38 Hans Humenberger, Jan H. Müller: Assessments of traffic situations INTRODUCTION

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1 38 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations Assssmnt of traffic situations daling with braking distancs and rmaining locitis Hans Humnbrgr Unirsity of Vinna Jan H. Müllr Riius Gymnasium Attndorn Tchnical Unirsity of Dortmund Abstract: W prsnt workshts with which studnts of grad 11 can gt important knowldg about brak applications, raliz crucial phnomna, and dlop formulas concrning braking applications primarily by indiidual work. In Grman litratur thr ar also sral othr articls by othr authors (s th rfrncs) daling with th phnomnon of rmaining locitis. Hr in our papr th focus lis on th autonomous work of studnts: How can w prpar workshts so that studnts can com to dpr insights by working indpndntly in groups? W ha alrady publishd a possibl rsion in th last projct publication ( planting mathmatics, s []) but Jan Müllr has changd som itms so that w can prsnt anothr rsion which is mor rlatd to ral data, dscripti statistics, and fitting of functions. H also tstd this matrial in grad 11 in th last yar quit succssfully. Th aim of th following larning nironmnt is to snsitis studnts for problms in traffic ducation: assssing som gin and spcifid or slf constructd situations of dangr in traffic. INTRODUCTION ginning at th ag of 15 studnts attnd courss at driing schools in ordr to gt thir driing licnc for mopds, motorbiks and latr also for cars. In ths courss th problm of th dpndnc of th braking distanc on th initial locity is, of cours, dalt with. ut du to th htrognity of th participants only simpl mpirical formulas ar usd. E.g. 1 gis approximatly th braking distanc in mtrs if dnots th locity in km/h. In most cass such formulas ar not xplaind at driing schools. Tim is not spnt on answring th qustion whr this formula coms from, how can on undrstand it thoroughly? It is a ry usful formula bcaus on can asily stimat th braking distanc by looking at th spdomtr: If you dri at a locity of 5 km/h thn a short had calculation shows that th braking distanc is approx. 5² = 5 mtrs. In th abo form th formula also disrgards th arity for th possibl alus of th so calld braking dclration b. Th larning nironmnt w want to prsnt hr should nabl th studnts to dlop such a formula by thmsls, to undrstand it, to analys and to assss traffic situations. In addition th studnts should b snsitisd for th dlopmnt of th locity in trms of th distanc cord (instad in trms of th tim lapsd). This functional rlation locity distanc is a bit mor complicatd to analys than th rlation locity tim but it lads to dp and important insights

2 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 39 concrning rmaining locitis. You can fac th dramatic and normous consquncs of too high spd in traffic much bttr whn haing a look at rmaining locitis than by mrly daling with th formula for th braking distanc 1 s = and larning it by hart, n if on intrprts th b formula corrctly: doubl locity causs four tims th braking distanc. Considring th rmaining locity in cas of a collision is a propr way to show th striking consquncs of too high spd, spcially in cas of accidnts whr popl ar inold (safty in traffic)! Th larning nironmnt had thr parts: PART A: THE RAKING TIME AND THE RAKING DISTANCE IN TERMS OF THE INITIAL VELOCITY SEE WORKSHEET A1 In this part th formula s (, b) = for th braking distanc s should b b drid (in trms of th initial locity and th braking dclration b). W assum a stady brak application that mans a constant braking dclration b, which is dtrmind by th powr of th braks, by th road surfac, by wathr conditions, tyr profils tc. Th modlling is orintd at a gomtric illustration of th brak application in th -t-diagram: Th braking tim (tim btwn bginning of th brak application and stop) is dnotd as t. It is clar that th car would cor th distanc t without braking. This product i.. th braking distanc can b intrprtd as th ara of a rctangl in th -t-diagram. Whn braking stadily th locity will dcras stadily during th braking tim until it is. Th -tdiagram thn is no mor a constant function but a linar function with ngati slop. Th usual considration with small tim intrals xplains plausibl that also in cas of a non constant locity th ara undr th -tgraph gis th distanc cord ( s th gry triangl in th figur). 1 Th Indx for th distanc s coms from braking. dnots th initial locity and b dnots th (constant) braking dclration.

