Avalable onlne at www.scencedrect.com Proceda Proceda Engneerng 00 (010) (009) 000 000 305 310 Proceda Engneerng www.elsever.com/locate/proceda 8 th Conference of the Internatonal Sports Engneerng Assocaton (ISEA) A mathematcal soluton of the optmum takeoff angle n long jump Kazuhro Tsubo a, * a Dept. of Intellgent Systems Engneerng, Ibarak Unversty, Naka-narusawa 4-1-1, Htach, 316-8511 Japan Receved 31 January 010; revsed 7 March 010; accepted 1 March 010 Abstract We successfully derve the optmum takeoff angle n long jump based on the maxmzaton problem of the flght dstance. The takeoff model proposed here ncludes three parameters: the horzontal speed of the centre of mass, the takeoff speed and the takeoff angle. The optmum takeoff angle s determned explctly as a functon of these parameters. The estmaton of the optmum angle shows good agreement wth the measured data of Mke Powell and Carl Lews, whch demonstrate the effectveness of the present model and ts soluton. c 009 010 Publshed by by Elsever Ltd. Ltd. Open access under CC BY-NC-ND lcense. Keywords: long jump; takeoff; optmum angle; mechancal model; three degrees of freedom; maxmzaton problem; analytcal soluton; projectle; flat aeral path 1. Introducton Long jump conssts of four phases; namely, run-up, takeoff, aeral and landng phases. The three phases except the run-up affect the total dstance of the jump. In other words, the total dstance L s dvded nto three parts of L 1, L and L 3 [1]. The frst part L 1 comes from the takeoff phase, and t s the horzontal dstance between the front edge of a takeoff board and the centre of mass of a jumper. The second part L s the horzontal dstance of the aeral phase n whch the jumper s centre of mass moves n the ar. The landng phase generates the last part L 3 that s the horzontal dstance between the jumper s centre of mass and the landng poston. The dstances L 1 and L 3 are bascally determned from the physcal sze of a jumper, and the sum of the dstances s approxmately 10% of the total dstance L. In contrast, the dstance L occupes 90% of the total dstance [1]. Ths means that the success of long jump depends on the aeral phase and therefore ths phase s the most sgnfcant factor for mprovng the record of long jump. For smplcty, we call the dstance L the flght dstance hereafter. Durng the aeral phase, forces actng on a jumper are gravty and ar resstance. Then, the moton of a jumper n ths phase can be descrbed by the moton equaton of a pont partcle and the jumper s centre of mass s movng * Correspondng author. Tel.:+81 94 38 507; fax: +81 94 38 507. E-mal address: ktsubo@mx.barak.ac.jp. 1877-7058 c 010 Publshed by Elsever Ltd. do:10.1016/j.proeng.010.04.133 Open access under CC BY-NC-ND lcense.
