COMPARATIVE STUDY OF VELOCITY REDUCTION ON FEATHER AND SYNTHETIC SHUTTLECOCKS USING CORRECTED INITIAL VELOCITY DURING OVERHEAD SMASH

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1 Journal of Engineering Science and Technolog Special Issue on AASEC 6, Ocober (7) 9-5 School of Engineering, Talor s Uniersi COMPARATIVE STUDY OF VELOCITY REDUCTION ON FEATHER AND SYNTHETIC SHUTTLECOCKS USING CORRECTED INITIAL VELOCITY DURING OVERHEAD SMASH AGUS RUSDIANA, *, ASRIL PRAMUTADI ANDI MUSTARI Spor Science Sud Program, Uniersias Pendidikan Indonesia, Jl. Dr. Seiabudi no 9, Bandung 454, Indonesia Phsics Sud Program, Insiu Teknologi Bandung, Jl. Ganesha No., Lb. Siliwangi, Coblong, Koa Bandung, Jawa Bara 43, Indonesia *Corresponding Auhor: agus.rusdiana@upi.edu Absrac Oerhead smash in badminon is a comple condiion where high eloci is aking place. Mos of he inesigaions a his oerhead smash ma find difficul in finding he iniial eloci (afer impac) of a shulecock. In his sud, he fligh performance of hree shulecock models namel Feaher Yone, Nlon Mizuho, and Nlon Gozen was inesigaed during oerhead smash. Addiional comparison beween sanding and jumping smashes was also inesigaed. Correced iniial eloci b using regression equaion mehod was performed and he reducion of eloci a cerain disances was inesigaed. The record of racque swing and shulecock fligh afer impac from high speed camera were obsered. The shulecock release poin was drawn in he X-Y coordinae in a linear form. The equaion for he racque swing was Y = a X + b and he pah for shulecock was (Y = a.x + b ). I was found ha Nlon Mizuno had he lowes iniial eloci, while Feaher Yoned and Nlon Gozen had similar alue. Afer raeling for 5 meers, he eloci of boh sanding and jumping smashes reduced ino 3-4% of he iniial eloci. Furhermore, he inesigaion beween sanding and jumping smashes showed ha he iniial eloci of jumping is higher han ha of sanding smash. Thus, jumping smash can be considered as he main weapon when plaing badminon. Kewords: Iniial eloci, Shulecock, Oerhead smash, Badminon, Regression.. Inroducion Badminon is one of he mos popular and oldes spors on earh. I is repored ha oer million people plaed badminon around he world []. Badminon game 9

2 9 A. Rusdiana and A. P. A. Musari is a racque spor plaed wih a feaher shulecock. The shulecock is aerodnamicall shaped and has low mass, hus i deceleraes er quickl as i flies in he air due o he drag force of he air []. Some shulecocks are made of naural feaher and some ohers are made of snheic rubber. Generall, boh pes of shulecocks hae a mass around grams, diameer of he cork is 5 8 mm and he diameer of he circle is around 54 mm []. Mos amaeur plaers use a snheic shulecock because i is more durable and cheaper compared o feaher shulecock which is predominanl used b professional plaers. The fligh rajecor of he feaher and snheic shulecocks are significanl differen from he balls uilized in mos racque spors due o er high iniial speed [3,4]. For ears researchers hae sudied he naure of a shulecock regarding is rajecor, maerial, roaion, ec. Seeral sudies had been conduced o inesigae he parabolic rajecor of shulecock [,5]. A sud a differen Renolds number (Re) shows low drag coefficien for shulecock wihou a gap in comparison o he one wih gap [6]. Comparison beween naural feaher and snheic shulecocks shows lower drag force for naural feaher when esed in wind unnel [,3,5,7-9]; howeer, one should keep in mind ha calculaion of drag force, heoreicall, is highl affeced b iniial eloci (afer impac). Thus, iniial eloci should be obsered carefull o obain he correc number of drag force. Among he badminon skills, he oerhead smash sroke is he mos powerful sho and comple because of is speed and seep rajecor []. The speed of shulecock decreases significanl due o drag force ha is highl influenced influenced b iniial eloci []. Seeral mehods hae been used o measure raeling speed such as radar gun and he high-speed camera. Howeer, boh mehods canno measure iniial eloci. Radar gun required highl capable operaor so i can be used effeciel. The radar gun also canno differ beween arges (racque and shulecock), hus operaor plas an imporan role []. And mos imporanl, he radar gun is measuring insananeous speed raher han iniial speed. On he oher hand, he high speed camera is commonl-used for higher specs. And speed calculaion using high-speed camera is mos likel o produce aerage or insananeous because of he difficul of obaining figure eacl when impac beween racque and shulecock occur [8]. The posiion of shulecock when i jus releases from he ne of a racque is reall hard o decide. In addiion, low picure quali and insufficien sampling frequenc are among he lis of he problems[]. Forunael, for real case (no in he wind unnel), mos of he sudies inesigaed in he area of low eloci of shulecock like a drop sho, so inaccurac of iniial speed can be negligible. Since undersanding in he area of a high-speed siuaion during oerhead smash is limied, he main objecie of his work is a comparaie analsis of eloci reducion of naural feaher and snheic using a regression equaion o obain iniial eloci. In addiion, comparison of he iniial eloci of sanding and jumping smashes on feaher shulecock was gien. This sud is imporan since here hae no been scienific references of shulecock speed release using phsics analsis. There are acuall seeral sudies on speed release using machines merel; howeer, in he real life, here is Journal of Engineering Science and Technolog Special Issue /7