3 4 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations That mans th distanc cord during th brak application quals half th 1 distanc in th cas of not braking during th tim t : s = t (*) Th braking tim is t =. Within ry scond th locity gts lss by b b m/s. This can b insrtd in (*) which lads to: 1 1 s(, b) = t = b =. b If th initial locity is doubld ( ) you can s in th -tdiagram that also th braking tim is doubld ( t t ). Gomtrically spokn a cntral dilatation with factor happns to th considrd braking triangl. Thrfor th ara (i.. braking distanc) is multiplid by th factor 4! Aftr working with worksht A1 th studnts subsquntly did som adquat xrciss (s A) which ld to an assssmnt of gin situations by using numrical and graphical mthods dlopd bfor. Tak for xampl task 3 of A and you will s: Du to th longr braking distancs it is clar that th powr of th braks gts lss. So it is possibl to dtrmin a limit byond which th braking systm of th car cannot b sn as saf. It is also possibl to dal mor dply with th abo formula of driing schools s 1. Which alu of b is usd hr3? PART : THE VELOCITY DURING A RAKING PROCESS AS A FUNCTION OF THE DISTANCE COVERED SEE WORKSHEET 1 In th scond part th studnts should by indiidual work dri th formula ( s) = bs for calculating th locity in trms of th In th first lssons th intrprtation of b was pr scond th locity gts lss by b m/s, w did not us th unit m/s² in th bginning. 3 In th class of Jan H. Müllr this was don at th nd of th cours.

4 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 41 distanc s cord, th initial locity and th braking dclration b. This is don in th following way (s 1, task 1): Studnts should draw a fr hand sktch. Thy ar askd to spculat on how th locity dlops in trms of th distanc cord. Th rsult was that many studnts wr unsur whthr thy should tak a linar function ( straight lin ) or a lin slightly curd upwards. To clarify this uncrtainty th nxt stp in this xrcis is to dal with ral data. This shows that th non linar modl (th curd upwards on) fits bttr hr. Th trm ( s) = bs of th function which is aimd at is drid by intrprting th diffrnc b b (with < ). oth b and b gi braking distancs, on th on hand with initial locity and on th othr with initial locity <. Thrfor th diffrnc b b dnots th distanc s which is ncssary to rduc th locity from to : s = b b. Soling this quation for yilds ( s) = bs. This squar root trm may hlp to bttr undrstand th non linarity of th rlation btwn th locity and th distanc cord (during a braking application) s abo: fr hand sktch. To point out th adantag or usfulnss of this function th studnts should calculat (, task 3). y which prcntag has locity dcrasd aftr 1%, %,..., 9%, 1% of th braking distanc. Hr you can s drastically that aftr half th braking distanc only ¼ of th locity is rducd, and that you nd ¾ of th braking distanc to rduc half th locity. This shows that locity is rducd primarily at th nd of th braking distanc (only in trms of th tim lapsd th locity slows down stadily). PART C: ASSESSING A CERTAIN TRAFFIC SITUATION SEE WORKSHEET C