306 K. Tsubo / Proceda Engneerng (010) 305 310 K. Tsubo / Proceda Engneerng 00 (010) 000 000 along the trajectory of a projectle []. Ths fact suggests that there exsts the optmum angle of launchng velocty. The takeoff wth a launchng velocty of ths angle brngs the longest flght dstance n the same launchng speed. It s known well that the optmum angle of a projectle s 45 n a vacuum. However, t s mpossble for long jumpers to take off at such a large angle. Long jumpers utlze the knetc energy of run-up n order to take off wth large launchng velocty. The process of takeoff converts the horzontal momentum n run-up nto the momentum n launchng drecton. Ths converson determnes the launchng angle. In fact, most jumpers use the angles of 0 30 at ther takeoff. A crude estmaton wth horzontal and vertcal speeds qualtatvely explans the reason why the launchng angle n long jump les n ths range [3]. However, ths estmaton cannot lead to any quanttatve results such as the flght dstance and the optmum launchng angle. In order to clarfy the optmum launchng angle n long jump, several attempts have been acheved. For example, the energy conservaton law at takeoff was utlzed and a soluton of the optmum angle was derved theoretcally [4]. The result, however, showed rather large values of the launchng angle. As another approach, the launchng speed was obtaned as a functon of the launchng angle based on practcal measurements [5]. It s true ndeed ths approach provded good results on the optmum launchng angle, but t needs measured data and cannot clarfy the mechancs of the takeoff process. In the present paper, we attempt to mathematcally derve a soluton of the optmum launchng angle n long jump based on the takeoff model that ncludes three parameters of horzontal and takeoff speeds and takeoff angle. We can regard ths model as a lnk mechansm of human body wth the least degrees of freedom. Based on ths model, a maxmzaton problem of the flght dstance s formulated and the optmum takeoff angle s obtaned n an analytcal form. Also, the optmum angle s estmated from measured data n several cases.. Trajectory of aeral phase In the aeral phase, the moton of the jumper s center of mass s descrbed by the moton equaton of a partcle, u& = α u u + v, v& = α v u + v g, (1) where u and v mean the horzontal and vertcal velocty components of the center of mass, and α ndcates the drag coeffcent per unt mass. The exact soluton of eq. (1) has not been obtaned [6] and then we start our modelng wth an approxmated trajectory of eq. (1), v g ( ) = + ( exp( ) 1), g y x x αx () u αu 4α u where (u, v ) denotes the launchng velocty, and the orgn of ths expresson s set to the poston of the center of mass at takeoff. In dervng eq. () from eq. (1), t s assumed that v s suffcently smaller than u and therefore eq. () s called flat aeral path [7]. In order to obtan a more useful expresson, we ntroduce another assumpton that the drag coeffcent α s also suffcently small. As shown later, the second assumpton s really vald. Under these assumptons, we can obtan a perturbed soluton of the trajectory n the followng form, v g g 3 y( x) = y0( x) + α xy1( x) = x x α x. u u (3) 3u When the vertcal dsplacement of the center of mass at landng s h (> 0), the equaton y(x) = h of the flght dstance X s obtaned and we have the followng soluton; gx ( v ), 0 + v + gh X1 =. u X = X 0(1 + α X1), X 0 = (4) g 3u v + gh In order to confrm the accuracy of eq. (4), we estmate the flght dstance based on some measured data shown n Table 1. In ths table, WS means the data used by Ward-Smth n hs paper []. Also, MP and CL denote the data measured n practcal performance by Mke Powell and Carl Lews n 1991 [8 9]. Some data wth the symbol of * mean corrected values: the dstances L * are modfed from the actual values of L accordng to the wnd effect [9]
