the correction term DL of the hammer throw distance L computedfor calmness (A) andan east-windwith a speedof 2 m =s (B) for a stadium at the latitude of Sodankyla(. j ¼ 67:51) if the reference stadium is at the latitude of Athens (j ¼ 381) where the reference release direction is North (b ¼ 01). The border lines of the alternating grey and white zones represent different DL values changing between the numerical values given at the margins of the annular maps with a step of DL ¼ 2 cm (A) and8 cm (B). Every point of the map gives the numerical value of distance DL; by which the hammer thrown in Sodankyla. wouldﬂy further or closer in comparison with an imaginary hammer thrown northwardin the reference stadium in Athens on horizontal ground with the same initial (release) and environmental conditions T ¼ 201C; p ¼ 101 325 Pa; H ¼ 0m: The inner black circle in the annular maps represents the male worldrecord Lr ¼ 86:74 m achievedin 1986. (C, D) Change of DL computed for the male world-record range Lr of the hammer throw as a function of release direction b:

4.2. Comparison of environmental factors influencing 4 cm (headwind),3 cm (tail wind),respectively. Thus, it range is pertinent to suppose that the victory of certain hammer throwers andshot putters might have been due We now discuss the influence of environmental factors to an advantageous wind. on the hammer throw andshot put range. Table 7 Altitude - gravity þ air density - air drag: Altitude summarizes the possible maximal change jDLjmax of the H influences range L indirectly (Table 2) through four throw distance L due to variations of different physical factors: andmeteorological variables. (A) Gravitational acceleration g decreases with alti- Air resistance: In a vacuum the worldrecordsof tude, which has two consequences: hammer throw (L ¼ 86:74 m) andshot put ( L ¼ (A1) The shot or hammer becomes lighter, thus the 23:12 m) wouldbe 88 :09 m (DL ¼ 135 cm) and thrower can release it with a greater initial velocity. 23:24 m (DL ¼ 12 cm), which wouldmean an enhance- Although reduced gravity increases vertical velocity of ment of 2.8% and0.6%, respectively. Air draghas a the implement at release, it wouldbe difficultto considerable effect on hammer throw and shot put calculate the size of this effect. Thus, we did not take ranges. it into consideration, restricting our calculations to Wind - air drag: Since air drag depends on the throws with the same intial velocity. velocity of the thrown implement relative to the air, (A2) In a weaker gravitational field at a higher altitude windstrongly influences the range. A relatively small the shot and hammer fly farther. The range is about 2 windvelocity of 2 m =s parallel to the release direction L ¼ v0=g for an approximately optimal release angle of results in a change of the hammer throw andshot put 451: The relative change DL=L of the range due to a range of about 66 cm (head wind), 62 cm (tail wind) and relative change Dg=g of the gravitational acceleration 792 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796

hammer throw in Papeete, reference stadium in Athens

step of ∆L=2cm -10.5 (a) (c) -11.0

of the -11.5

N -12.0

L [cm]

β ∆ -12.5 -12 -6 -8 -18 hammer throw cm cm calmness WE -13.0

n term S -13.5

d-record -14.0

worl

calmness correctio -14.5

male world record +60 (b) (d) +40

of the +20 east-wind of 2 m/s east-wind 0

L [cm]

∆

-104 -32 +24 +88 hammer throw cm cm cm cm -20

2m/s n term -40

d-record -60

worl correctio -80 step of ∆L=8cm NorthWest South East North release direction β

Fig. 8. As Fig. 7 if the stadium is at the latitude of Papeete (j ¼À17:51).

shot put in Sodankyla, reference stadium in Athens step of∆ L = 0.5 cm +5.30 (a) (c) +5.25

of the

+5.20

shot p

L [cm]

∆ +8.0 +7.5 +5.15 cm cm calmness

term

n +5.10

world-record +5.05

calmness correctio +5.00 male world record +10 (b) (d) +9

of the

ut +7 east-wind east-wind of 2 m/s shot p +6

L [cm]