Comparaie sud of eloci reducion on feaher and snheic.... 93 acuall difference especiall due o he impac of racques used. Besides, he selecion of hree differen racques is due o heir maerials.. Mehod.

3 Comparaie sud of eloci reducion on feaher and snheic acuall difference especiall due o he impac of racques used. Besides, he selecion of hree differen racques is due o heir maerials.. Mehod.. Procedure The paricipan was a righ-handed uniersi badminon plaer, ears old,.7m heigh and had a mass of 6 kg. Afer a general and a specific badminon warm-up, he subjec was asked o hi eigh shulecocks wih he maimal speed smash sho. The shulecock rajecor was recorded using a high-speed ideo camera (Phanom Miro-eX, USA) se a frames/sec and / eposure ime. A camera was posiioned a righ-cener side of badminon cour so ha he opical ais of he camera lens was almos perpendicular o cener of he ne posiion wih approimael horizonal disance m. Fig. shows op iew schemaic of badminon eperimen se-up. The cork of shulecock was digiized using Frame DIAZ IV sofware analsis. Fig.. Top iew schemaic of badminon eperimen se-up... Shulecocks descripion The shulecocks were seleced for his sud, namel he naural feaher (Feaher Yone) and snheics shulecocks (Nlon-Mizuno and Nlon-Gozen), see Fig.. Generall, all pes of shulecocks hae a mass around grams. Table Show he dimension of he feaher and snheic shulecocks in deail. Journal of Engineering Science and Technolog Special Issue /7

94 A. Rusdiana and A. P. A. Musari Fig.. (A) The naural feaher, (B) nlon- mizuno, and (C) nlon-gozen shulecock images. Table. Dimensional of he naural feaher and snheic shulecocks.

4 94 A. Rusdiana and A. P. A. Musari Fig.. (A) The naural feaher, (B) nlon- mizuno, and (C) nlon-gozen shulecock images. Table. Dimensional of he naural feaher and snheic shulecocks. Shulecock Characerisics Shulecock Models Feaher-Yone Nlon-Gozen Nlon-Mizuno Skir diameer, d (mm) Lengh of shulecock, H (mm) Cork diameer, dn (mm) Lengh of cork, I (mm) Mass, m (gram) Deelopmens of calculaion mehod There were seeral sages o deermine shulecock iniial speed b using a regression equaion approach. Firs, idenif he impac posiion beween racques and shulecocks. Second, draw seeral elociies a seeral posiions afer he impac b simpl using frame based calculaion, hen generae regression equaion. Finall, insering he informaion of posiion ino regression equaion. Figure shows an ahlee conducing oerhead swing in four frames (5 ms per frame). The impac beween racque and shulecock occurred somewhere beween frame 3 and 4. This also confirmed he difficul in capuring eac posiion of he impac. Figure 3 shows he racque swing and a shulecock fligh 5 frames afer shule release. The black bo mark was shulecock release poin. The coninues-sraigh line and disconinuous arrow line were he racque swing and a shulecock fligh analsis, respeciel (he firs o second phases or wo frames afer impac). The shulecock release poin coordinae (X, Y) deermined b a cross line ineracion beween he racque swing (X = a X + b ) and a shulecock pah (Y = a.x + b ) afer impac were analzed b regression equaion approach (Fig. 5). Journal of Engineering Science and Technolog Special Issue /7

Comparaie sud of eloci reducion on feaher and snheic.... 95 Fig. 3.