5 4 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations In th third part th studnts workd togthr in groups and chos on traffic situation (out of two) to analys, to dscrib mathmatically and to assss with rspct to th paramtrs considrd ( worksht C). In our sampl study th studnts had two wks tim, also for th parts 1 and togthr, so that th whol larning nironmnt lastd four wks (3 lssons a wk). Studnts plannd for thmsls which way of working thy prfrrd,.g. thy workd partly alon and at hom and discussd thir rsults with th othr mmbrs of th group during th lssons. This kind of taching organisation had th adantag of giing tachrs th chanc to hlp and ncourag ry wll both high-prformanc studnts and low-prformanc studnts indiidually Rsult: high-prformanc studnts wantd to dal with ry challnging problms and qustions, th othrs wantd to b closr to qustions of part 1 and. Th aim for all was to produc a Powr-Point prsntation of about 1 minuts, which was prsntd in th last two lssons.. Du to th formula ( s) = bs from part it was possibl to calculat so calld rmaining locitis : At th ry position at which a car driing initially at 3 km/h coms to stop 4, anothr car driing initially at a spd of 5 km/h bginning th brak application at th sam point has a rmaining locity of 4 km/h. Such a high collision locity is in most cass lthal for a child! With rspct to a zon whr cars ar allowd to dri 3 km/h at most (for instanc in many rsidntial aras in innr citis) this mans: Somon driing only km/h too fast has at th crucial position 5 a significantly highr locity than somon who is obying th spd limit and not braking at all! If on taks into account th diffrnt raction distancs of ths two drirs (lt s assum qual raction tims: 1 s) and a braking dclration of b 6 m/s² (this is a highly ralistic alu) thn w ha th fact that th 5 km/h car has n its full locity (5 km/h) at th crucial position! That mans it n has not yt startd to brak at this position! Such (or similar) scnarios wr mant whn w wrot at th bginning considring th rmaining locity in cas of a collision is a good way to point-out drastically th striking consquncs of too high spd, spcially in cas of accidnts whr popl ar inold (safty in traffic)! W ar sur that ths ralisations ha mor ffcts than phrass lik: Doubl locity mans four tims th braking distanc. So mathmatics can contribut to mor rsponsibility and cautiousnss in traffic. 4 E.g. bcaus of a carlss child ftching a ball. 5 Th position at which th 3 km/h car coms to stop.

6 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 43 CONCLUSION Th dscribd larning nironmnt was profitabl to th studnts for sral rasons: Du to th way of organisation th tachr had a bttr chanc to adis studnts indiidually,.g. to discuss adantags/disadantags of th working stratgy chosn, or to hlp thm in numrical and mathmatical analysis. High-prformanc studnts wr ncouragd to hlp othrs. Th working matrial (workshts and xrciss) mad it ncssary for th studnts to dcid for thmsls which parts had to b don at hom. A fdback aftr daling with this larning nironmnt showd that many studnts had troubls with this way of working whil suprising thmsls. Th whol topic was dalt with in grad 11. It was linkd to th topic of dscripti statistics. Thrfor idas lik linar rgrssion, fitting curs could b rpatd. In th physics lssons th studnts had dalt with stadily acclratd (dclratd) motions a fw wks bfor, so that this projct was also a good rptition and addition. If ral data and dscripti statistics ar not supposd to play such an important rol thr is also a rsion dscribd in th last publication planting mathmatics ( []). In this form th topic was (or should b) dalt with in sral othr countris: Austria, Dnmark, England, Hungary, Italy, Poland, Romania, Sloakia. W do hop that th xprimnt was (will b) succssful! REFERENCES Hrgt, W. (1995): Mobilität, Modllbildung Mathmatik! In: mathmatik lhrn 69, pp 4 7. Humnbrgr, H. (8): rak applications and th rmaining locity. In: W. Hnn, S. Mir: Planting Mathmatics. Dortmund, pp67 8. Humnbrgr, H. & Müllr, J. (9): Wi schätzt du di Vrkhrssituation in? rmswg und Rstgschwindigkitn rarbitn. In: mathmatiklhrn 153, pp5 55. Myr, J. (1995): Gschwindigkit und Anhaltwg. In: ISTRON-matrials, ol., Hildshim, Franzbckr. MUED: Untrrichtsinhit 1-4-3, Gschwindigkitsbrchnung. Wintr, H. (1989): Entdcknds Lrnn im Mathmatikuntrricht. raunschwig, Viwg, pp 7. Wintr, H. & Haas, N. (1995): Vrsthn als Modllbildn. ispil aus dm motorisirtn Straßnrkhr. In: mathmatiklhrn 68, pp47 53.

