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. 0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0021-9290(02)00029-5 786 F. Mizera, G. Horvath! / Journal of Biomechanics 35 (2002) 785–796 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 direc- tions 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 centri- fugal force effects which depend abstractly on latitude d2r dr and direction in much the same way as air density does m % ¼ mgðj; HÞþ2m %  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  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  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  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