∆ +1.5 +1.5 -6.0 -14.5 +5 cm cm cm cm

term +4

n 2m/s +3

world-record +2

correctio +1 step of∆ L = 0.5 cm NorthWest South East North release direction β

Fig. 9. As Fig. 7 for shot put. F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 793

shot put in Papeete, reference stadium in Athens step of∆ L = 0.5 cm -2.8 (a) (c)

the -2.9

ut -3.0

N shot p -3.1

∆ -4.0 -2.0 β -2.0 -5.0 calmness cm cm -3.2 WE

d-record S -3.3

worl -3.4

calmness correction term L [cm] -3.5 world record +1 (b) (d) 0

the

of -1

-2 east-wind east-wind of 2 m/s

shot p

-10.0 -4.0 0 +1.5 ∆ -3 cm cm cm cm -4

d-record 2m/s -5

worl -6

correction term L [cm] -7 step of∆ L = 0.5 cm NorthWest South East North release direction β

Fig. 10. As Fig. 8 for shot put.

Table 6 The extrema between which the correction term DL of the world-record range L of the male hammer throw (Figs. 7, 8) andshot put (Figs. 9, 10) varies computed for a stadium at the latitude of Sodankyla(. j ¼ 67:51) or Papeete (j ¼À17:51) if the reference stadium is in Athens (j ¼ 381) where the reference release direction is North (b ¼ 01), andthere is calmness or an east-windwith a speedof W ¼ 2m=s

Male hammer throw Male shot put

Papeete Sodankyla. Papeete Sodankyla.

Calmness À14:1pDLp À 10:9cm þ19:2pDLp þ 20:5cm À3:4pDLp À 2:9cm þ5:05pDLp þ 5:25 cm East-wind À72:5pDLp þ 53:0cm À40:7pDLp þ 85:8cm À6:3pDLp þ 0:6cm þ1:9pDLp þ 9:0cm

is therefore: temperature were the same as at the lower altitude. If the DL v2 Dg 2H temperature at the higher altitude was lower, then that ¼À 0 Dg ¼À E À if H5R : ð6Þ wouldbe taken into account separately as a pure L g2L g R Earth Earth temperature effect. (B) Air pressure p decreases with altitude. Average air The hammer andshot ranges increase with altitude pressure p decreases approximately exponentially with predominantly because of the decrease of the air density altitude H [see Eq. (2)]. The exact change of p depends with decreasing air pressure. The decreasing air tem- strongly on the meteorological conditions, nevertheless perature andgravitational acceleration on range is much the relative change of p can reach about 30% at altitudes smaller. of several thousands m. The decrease of p results in a Latitude - centrifugal force þ earth radius change: decrease in air density r; which decreases the air drag, Due to the centrifugal acceleration andthe increase of the consequence of which is an increase in distance the Earth radius from the pole to the equator, the net (Table 4). gravitational acceleration gðjÞ changes as a function of (C) Air temperature T decreases with altitude.We latitude j: On the basis of the Cassini formula (Caputo, excluded air temperature from the calculation of the 1967), the relative change of g between the equator and effect of altitude, because it was kept as a separate factor the poles is about 0.53%. This results in a change of (Table 5). In other words, our calculations ﬁgure out the hammer throw andshot put worldrecordsof about what wouldhappen in a competition at higher altitudeif 45 and11 cm ; respectively. However, there are no 794 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796

Table 7

The possible maximal change jDLjmax of the throw distance L due to the variation of different environmental factors in order of importance computedfor the male world-recordhammer throw andshot put