7 44 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations

8 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 45 A1) raking tim and braking distanc as a function of arious locitis Task 1: km/h and m/s ar two typical units for locity (spd) a) Which adantags and/or disadantags dos ach of ths ha? b) Conrt th locitis! km/h m/s c) Spcify a gnral formula for ach of th conrsions. Task : Th following tabl shows th data of ONE braking procss. Tim t [s] aftr bginning to brak Vlocity [m/s] a) Show th data in a graph. b) Calculat a alu b which roughly indicats by how much th spd pr scond dcrass on arag in this braking procss. How do you tchnically or colloquially call this alu? c) Th following tabl shows th data of VARIOUS braking procsss. Using th alu b from part b) calculat ach of th braking tims t for ach of th arious initial locitis. [m/s] t [s] d) Find a formula with which th braking tim t can b calculatd using th gin initial locitis and th brak rtardation b : t (, ) b Task 3: If you nithr brak nor acclrat, th locity will rmain constant (graph: thick lin). If you brak, th locity will dcras with tim (graph: data points). a) How can you calculat th distanc cord by a car drir who is driing t sconds at a spd of m/s? b) How could you compar th distanc that a drir would cor during a braking procss ( braking distanc ) with th distanc h would cor in th sam tim without braking? c) Which distanc s did th drir with th data in a) cor whil braking ( braking distanc )? d) How can you calculat this braking distanc s in gnral using and t or b : s (, ) b?

9 46 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations Possibl solutions for A1: 1a) Km/h is closr to ryday-lif and you can imagin th concrt locity much bttr; m/s is suitd bttr as a unit of masurmnt for calculating sinc th braking procsss happn on th tim scal of sconds and on th distanc scal of mtrs. b) km/h m/s c) in m/s = in km/h 3.6 a/b) Linar rgrssion proids a cofficint of corrlation of r.99. Thus th data indicat a linar corrlation btwn t and. Th trndlin linar quation is 5.81t According to this, th locity pr scond is rducd on arag by th absolut alu of th slop of th straight lin, thus b 6. This alu ( 6) is calld brak rtardation or braking dclration. or in km/h = 3.6 ( in m/s) ] /s 1 [m 1 d 8 6 S p 4 y = -5,89x + 19, c) [m/s] t b [s] b us m/s for and m/s² for b.,5 1 1,5,5 3 3,5 Tim t [s] d) t = ; b attnti to th units: if t should com out in sconds, thn you ha to 3a) If you dri at a constant spd of for a long priod of tim t, thn th distanc cord is calculatd using th product of locity tims tim, or in short: t. Th distanc cord is, gomtrically xplaind, th ara of th rctangl that is rstrictd by th coordinat axs and th thick lin. b) If you now conci a straight lin t () = b t through th data points, thn th distanc cord whil braking is barly half as larg as th on with constant locity. Th distanc cord is, gomtrically xplaind, th ara of th triangl that is rstrictd by th coordinat axs and th lin that runs through th data points. c) From th trndlin linar quation -5.81t you can xtract that amounts to: =19.14m/s. t is calculatd according to task as Th braking 5.81 distanc is thus roughly m 3.3s 3 m long. s d) Th distanc cord whil braking ( braking distanc ) is th surfac ara of th 1 1 dscribd triangl: s = t = b = b s

10 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 47 A) Exrciss: Task 1: Th following tabl shows th data of on braking procss. Tim t [s] Vlocity [m/s] Which distanc did th car drir approximatly cor whil braking? s Task : Th alu m 7 s. b for th rtardation of a car whil braking is roughly a) Complt th tabl. [km/h] b) Show th data in a graph. s [m] c) Compar th alus. Dduc statmnts that nw drirs should b awar of. Task 3: Assum that a car tralling at a locity of = 7 km/h is suddnly brakd by th drir and h nds th gin braking distancs. a) Complt th tabl. b) Show th data in a graph. s [m] c) Compar th alus. According b [m/s²] to th Road Traffic Licnsing Rgulation in Grmany (calld StVZO) 41 (4): braks and whl chocks, automobils must on arag bcom slowr at a rat of at last 5 m/s. What would you say to that as an MOT official? Task 4: Assum you could impro th rtardation alu of th braks using tchnical masurs. Would this hlp th braking distanc? Impromnt of th rtardation alu b by s 1% % 3% 4% 5% 1% Rduction of braking distancs also in % a) Complt th tabl. b) Show th data in a graph. c) Compar th alus. Dduc statmnts that nginrs should b awar of. Task 5: Using th lngth of th braking distanc skid mark which is mad by a car at an accidnt, xplain whthr you can draw any conclusions about th locity at which th accidnt happnd. If ys, spcify.g. a formula with which th locity can b rconstructd. If no, gi rasons why!