K. Tsubo / Proceda Engneerng (010) 305 310 307 K. Tsubo / Proceda Engneerng 00 (010) 000 000 whle the vertcal dstances h are estmated by the author from the measured data of the maxmum heght n aeral phase and the heght at landng [10]. Table 1 Measured data of long jump for reference WS MP CL L [m] L * [m] u [m/s] v [m/s] α [1/m] h [m] 8.03 8.03 10.10.87 0.003 0.50 8.95 8.9 * 9.7 4.6 0.01 0.61 * 8.91 8.6 * 9.11 3.37 0.011 0.78 * The estmated results are summarzed n Table. The dstance L s obtaned from 90% of L * n Table 1 and X RK ndcates numercal evaluatons wth the Runge-Kutta method of eq. (1). In the case of WS, the agreement between L and X s excellent. In contrast, the comparson n the cases of MP and CL shows a dfference n tens of centmeters n each case. However, the relatve errors of the dfference are 6% n MP and % n CL respectvely, and t seems to be suffcently small. Thus, the approxmated dstance X of eq. (4) works well n estmatng the flght dstance n long jump. Table Estmaton of the flght dstance based on the data n Table 1 L [m] X 0 [m] X [m] X RK [m] WS 7.3 7. 34 7. 4 7.4 MP 8.03 9. 8.6 8.64 CL 7.76 7. 93 7. 55 7.56 3. The optmum takeoff angle Fgure 1 shows the takeoff model of the present study and t s characterzed by three parameters V, w and ψ. The parameter V means the horzontal speed of the centre of mass at takeoff and therefore represents the speed of the trunk of a jumper. The speed w s named the takeoff speed and a push by a jumper s leg generates t. The takeoff angle ψ s defned here as the angle between V and w. Thus, ths takeoff model has three degrees of freedom and provdes a lnk mechansm of human body wth the least degrees of freedom. These parameters gve the followng expresson of the launchng velocty (u, v ); Fg. 1 Takeoff model n long jump. The fgure s drawn by the author based on a fgure n the reference [11]. u = V + wcosψ, v = wsnψ, (5) and we use the symbol θ for representng the angle of the launchng velocty gven as v wsnψ tanθ = =. (6) u V + wcosψ Substtutng the expresson of eq. (5) nto eq. (4) of the flght dstance, we obtan the dstance X as a functon of the takeoff angle ψ nstead of the launchng velocty (u, v ). Ths defnes the followng maxmzaton problem of the flght dstance; o o maxmze X ( ψ ) under the condton of 0 ψ 180 for a gven w and V. (7) The maxmzaton problem (7) s equvalent to the followng requrement dx dx = 0, (8) dψ dψ0
308 K. Tsubo / Proceda Engneerng (010) 305 310 K. Tsubo / Proceda Engneerng 00 (010) 000 000 where the effect of ar resstance to the angle s neglected here because t s of the order of 0.01 [10]. Substtutng the dstance X(ψ) nto eq. (8) and after some manpulatons, the requrement (8) reduces to the followng cubc equaton of cosψ ; ( w + V + gh) cos ψ ( w + ) = 0. 3 wv cos ψ + gh (9) In order to nvestgate the solutons of eq. (9), we defne the cubc functon F(cosψ) that equals to the left hand sde of eq. (9). Then, t s obvous that F( ) = and F(+ ) = +. Furthermore, the functon F has the followng propertes; F ( 1) = ( V w) 0, F(0) = ( w + gh) < 0, F(1) = ( V + w) > 0. These propertes lead to the concluson that eq. (9) has three solutons n real number, one of whch has a postve value and the others have negatve values. Also, snce one of the negatve solutons s smaller than 1, t s obvously nvald as the present soluton. Consequently, we have two solutons of eq. (9) n the range from 1 to 1, and the postve one gves the optmum soluton of the maxmzaton problem (7). In partcular, t s noted that eq. (9) ncludes the specal soluton n V = 0. In ths case, we have w + gh cosψ opt =, (10) ( w + gh) whch follows the well-known soluton n a projectle; w tanψ opt =. (11) w + gh Accordng to the formula of cubc equaton [1], we can obtan the postve soluton of eq. (9) as follows; 1 V 1 w gh ϕ cosψ opt = + + cos 1, (1) 3 w V V 3 where 3 7 w 1 w gh 1 cos 1 w gh ϕ = + + + +. (13) V V V V V The present soluton mples that two non-dmensonal parameters w/v and gh/v determne the optmum angle. The former s the rato of the takeoff speed to the horzontal speed at takeoff whle the latter means the rato of the potental energy obtaned n aeral phase to the knetc energy n run-up. The optmum angle ψ opt s shown n Fg. as the functon of these parameters. The angle ψ opt monotonously decreases wth ncreasng the two parameters. 