Racque swing and a shulecock fligh before and afer impac b he sick picure image.

5 Comparaie sud of eloci reducion on feaher and snheic Fig. 3. Racque head swing and a shulecock before and afer a impac seup speed of camera fps and / shuer speed (-4), beween he number 3 and 4 occurred he shulecock impac in he smash sho. Fig. 4. Racque swing and a shulecock fligh before and afer impac b he sick picure image. From he firs o fifh phases were he racque and a shulecock around m disance from he impac. The black bo mark was shulecock release poin. The sraigh and disconinuous lines were he racque swing and a shulecock fligh analsis, respeciel. Journal of Engineering Science and Technolog Special Issue /7

96 A. Rusdiana and A. P. A. Musari Fig. 5. The sraigh and disconinuous lines showed he racque swing and a shulecock fligh afer shulecock release (he firs and second phases, respeciel).

6 96 A. Rusdiana and A. P. A. Musari Fig. 5. The sraigh and disconinuous lines showed he racque swing and a shulecock fligh afer shulecock release (he firs and second phases, respeciel). From he descripion of he Fig. 5, assised b he saisical mehod leassquares mehod approach, i will ge he equaion of a sraigh solid line for shulecock and dashed line racque as follows: a a () b () b Number is a represenaion of he posiion of he shulecock and he number is a posiion of he racque. The posiion of he objec - (shulecock) can be epressed mahemaicall as follows: a (3) a (4) The posiion of he objec - (racque) can be epressed mahemaicall as follows: a (5) a (6) The impac of he racque and shulecock is assumed o be a = s, assuming he acceleraion of each objec is zero, hen he aboe equaion will change o: (7) (8) Journal of Engineering Science and Technolog Special Issue /7

7 Comparaie sud of eloci reducion on feaher and snheic Journal of Engineering Science and Technolog Special Issue /7 (9) () Subsiuing beween equaions 7 and 8 as well as for he equaions 9 and, hen he equaions ransform ino: () () B using posiion daa of racque and shulecock a some ime,, hen he alue will be m, b, m, and b, where m is gradien while b is a cross line a he -ais. Equaion of sraigh line will be obained, as in Fig. 4. B finding he poin of inersecion of wo lines, we can obain he poin of inersecion of he wo lines, which he poin of impac beween he racque and shulecock. Poin P( (), ()) are he same poin wih P( (), ()). This siuaion causes boh objecs, racque and shulecock, o be a he same poin a = s is he poin P. Because ; (3) b b a a ; ; (4) We will ge 4 equaions wih 4 unknown parameers, namel: V, V, V, and V (5) The four parameers will be obained. Then, he eloci of he racque and shulecock a = s a he poin P can be known. The ne analsis was a regression equaion using second-order polnomial funcions from he firs o fifh phases or 5 frames afer impac beween X horizonal ais (m) and shule eloci afer shulecock release as shown in Fig. 6. Finall, he iniial eloci of shulecock deermined b X-ais alue of release poin coordinae and he coefficien of he second order of he regression equaion beween shulecock eloci and X ais (m). All he calculaion were applied o hree models of shulecock, hen he resuls will be compared. Ealuaion was conduced afer 5 and 9 m raeling disance. Racque iniial eloci also will be ealuaed.

98 A. Rusdiana and A. P. A. Musari Fig. 6. The regression equaion using second order polnomial funcions o deermine he iniial shule speed ( m was he ne line). 3. Resuls 3.

All daa obained from 8 imes rial using differen pes of shulecocks. Table shows ha aerage iniial eloci, i.e. using regression equaion mehod, of feaher shulecock was abou 86. m/s.