11 48 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations Possibl solutions for A: 1) r.99, t Th tim until th car coms to a halt is m/s t = 3s. Thus th braking distanc is m 3s 37m long m/s s ( / 3.6) ) Th wantd alus can b calculatd quickly using s = [km/h] [m/s] s [m] In th last lin, th intrmdiat data (lin ) was usd for furthr calculation with full calculator accuracy. In th diagram, th point (/) is additionally ntrd sinc, of cours, no braking distanc can occur at km/h. You can clarly rcognis (also by simply using th 6 data) that th braking distanc dos not proportionally dpnd on th 5 ) locity. If you considr th diagram (m 4 in connction with th data, thn it c n n bcoms idnt that th ta 3 braking distanc roughly quadrupls is d whn th spd is doubld. This can g also b asily rcognisd with th k in 1 ra formula s =, sinc b 5 1 ( ) 4 s = = = 4. Nw Spd (km/h) b b b drirs should.g. tak into considration that an incras in locity considrably incrass th lngth of th braking distanc orproportionally. 3) Th wantd alus for th rspcti braking rtardations can b calculatd quickly using b =. s Whn dpicting th data points you can clarly s that with th rduction of th braking distanc, th alu of th rtardation has to bcom considrably bttr orproportionally (you s this at th x-axis from right to lft). As an MOT official you should notic that braking distancs of roughly 4 mtrs and mor at a locity of 7km/h ar not allowd by th Road Traffic Licnsing Rgulation in Grmany! s [m] b [m/s²] ²) /s 1 (m n tio 1 a 8 tard R raking distanc (m)

12 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 49 4) Th wantd alus for rspcti impromnts in th lngth of th braking distanc in % can b calculatd by comparing th braking distanc lngth with improd braks with th old braks. If you compar both alus by.g. diiding thm, thn you can rad th impromnt in % from th quotint. An xampl for a 1% impromnt: simprod 1.1b b 1 = = =,91, that mans an impromnt of th braking sold 1.1 b 1.1 b 1 distanc by 9%; for % th outcom is.83, thus, an impromnt of 17%, tc.: 1. Impromnt in th rtardation alu b [m/s²] by Impromnt in th braking distanc lngth s [m] by 1% % 3% 4% 5% 1% 9% 17% 3% 9% 33% 5% From th data and its dpiction, you 5,% can clarly ) rcognis that a 45,% (% crtain rlati 4,% c impromnt in n braking fficincy ta 35,% (in %) dos not d is 3,% g caus th sam 5,% rlati k in ra,% impromnt in f b 15,% braking distanc. If you impro.g. n t o 1,% th rtardation 5,% alu by % from m,% 6m/s² to 7.m/s², p ro thn th braking Im % % 4% 6% 8% 1% distanc dcrass Impromnt of braks (%) by only 17%. On th othr hand, ry impromnt that can sa human lis is, of cours, worth it. 5) It can b ry asily ralisd that in most cass, th lngth of a braking distanc skid mark dos not allow any conclusions to b drawn about th tralld locity at th momnt of an accidnt: at th momnt w assum that two cars collid or a car hits a stationary obstacl, th car no longr has th chanc to continu its skid mark. Th skid mark is thus incomplt and dos not allow any conclusions to b drawn about th locity tralld. ut also in th cas of bodily injury, that mans a car hits.g. a pdstrian and can continu its braking distanc (almost) without intrruption, thus th lngth of th skid mark is not only dpndnt upon th rtardation of th braks, but also on.g. th road surfac, moistur, las on th roadway, tyr trad, tc. According to th statmnt from th polic station in Olp, Grmany, in srious cass (.g. accidnts rsulting in dath), th dformation of th car body and th chassis is xamind by xprts in ordr to assss th locity at th momnt of th accidnt. This data is thn compard to data achid in a sris of tsts.