4. Dscusson As shown n Fg. 1, the present takeoff model ncludes the parameters w and V. The estmaton of w and V s, therefore, nevtable n order to determne the optmum takeoff angle ψ opt. In evaluatng the horzontal speed V, we employ the fact that a run-up speed decelerates to approxmately 80 90% of the full speed at just the moment of takeoff [1]. For example, world class athletes are able to run 100 m n 10 seconds, whch leads to the estmaton of V=8.0 9.0 m/s. On the other hand, no data on the takeoff speed w have been found, and thereby we ntroduce an assumpton to ths parameter. Elmnatng the angle ψ n eq. (5), we have the followng relaton; w = ( u V ) +, (14) v ψ opt [deg.] gh V Fg. The effect of non-dmensonal parameters to the optmum takeoff angle. w /V
K. Tsubo / Proceda Engneerng (010) 305 310 309 K. Tsubo / Proceda Engneerng 00 (010) 000 000 whch mples that the takeoff speed w s evaluated from the launchng velocty and the horzontal speed V. Here, employng the expresson (14), we show that the angle defned by eq. (1) s actually the optmum value. When measured data of the launchng velocty (u, v ) and a value of the horzontal speed V are gven, eq. (14) provdes the correspondng value of the takeoff speed w. Then, a value of ψ n eq. (5) yelds another launchng velocty nstead of the used launchng velocty. Ths launchng velocty and the vertcal dsplacement at landng predct the flght dstance X accordng to eq. (4). The results of V=7.0 m/s and 9.5 m/s are shown n Fg. 3, n whch the data of WS n Table 1 are used n the measured launchng velocty and the vertcal dsplacement at landng. Equaton (1) gves the optmum angles of 61.6 and 68.6 to V=7.0 m/s and 9.5 m/s, respectvely. We can confrm n Fg. 3 that the maxmum flght dstance actually appears at the value n each case. Now, we summarze the procedure for evaluatng the optmum takeoff angle. As mentoned above, a takeoff speed w s estmated from eq. (14) wth values of the horzontal speed V and the measured launchng velocty (u, v ). Then, t s possble to obtan the optmum takeoff angle ψ opt n eqs. (1) and (13) wth a value of the vertcal dsplacement h. The values of V, w and ψ opt determne the optmum launchng velocty (u, v ) opt whch dsagrees wth the orgnal launchng velocty n general. Then, the optmum launchng angle θ opt s obtaned from eq. (5), and consequently eq. (4) yelds the optmum dstance X opt n ths case. Accordng to ths procedure, we attempt to estmate the takeoff angle from the data of WS n Table 1. In order to clarfy the feature of eq. (1), the horzontal speed V s vared n the range from 0 to u. Also, eq. (5) defnes the actual takeoff angle ψ pr ; v tanψ pr =, (15) u V whch gves the measured launchng velocty n a gven value of V. The result s shown n Fg. 4, n whch the varaton of the actual takeoff angle s also shown wth a broken lne. Snce the velocty u s equal to wcosψ n the case of V = 0, the takeoff angle ψ pr s consstent wth the launchng angle θ and the value becomes 15.87. Then, the optmum takeoff angle ψ opt has the value of 43.96. In contrast, when V agrees wth u, w s dentcal wth v, and thus ψ pr becomes 90. The value of ψ opt n ths lmt becomes 69.6. The curves of ψ pr and ψ opt ncrease monotonously wth ncreasng the value of V, and the range of the former s larger than that of the latter: for example, ψ pr vares from 15.87 to 90 whereas ψ opt from 43.96 to 69.6. As the result, these curves has the ntersecton at V = 8.9 m/s ( V c ). Ths means that f the horzontal velocty at takeoff was equal to ths value n the practce, the jump was performed wth the optmum takeoff angle. Fg. 3 Varaton of the flght dstance wth the takeoff angle. Fg. 4. The optmum takeoff angle and the flght dstance n the data of WS n Table 1. Furthermore, the results of the optmum takeoff angle obtaned from the data of MP and CL n Table 1 are summarzed n Table 3. Here, we assume that V vares from 7.0 m/s to 9.0 m/s. In ths range of V, the optmum launchng angles of M. Powell and C. Lews are n the range from 19.6 to 4.0 and 16.3 to 0.9, respectvely.