8 98 A. Rusdiana and A. P. A. Musari Fig. 6. The regression equaion using second order polnomial funcions o deermine he iniial shule speed ( m was he ne line). 3. Resuls 3.. Sanding Smash afer 5 m raeling disance Table unil Table 4 show iniial and final elociies of feaher, Nlon Mizuno, and Nlon Gozen, respeciel, a 5 m raeling disance from he oerhead smash posiion. All daa obained from 8 imes rial using differen pes of shulecocks. Table shows ha aerage iniial eloci, i.e. using regression equaion mehod, of feaher shulecock was abou 86. m/s. one he oher hand, he aerage final eloci was 9.8 m/s, i is abou 34.7 % of iniial eloci. Table 3 shows a alue of 79.9 m/s for aerage iniial eloci and 5.5 m/s for final eloci. Meaning ha he final eloci was 3.8% of iniial eloci. The iniial eloci of nlon gozen was abou 85.9 m/s, see Table 4. Figure 7 shows he comparison of shulecocks iniial elociies of feaher, nlon Mizuno, and nlon Gozen. I can be seen ha Nlon-Mizuno has he lowes iniial eloci, while feaher and Nlon-Gozen hae similar alue. Figure 8 shows he final eloci of all shulecocks. The Nlon-Mizuno has he lowes alue, while he nlon gozen has he highes alue. These resuls show ha drag force affeced he nlon Mizuno he mos. And he nlon gozen was less affeced b drag force of air. Journal of Engineering Science and Technolog Special Issue /7

9 Comparaie sud of eloci reducion on feaher and snheic Table. Means of iniial speed, final speed and speed reducion (%) of feaher shulecock (5 m disance). No Disance Iniial eloci Final eloci (m) % Aerage Table. Means of iniial speed, final speed and speed reducion (%) of Nlon Mizuno (5 m disance). No Disance Iniial eloci Final eloci (m) % Aerage Table 3. Means of iniial speed, final speed and speed reducion (%) of Nlon gozen (5 m disance). No Disance Iniial eloci Final eloci (m) % Aerage Journal of Engineering Science and Technolog Special Issue /7

A. Rusdiana and A. P. A. Musari Fig. 7. Iniial eloci comparison of feaher, Mizuno and Nlon-Gozen. Fig. 8. Final eloci comparison of feaher, Mizuno and Nlon-Gozen. 3.

All daa obained from 8 imes rial using differen pes of shulecocks. Table 5 shows ha aerage iniial eloci, i.e. using regression equaion mehod, of feaher shulecock was abou 86. m/s.

10 A. Rusdiana and A. P. A. Musari Fig. 7. Iniial eloci comparison of feaher, Mizuno and Nlon-Gozen. Fig. 8. Final eloci comparison of feaher, Mizuno and Nlon-Gozen. 3.. Jumping smash afer 9 m raeling disance Tables 5-7 show iniial and final elociies of feaher, Nlon Mizuno, and Nlon Gozen, respeciel, for 9 m disance. All daa obained from 8 imes rial using differen pes of shulecocks. Table 5 shows ha aerage iniial eloci, i.e. using regression equaion mehod, of feaher shulecock was abou 86. m/s. One he oher hand, he aerage final eloci was 9.8 m/s, i is abou 34.7 % of iniial eloci. Table 6 shows alue of 79.9 m/s for aerage iniial eloci, while alue of 5.5 m/s for final eloci. And his means ha he final eloci was Journal of Engineering Science and Technolog Special Issue /7

11 Comparaie sud of eloci reducion on feaher and snheic % of iniial eloci. The iniial eloci of nlon gozen was abou 85.9 m/s, see Table 7. Table 4. Means of iniial speed, final speed and speed reducion (%) of feaher shulecock (9 m disance). No Disance Iniial eloci Final eloci (m) % Aerage Table 5. Means of iniial speed, final speed and speed reducion (%) of Nlon Mizuno (9 m disance). No Disance (m) Iniial eloci Final eloci Aerage % Table 6 Means of iniial speed, final speed and speed reducion (%) of Nlon gozen (9 m disance). No Disance (m) Iniial eloci Final eloci Aerage % Journal of Engineering Science and Technolog Special Issue /7

12 A. Rusdiana and A. P. A. Musari Figure 9 shows final eloci of all shulecocks afer 9 m raeling. Oerall, i has similar rend wih resul for 5 m (Fig. 8). Howeer, afer 9 m raeling he alue reduce abou 4% of eloci afer 5 m. Fig. 8. Final eloci comparison of feaher, nlon Mizuno, and Nlon-Goze Sanding s. Jumping smash eloci (feaher) Figure shows comparison of eloci profile beween sanding and jumping smashes using feaher shulecock. I can be seen ha iniial eloci of jumping smash is slighl higher han sanding smash. Afer raeling for 5 m and 9 m, sanding smash shows slighl higher han jumping smash. From he graph, i can be seen ha eloci reducion was no linear. This is predicable since formulaion of drag force uilizes quadraic eloci. Fig. 9. Comparison of eloci profile beween and jumping smashes. sanding Journal of Engineering Science and Technolog Special Issue /7