13 5 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 1) Th locity during a braking procss as a function of th distanc cord Assum you ar driing a car with an approximatly constant locity and you suddnly ha to slam on th braks. Or th cours of th distanc s tralld whil braking your locity will obiously continu to dcras until you ultimatly com to a complt stop. Task 1: a) ut HOW do you think th locity s () will dcras? At first, sktch th graph ( frhand ) of th dlopmnt of th locity as you think it could go without calculating. Not in bullt point form why you think that th locity dlops th way in which you drw it. b) Th following tabl shows th data of such a braking procss: Distanc s [m] Vlocity [m/s] Entr th data carfully into th diagram. Scal th axs for this data snsibly/appropriatly. Rdraw th diagram if ndd, so that it is tidir. c) Compar your graph with th data. Dscrib in abbriatd form what you notic hr. Task : Considr a braking procss with th locity of stop; what dos b man hr? What dos man during a braking procss ( < b ) stop? What dos th diffrnc s = b b man? If you sol this quation for, you rci th function for th locity (initial locity as a function of th tralld distanc s: = ( s) xplain why! Task 3: a) How do you ha to choos and b in th formula from task in ordr for th graph of th function (s) to run xactly through th points (/15) and (/)? b) Draw th function graphs for (s) for ths alus.

14 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 51 Possibl solutions for 1: 1) Hr only th data ar prsntd. It is intrsting that th locity dos not sink uniformly (linar): at th bginning, th locity actually slightly dcrass and thn (bginning at roughly 15 mtrs) it dcrass astonishingly ry rapidly! 16 ) 14 /s 1 (m 1 d 8 p 6 a l s u 4 s id R raking distanc s (m) ) Th maning of is braking distanc b at an initial locity of. Th maning of is braking distanc at b an initial locity of <. Th maning of th diffrnc s = b b is th ncssary distanc for th rduction of th locity. Soling for rsults in = () s = bs. With this formula/function, you can calculat th rsidual locity of a braking car as a function of th distanc s tralld whil doing so (if you know and b ). 3a/b) You larn from th data that 16 =15 must b 14 applid. So w can ) /s 1 conclud that (m 1 s ( ) = 5 bs. 8 Thrwith it is d 6 nsurd that th graph a l s p passs through th u 4 point (/15). For s id (/) th ansatz R follows = 5 b. raking distanc s (m) Soling this quation for b rsults in b = Thus it follows s ( ) = s. Tip: With Excl you can also click on othr typs of rgrssions instad of th trndlins for th data, in ordr to ralis a data rgrssion plas try it and find a rasonabl answr!

15 5 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations ) Exrciss: Task 1: Th following tabl shows th data of a braking procss. Distanc s [m] Vlocity [m/s] a) Dscrib (at last) possibilitis of how you can dtrmin a function (s). Explain th adantags and disadantags of your possibilitis. b) Dtrmin a function (s) and draw its graph. Task : Assum a car is suddnly brakd by th drir at a locity of = 1km/h. Th alu b for th rtardation of th car whil braking m amounts to roughly 7. s a) Sktch th graph of th dlopmnt of th locity whil braking as you think it could go. Entr at last 3 data points in your sktch which you think th graph passs through. b) Complt th tabl. Distanc s [m] Vlocity [m/s] c) Show th data point and th ntir function (s ) in a graph. Task 3: Assum a car is suddnly brakd by th drir at a locity of = 8 km/h. Th alu b for th rtardation of th car whil braking m amounts to roughly 8. s a) What prcntag of th initial locity is rducd aftr 1%, %,... of th braking distanc? Complt th tabl. raking % 1% % 3% 4% 5% 6% 7% 8% 9% 1% distanc tralld s (%) Rducd % 1% locity (%) b) Show th data in a graph. c) Rflct on your rsults: What should a nw drir b awar of in this rspct? Task 4: Th following tabl shows th data of a braking procss. Tim t [s] Vlocity [m/s] Dtrmin and draw (s). How long is th braking distanc s?