Environmental variable Change of the variable Physical factor(s) Male world-record Male world-record affecting the range hammer throw shot put (L ¼ 86:74 m) (L ¼ 23:12 m)

jDLjmax jDLjmax

3 3 Air density r0 ¼ 1:23 kg=m Dr ¼ 1:23 kg=m Air drag 477 cm 13 cm air - vacuum

Windvelocity W0 ¼ 0m=s DW ¼ 72m=s parallel Air drag 66h þ 62t ¼ 128 cm 4h þ 3t ¼ 7cm to the release direction h: headwind t: tail wind Altitude H0 ¼ 0m DH ¼ 1000 m Gravity, air density 55 cm 2 cm (air drag)

Latitude j Dj ¼ 901; poles - equator Centrifugal force þ 45 cm 11 cm Eastwardthrow ( b ¼ 2701) Earth radius change

Latitude j Dj ¼ 67:51 À 17:51 ¼ 501; Centrifugal force þ 34 cm 9 cm Sodankyla. - Papeete Earth radius change eastwardthrow ( b ¼ 2701)

Air temperature T0 ¼ 201C DT ¼ 7101C Air density (air drag) 2 Â 17 ¼ 34 cm 2 Â 0:5 ¼ 1cm

Groundobliquity o0 ¼ 0 Domax ¼ 70:001 Gravity 2 Â 8:7 ¼ 17:4cm 2Â 2:3 ¼ 4:6cm

Air pressure p0 ¼ 101 325 kPa Dp ¼ 72 kPa Air density (air drag) 2 Â 8 ¼ 16 cm 2 Â 0:2 ¼ 0:4cm

Release direction b Db ¼ 1801; East - West Coriolis force 3:4cm 0:6cm throw on the Equator (j ¼ 01)

Release direction b Db ¼ 401; left - right border Coriolis force 1:5cm 0:2cm of the sector, northwardor southwardpointing centre line

world-class competitions outside the latitude range the groundof 72:3 cm and 78:7 cm for the shot and 17:51ojjjo67:51: The hammer throw andshot put hammer worldrecordsof 23.12 and86 :74 m; respec- range difference between latitudes of 17:51 and67 :51 is tively. Since both hit the groundat an angle of about about 34 and9 cm ; respectively. 451; this means that above such an oblique groundthe Air temperature - air density - air drag: Air shot andhammer flies 2.3 and8 :7 cm more or less temperature T influences range (Table 5) indirectly horizontally. Thus, a shot putter or hammer thrower through density [see Eq. (5)]. Usually, T changes by less throwing on a sinking ground(with o ¼À0:001) has an than a few tens Kelvin; for track andfieldcompetitions advantage of about 2 Â 2:3 ¼ 4:6cm; or 2 Â 8:7 ¼ a range of 151CpTp351C is reasonable. For example, 17:4 cm against a shot putter or hammer thrower the hammer worldrecordof 86 :74 m wouldincrease by throwing on a rising ground(with o ¼þ0:001). about 17 cm; if the throw were performedat 30 1C Air pressure - air density - air drag: Range is insteadof 20 1C: The implication is that, unless one influencedby the air pressure (Table 4) indirectly travels to a town at different altitude, the effect of through air density, and thus via the air resistance temperature variation may be larger than that of being proportional to the air density r: Air pressure pressure variation. varies about 2 kPa among cities at similar altitudes. A Groundobliquity - gravity: It can easily happen 72 kPa pressure change wouldhave an effect in the that the throwing groundis not exactly horizontal. Since order of 16 cm on hammer range. Air pressure has an due to the gravitation the time of flight and the range of important effect on range only when it varies by a large the shot or hammer depends on the degree of sinking or amount, but this occurs only when the two cities are at rising of the ground(Table 3), groundobliquity must be very different altitudes. Since the not-altitude-dependent limited. The maximal obliquity of 70:001 permittedby part of the air pressure variation is quite small, it has the IAAF is equivalent to a maximal change in level of only a small effect on distance. F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 795