13 Comparaie sud of eloci reducion on feaher and snheic Racque eloci comparison Table shows comparison of racque eloci afer impac wih shulecocks b sanding smash. Roughl, he comparison shows sligh differen among all elociies. Howeer, eloci afer impac wih feaher has he lowes aerage eloci, while nlon mizuno has he highes eloci. Jumping smash on feaher shows increase of racque eloci in comparison wih sanding smash. Table 7. Comparison of racque eloci righ afer he impac wih paricular shulecocks. Shule Feaher Nlon mizuno Nlon gozen Jumping smash Aerage Discussion The presen inesigaion has deeloped a regression approach based calculaion for obaining iniial eloci during oerhead smash. This approach enabled he deerminaion of racque-shulecock impac posiion and iniial eloci for oerhead smash whils oercoming he difficulies presen b oher echnolog. Mos preious sudies inesigae he rajecor of shulecock during drop sho where eloci is no so high [,3]. McErlain-Nalor e al deeloped mehod using cure fiing mehod which is close o our mehod [4]. Howeer, in his mehod, i was assumed ha seeral posiions a he beginning, jus afer impac, was sill fligh in sraigh pah [5]. The resul of iniial eloci (Fig. 7) reealed ha afer impac feaher shulecock had high iniial eloci ogeher wih nlon gozen (small difference). This resul ma srongl relaed o maerial of shulecock s head. Theoreicall, parameers ha ma affec iniial eloci are mass, shooing power, resiuion consan of he maerial. Since all of he esed shulecocks had he same mass hen his reason should be omied. Thus, he maerial of he head had a high possibili of affecing is iniial eloci. To elaborae more regarding his reason, furher inesigaion should be conduced. Tpical iscoelasic impac characerisaion ma appropriae for esing [6]. Since he es conduced using he same racque for eer rial, hus effec of resiuion of ne of he racque can be eliminaed. The final eloci of shulecock afer 5 and 9 m seems o be following negaie eponenial paern. I was heoreicall acceped since reducion of eloci is mos likel due o drag force, which is affeced b quadra eloci. On he oher hand, difference of percenage among shulecock ma be due siffness of he maerial. Afer he impac, shulecock will be in opposie direcion wih ecor eloci, afer hen i will roae and hae he same direcion wih eloci. During his process, due o air resisance, shulecock will be deformed a lile bi, i.e. ge wider, as consequence drag force will increase sharpl. Maerial wih high deformaion (bigger dimension) will eperience higher drag force. According o he resul, i seems ha nlon mizuno has he highes deformaion. And he leas deformaion ehibi b nlon gozen or in oher word his maerial had high siffness leel. Howeer, furher inesigaion is needed o confirm his Journal of Engineering Science and Technolog Special Issue /7