310 K. Tsubo / Proceda Engneerng (010) 305 310 K. Tsubo / Proceda Engneerng 00 (010) 000 000 The measured angles of 4.6 and 0.3 le on the upper lmts of the range and ths result supports the valdty of the present optmum angle. The comparsons of the optmum launchng angle θ opt and the actual takeoff angle ψ pr show that the results of M. Powell and C. Lews are dfferent n each case. However, the range of the optmum takeoff angle ψ opt n both the results agrees wth each other. Although the reason has not been evdent yet, t s new fndng and an nterestng result of the present study. Table 3 Estmated results on the optmum takeoff angle of M. Powell and C. Lews based on the measured data n 1991. ψ pr [ ] ψ opt [ ] θ opt [ ] θ [ ] V c [m/s] MP CL 61.9-86.4 57.9-88.1 60.0-64.5 59.9-64.6 4.0-19.6 0.9-16.3 4.6 0.3 6.8 7. 5. Concluson In order to analytcally obtan the optmum takeoff angle n long jump, a perturbed soluton of the trajectory s derved from the flat aeral path that s an approxmate soluton of the moton equaton wth the quadratc resstance law. The approxmate soluton of the trajectory ncludes the effect of ar resstance n the lnear form. Snce the effect of ar resstance s suffcently small n long jump, the comparson of ths soluton wth measured data shows good agreement. Based on the approxmate trajectory, we formulate a maxmzaton problem of the flght dstance wth a takeoff model. The present takeoff model ncludes three parameters: the horzontal speed of the center of mass (V), the takeoff speed (w) and the takeoff angle (ψ). As the soluton of the maxmzaton problem, the optmum takeoff angle s obtaned analytcally and gven explctly by a functon of w/v and gh/v. The flght dstance and the takeoff angle are evaluated usng the measured data of M. Powell and C. Lews. The evaluated dstances are n suffcent agreement wth the measured values. Also, the present soluton of the optmum takeoff angle provdes well-known values of the launchng angle. The comparsons of the evaluated results wth the measured values demonstrate the effectveness of the present approach and ts soluton. In partcular, the present study suggests that the optmum range of the takeoff angle (ψ opt ) becomes the same n both the famous world-class jumpers, although the launchng angles are dfferent n ther practcal performance. It s new fndng and an nterestng result of the present study. Fnally, the followng pont should be noted. In the case that we are nterested n the flght dstance of long jump, the launchng velocty at takeoff can provde the suffcent estmaton. However, n the estmaton of the optmum takeoff angle, further nformaton on the takeoff phase (the horzontal and takeoff speeds n the present model) s necessary. Ths s the concluson of the present study and the measurement of these speeds wll make the estmaton of the optmum takeoff angle more accurate. References [1] Hay JG, Mller JA, Canterna RW. The technques of elte male long jumpers. J Bomechancs 1986; 19-10: 855 66. [] Ward-Smth AJ. Calculaton of long jump performance by numercal ntegraton of the equaton of moton. J Bomech Eng 1984; 106: 44 8. [3] Brancazo PJ. Sports Scence. New York: Smon & Schuster Inc; 1984. [4] Tan A, Zumerchk J. Knematcs of the long jump. The Phys Teacher 000; 38: 147 9. [5] Lnthorne NP, Guzman MS, Brdgett LA. Optmum take-off angle n the long jump. J Sports Sc 005; 3-7: 703 1. [6] de Mestre N. The mathematcs of projectles n sport. Cambrdge Unversty Press; 1991. [7] Lamb H. Dynamcs. Cambrdge Unversty Press; 193. [8] Vorobev A, Ter-Ovanessan I, Arel G. Two world s best long jumps: Comparatve bomechancal analyss. Track & feld quarterly rev 199; 9 4: 6 10. [9] Guskov A, Vorobev A, Arel G. Body aerodynamcs: contrbuton to long jump performance. Track & feld quarterly rev 199; 9 4: 11 3. [10] Tsubo K. The optmum angle of takeoff n long jump wth vertcal dsplacement at landng. Trans of the Japan Soc Mech Engneers C 008; 74-743: 161 8 (n Japanese). [11] Fukashro S. Scence of Jump. Tokyo: Tasyukan-syoten; 1990, p. 34 (n Japanese). [1] The unversal encyclopeda of mathematcs, George Allen & Unwn Ltd; 1964.