Release direction - Coriolis force: The release would change. Since meteorological conditions during direction influences range because of the Coriolis force, earlier athletic competitions are usually unknown, the which is maximal on the equator andzero at the poles most one couldreconstruct subsequently—on the basis (Figs. 2, 4–6, 7–10C). In the case of eastwardor of the latitudes of the stadia and the orientations of the westwardrelease directions the range is increasedor throwing sectors—is the extent to which the Earth decreased, respectively, by the Coriolis force in both the rotation helpedor hinderedthethrowers. northern andsouthern hemispheres. For northwardor For example, the advantageous or disadvantageous southward release directions the Coriolis force does not influence of latitude on throw distance could be change the range. Using computer simulation, we eliminatedin the future only in that case, if the major establishedthat at the equator the Coriolis force international athletic competitions wouldbe organized changes the hammer andshot ranges by about 3 :4cm in stadia placed always at the same latitude in such a and6 mm ; respectively, which is smaller by about one way that the orientation of the centre line of the order of magnitude than the change due to a latitude throwing sectors wouldalso be the same. Since in all variation of 901: Hence, a hammer thrower or shot likelihoodthis rigorous rule cannot be introducedbythe putter throwing eastwardhas an advantage of about IAAF due to political, diplomatic and technical reasons, 3:4cmor6mm; respectively, against throwing west- we suggest the following, perhaps more acceptable, ward. Since the range is measured with an accuracy of procedure for a fair and physically correct approval of mm; but is rounded to cm; the effect of the Coriolis force worldrecordsin the hammer throw andshot put. on the shot range can be neglected, but it has a Using computer simulations, maps of the correction measurable influence on hammer range. terms for the hammer throw andshot put (Figs. 7–10) The above results holdfor both the northern (positive can be determined as a function of latitude. On the basis latitudes) and southern (negative latitudes) hemispheres of these correction maps, range achievedat a given of the Earth. Although the horizontal component of the latitude with a given release direction can be corrected in Coriolis force has different signs at positive and negative such a way as if the throw hadbeen performedin a latitudes (resulting in clockwise or counter-clockwise reference stadium (e.g. Athens) with a reference release slight deviations of the thrown hammer and shot on the direction (e.g. northward). After the throw distance is northern andsouthern hemisphere, respectively), the corrected for latitude and release direction in this way, it influence of Earth rotation on range at a given positive can be correctedfor other relevant environmental latitude is practically the same as that at the same factors such as wind, altitude, ground obliquity, air negative latitude, because it is dominated by the pressure andtemperature as if the throw hadbeen centrifugal force (being independent of the latitude sign) performed under standard physical and meteorological rather than by the much weaker Coriolis force. conditions: at sea level (H0 ¼ 0 m), on a horizontal throwing ground( o0 ¼ 0), at normal air pressure (p0 ¼ 4.3. Normalization of world-record ranges 101 325 Pa), at normal air temperature (T0 ¼ 201C) and under wind-less conditions (W0 ¼ 0m=s). The influence On the basis of the results of the computer simula- of wind, altitude, ground obliquity, air pressure and tions presentedin this work we can establish the temperature can be characterizedby relatively simple following: If the influences of environmental factors formulae (Fig. 3, Tables 2–5). All these corrections (in order of importance: wind, altitude, latitude, air couldbe automatically performedon-line by a computer temperature, groundobliquity, air pressure andrelease in a competition. direction) on throw distance are not taken into It wouldbe logical to accept these correctedthrow consideration in the approval of the new world records, distances as the real performances of the throwers, and the hammer andshot championship tables may incor- these correctedranges can fairly andcorrectly be rectly represent the relative