16 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 53 Possibl solutions to : 1a) 1st possibility: As in task you slct points of 1 and you carry out th algorithm analogously. Adantag: quick algorithm; Disadantag: you only considr data points. nd possibility: With Excl you can dpict th data as a diagram. If you click on diagram typ:.g. polynomial, thn you gt th quation as a rgrssion function -.45s² s + 5. Th cofficint of corrlation is r.98. thus this is also not a bad choic. Howr, this dpiction has th disadantag that th cur dfinitly dos not pass through (/). It is also th wrong typ of function (parabola instad of squar root function). b) You larn from th data that = 5 must b applid. Hnc: s ( ) = 65 bs. Thrwith it is nsurd that th graph passs through th point (/5). For (/) w gt = 65 b. Soling this quation for b rsults in b 14.. Hnc: s ( ) = 5 8.4s. ) 1km/h 7.8m/s, from this follows th function s () s Distanc s [m] Vlocity [m/s]

17 54 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 3) = 8km/h.m/s, from this it follows sb 31m raking distanc % 1% % 3% 4% 5% 6% 7% 8% 9% 1% tralld s (prcntag) raking distanc tralld s (m) (s) (m/s) Rducd locity (prcntag) % 5% 11% 16% 3% 9% 37% 45% 55% 68% 1% It is asy to rcognis in th diagram that aftr mor than 8% of th tralld braking distanc only about 5% of th original locity is rducd. Thrfor th rduction of th rmaining 5% of th locity happns in th last 6 mtrs (from a total of 3!). That undrlins onc again th pla of th driing schools and road patrols for carful, cautious and anticipatory driing!!!! 4) Th trndlin linar quation is 8.193t (with a cofficint of corrlation of r.998!). As a rsult, thr is a braking rtardation of b 8.m/s² and a bginning locity of 3.4 m/s. From this follows th formula s ( ) s. Th braking distanc lngth is calculatd using 3.4 s 8. 64m.

18 Hans Humnbrgr, Jan H. Müllr: Assssmnts of traffic situations 55 C) Two typical situations in road traffic Situation 1: A car is driing in a 3km/h zon and is ortakn by anothr car. As both cars ar roughly on ll with ach othr, childrn run onto th strt without looking. Th slowr car barly coms to a halt in front of th childrn. Situation : A car is ortakn on a country road by anothr car. As both cars ar roughly on ll with ach othr, anothr car appars in th oncoming traffic. Task: Choos a situation and analys it basd on your knowldg. That mans: Find schoolmats who ha also chosn th sam situation. Tip: Not too many!!! Othrwis, spaking from xprinc, th tamwork dos not work as wll. Think of as many qustions as possibl that sm rlant to your situation (brainstorming collcting idas!). Dcid on at last on of your qustions and try to analys it. Th xpctations of th analysis ar: Analys th qustion basd on a concrt situation (.g. two assumd locitis for both cars). Analys th qustion using diffrnt locitis for ONE car. Gnralis th diffrnt locitis into a function and xamin its proprtis in ordr to mak conclusions for your issus/qustions. Prpar a ca. 1- minut prsntation (adantagous: Powr-Point!) Tip: In all of th workshts, th raction tim of th drir or th so-calld momnt of shock was not accountd for! If you considr th raction tim rlant, thn you should also considr this in your analysis. In th pictur: Raction distanc + braking distanc = orall stopping distanc A group of studnts prsntd th contnts of th road safty programm in Cologn ystrday. With roundabout 4, road safty programm nts, th ADAC (Gnral Grman Automobil Association) wants to prpar or 1, fifthgradrs for th dangrs in road traffic. In th on and a half hour lssons, th pupils ar shown quit plainly th connction btwn raction tim, braking distanc and orall stopping distanc. According to th ADAC th pupils ar, for xampl, allowd to rid in a car and xprinc full brak application at 5 km/h.