14 4 A. Rusdiana and A. P. A. Musari hpohesis. Inesigaion on eloci before shulecock roae and poining o he same direcion as eloci ecor. Comparison beween sanding and jumping smashes using he feaher as shown in Fig. shows quie similar paern and alue. In anoher word, an onl small difference occurs for boh mechanisms. Howeer, in deailed obseraion, jumping smash ehibis higher iniial eloci han sanding posiion. Afer he impac racque eloci comparison shows small discrepanc among all alues. This shows ha racque eloci was less affeced b he pe of shulecock. This migh be due o he reali ha he racque ha conneced o human bod had much higher momen ineria. The mehod being used o calculae iniial eloci was er useful o oercome difficul of comple (skill demanding) mehod and high cos mehod. More imporanl, he mehod proides beer obseraion of iniial eloci, which will affec calculaion of drag (air) force. This mehod can be used using wih a low capuring speed camera. Thus, his will conribue o improemen of raining mehod of badminon around he world. 5. Conclusion Calculaion of iniial eloci (afer impac) in oerhead sanding and jumping smashes was successfull conduced. Comparison on eloci reducion of naural feaher and snheic was conduced using iniial eloci daa using regression approach. The comparison shows ha Nlon-Mizuno had highes eloci reducion afer cerain disances (5 and 9 m). In he meanime, Feaher Yone and Nlon-Gozen hae similar iniial eloci. Howeer, Nlon-Gozen was less affeced b air resricion (drag force) in comparison o feaher. Furhermore, he inesigaion beween sanding and jumping smashes showed ha he iniial eloci of jumping is higher han ha of sanding smash. Thus, jumping smash can be considered as he main weapon when plaing badminon. Acknowledgemens RISTEK DIKTI (gran-in-aid in Peneliian Unggulan Perguruan Tinggi Negeri (PUPTN)) and Uniersias Pendidikan Indonesia were acknowledged for supporing his research. References. Alam, F.; Chowdhur, H.; Theppadungporn, C.; Moria, H.; and Subic, A. (). A comparaie sud of feaher and snheic badminon shulecock aerodnamics. 7 h Ausralia Fluid Mechanics Conference. Ausralia, -4.. Maemow, S. (9). Tha's a Drag : The Effecs of Drag Forces. Undergraduae Journal Mahemaical Modelling: one + wo, (), Alam, F.; Chowdhur, H.; Theppadungporn, C.; and Subic, A. (9). A Sud of Badminon Shulecock Aerodnamics. In The 8 h Inernaional Conference on Mechanical Engineering. Bangladesh, 6-8. Journal of Engineering Science and Technolog Special Issue /7

15 Comparaie sud of eloci reducion on feaher and snheic Alam, F.; Chowdhur, H.; Theppadungporn, C.; and Subic, A. (). Measuremens of aerodnamic properies of badminon shulecocks. Procedia Engineering, (), Le Personnic, J.; Alam, F.; Le Gendre, L.; Chowdhur, H.; and Subic, A. (). Fligh rajecor simulaion of badminon shulecocks. Procedia Engineering, 3, Kia, S.; Hasegawa, H.; Murakami, M.; and Obaashi, S. (). Aerodnamic properies of a shulecock wih spin a high Renolds number. Procedia Engineering, 3, Chan C.M.; and Rossmann, J.S. (). Badminon shulecock aerodnamics: Snhesizing eperimen and heor. Spors Engineering, 5(), Lin, C.S.H.; Chua, C.K.; and Yeo, J.H. (3). Turnoer sabili of shulecocks - Transien angular response and impac deformaion of feaher and snheic shulecocks. Procedia Engineering, 6, Cao, X.; Qiu, J.; Zhang, X.; and Shi, J. (4). Roaion properies of feaher shulecocks in moion. Procedia Engineering, 7, Teu, K.K.; Kim, W.; Tan, J.; and Fuss, F.K. (5). Using dual Euler angles for he analsis of arm moemen during he badminon smash. Spors Engineering, 8(3), Escor Radar (5). The Truh abou speed enforcemen. Rerieed Januar 5,, from Ongises, A.; and Xu, X. (3). Shulecock eloci of a badminon drop sho. Inernaional Scholasic Journal of Science, 7, Chen, L.M.; Pan, Y.H.; and Chen, Y.J. (8). A sud of shulecock s rajecor in badminon. Journal of Spors Science & Medicine, 8(4), McErlain-Nalor, S.; Miller, R.; King, M.; and Yeadon, M.R. (5). Deermining insananeous shulecock eloci: oercoming he effecs of a low ballisic coefficien. In Proceedings of he 4 h ITFF Spors Science Congress and 5 h World Racque Spors Congress. Suzhou, Cohen, C.; Teier, B.D.; Quéré, D.; and Clane, C. (5). The phsics of badminon. New Journal of Phsics, 7(6). 6. Collins, F.; Brabazon, D.; and Moran, K. (). Viscoelasic impac characerisaion of solid spors balls used in he Irish spor of Hurling. Spors Engineering, 4(), 5 5. Journal of Engineering Science and Technolog Special Issue /7

Using Rates of Change to Create a Graphical Model. LEARN ABOUT the Math. Create a speed versus time graph for Steve s walk to work.

Using Rates of Change to Create a Graphical Model. LEARN ABOUT the Math. Create a speed versus time graph for Steve s walk to work. 2.4 Using Raes of Change o Creae a Graphical Model YOU WILL NEED graphing calculaor or graphing sofware GOAL Represen verbal descripions of raes of change using graphs. LEARN ABOUT he Mah Today Seve walked