performances. Ranges comparedwith each other. Of course, the prerequisite achievedby different hammer throwers or shot putters for such corrections is that wind speed, latitude, with different release directions in stadia at different orientation of the sector centre line, altitude, ground latitudes, altitudes, with different wind speeds, air obliquity, andair pressure andtemperature are mea- pressures andtemperatures cannot be directlycompared sured during the throwing events. Nowadays this can with each other. The direct comparison done until now easily be performedwith standardphysical andmeteor- is physically incorrect, because throwers or putters may ological measuring instruments. possess an unfair advantage of several decimetres or To our knowledge, the IAAF has never allowed any centimetres, respectively, against other athletes throwing correction to any result in any event. The most that it in more disadvantageous conditions. has done is to declare illegal for record purposes the In principle it is imaginable that after taking into times of sprint races andlengths of horizontal jumps account the influence of environmental factors on range, made with excessive tail wind (above 2 m=s). They do the hammer throw andshot put championship tables not adjust the times of these races or the distances of 796 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 these horizontal jumps; they simply throw them away. References They allow quite a bit of environmental variability with no restrictions. For instance, they do not worry about Caputo, M., 1967. The Gravity Fieldof the Earth. From Classical and altitude effects for any events, and they do not take into Modern Methods. Academic Press, New York, London. de Mestre, N., 1990. The Mathematics of Projectiles in Sport. account the influence of favourable winds in the discuss Cambridge University Press, Cambridge. or javelin throw, which can have tremendous effects on Garfoot, B.P., 1968. Analysis of the trajectory of the shot. Track performance. Technique 32, 1003–1006. Due to all this, we do not hope that the IAAF will Hay, J.G., 1985. Biomechanics of Sports Techniques, 3rdEdition. adopt our recommendations for the adjustment of Prentice-Hall, EnglewoodCliffs, NJ. Heiskanen, W.A., 1955. The Earth’s gravity. Scientific American 193 records, since this probably will not happen. Instead, (3), 164–174. in this work we have focussedon the fact that certain Hubbard, M., 1989. The throwing events in track and field. In: environmental factors affect substantially hammer Vaughan, C.L. (Ed.), Biomechanics of Sport. CRC Press, Boca throw andshot put results. The readersandthe officials Raton, FL, pp. 213–238. of the IAAF should decide what to do with this Hubbard, M., de Mestre, N.J., Scott, J., 2001. Dependence of release variables in the shot put. Journal of Biomechanics 34, 449–456. information. Landau, L.D., Lifschitz, E.M., 1974. Theoretical Physics I. Mechanics. Tankonyvkiad. o,! Budapest. Lichtenberg, D.B., Wills, J.G., 1978. Maximizing the range of the shot put. American Journal of Physics 46, 546–549. Acknowledgements Megede, E., Hymans, R. (Eds.), 1991. Progression of World Best Performances andIAAF ApprovedWorldRecords.International This work was supportedby a Humboldtresearch Athletic Foundation. fellowship of the German Alexander von Humboldt Mizera, F., 1999. Throwing events on the rotating Earth. Diploma ! . . Foundation and by an Istvan! Szechenyi! fellowship of work (supervisor: Dr. G. Horvath), Eotvos University, Faculty of Natural Sciences, Department of Biological Physics, Budapest, p. the Hungarian Ministry of Education to G. Horvath.! ! ! 47 (in Hungarian). We thank Peter Nemeth for the meteorological Tutevich, V.N., 1969. Teoria Sportivnykh Metanii. Izdatelstvo data used in the computer simulations. The valuable Fizkultura i Sport, Moscow (in Russian). discussions with Akos! Horvath,! Jozsef! Szecs! enyi! and Zatsiorsky, V.M., 1990. The biomechanics of shot putting technique. Peter! Tasnadi! are gratefully acknowledged. Many In: Bruggeman, G.-P., Ruhl, J.K. (Eds.), Techniques in Athletics. Deutsche Sporthochschule, Koln.. thanks are due to two anonymous referees for Zatsiorsky, V.M., Lanka, G.E., Shalmanov, A.A., 1981. Biomechani- their valuable andconstructive comments on our cal analysis of shot putting technique. Exercise andSports Science manuscript. Reviews 9, 353–389.