s= d/t Kinematics 1 Page

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1 Knmac Th fr, and m fundamnal, qun f phyc rla mn. Many f h fr wrng f phyc ar n h pc and da back huand f yar. Th udy f mn calld knmac. I cm frm h Grk wrd knma, whch man mn. Alm vryhng w larn n phyc wll nvlv h mn f bjc. S, knmac mu b undrd wll n rdr undrand hr pc w wll b udyng n h fuur. Un f Tm and Danc In rdr ha ppl can cmpar hr maurmn wh h akn by hr; an nrnanal ym f maurmn wa agrd upn. Th Sym Inrnanal (SI) ud by vrually all cn n h wrld. In ha ym, h bac un f lngh h mr and h bac un f m h cnd. Danc maurd n mr (m). Tm maurd n cnd (). Cnan Spd If vryhng n h wrld ju d ll, w d nd maur m and danc paraly and w d b dn. Bu ha d b a pry brng wrld. Many f h m nrng hng nvlv mn; bjc mvng frm n lcan anhr n a cran amun f m. Hw fa hy d h h pd f h bjc. Spd n a fundamnal prpry f h wrld, lk danc and m, bu a human nvnn. I dfnd a h ra f h danc ravld dvdd by h m k ravl ha danc. Spd Danc/Tm r = d/ **Th qual gn wh hr paralll ln ju pn u ha h a dfnn. W mad up h wrd pd and hn dfnd man h ra f danc and m. Un f Spd Th un f pd can b drvd frm frmula: = d/ Th SI un f danc mr (m) and m n cnd (). Thrfr, h un f pd m/. 1 Pag

2 Whn lvng phyc prblm hr a r f p ha huld b fllwd. In h arly prblm ha w ll b dng, ll b pbl kp m p and ll g h crrc anwr. Bu ha wn gv yu a chanc pracc h mhd ha yu ll nd lv mr dffcul prblm. I w larn hw wm n h hallw nd f h pl, bu f all yu d and up hr, wn b much hlp whn h war g dp. S, pla u h fllwng apprach n all h prblm yu lv rgh frm h bgnnng. I ll pay ff n h nd. 1. Rad h prblm carfully and undrln, r mak n f, any nfrman ha m lk may b uful. 2. Rad h prblm hrugh agan, bu nw ar wrng dwn h nfrman ha wll b f valu yu. Idnfy wha bng akd fr and wha bng gvn. 3. If apprpra, draw a kch. 4. Idnfy a frmula ha rla h nfrman ha yu v bn gvn h nfrman yu v bn akd lv fr. 5. Rarrang h frmula lvd fr h varabl yu r lkng fr. Th man, g ha varabl b aln n h lf d f h qual gn. 6. Subu n h valu yu v bn gvn, ncludng un. 7. Calcula h numrcal rul. 8. Slv fr h un n h rgh d f h quan and cmpar h h un ha ar apprpra fr wha yu r lvng fr. Fr nanc, f yu r lvng fr danc, h un huld b n mr n mr pr cnd. 9. Rrad h prblm and mak ur ha yur anwr mak n. I bn hwn ha uccful phyc udn rad ach prblm a la hr m. Exampl 1: Rdng yur bk a a cnan pd ak yu 25 cnd ravl a danc f 1500 mr. Wha wa yur pd? Gvn Frmula S Up Slun Exampl 2: Hw far wll yu ravl f yu ar drvng a a cnan pd f 25 m/ fr a m f 360? Gvn Frmula S Up Slun Exampl 3: Hw much m wll ak yu ravl 3600 m f yu ar drvng a a cnan pd f 20 m/? 2 Pag

3 Inananu Spd Thr an ld jk abu a prn wh pulld vr fr pdng. Th plc ffcr ll h pdr ha h wa gng 60 ml pr hur n a fry ml pr hur zn. Th pdr rpn ha h culdn hav bn gng xy ml pr hur nc h d nly bn drvng fr ffn mnu. Inananu pd h pd ha yu rad n yur pdmr r ha a plcman rad n h radar r lar gun. Avrag Spd Whl ravlng alng, yur var; g up and dwn alng h way. Yu mgh vn p fr a whl hav lunch. Yur nananu pd a m mmn durng yur rp and yur avrag pd fr h al rp ar fn n h am. Yur avrag pd calculad by drmnng h al danc ha yu ravld and dvdng by h al m ha k yu ravl ha danc. Exampl 4: Yu rd yur bk hm frm chl by way f yur frnd hu. I ak yu 7 mnu ravl h 2500m yur frnd hu. Yu hn pnd 10 mnu hr. Yu hn ravl h 3500 m yur hu n 9 mnu. Wha wa yur avrag pd fr yur al rp hm? Gvn Frmula S Up Slun Exampl 5: Yu run a danc f 210 m a a pd f 7 m/. Yu hn jg a danc f 200 m n a m f 40. Fnally, yu run fr 25 a a pd f 6 m/. Wha wa h avrag pd f yur al run? Gvn Frmula S Up Slun Pn, Dplacmn and Vlcy S far ur analy ha n rqurd, r vn allwd, u knw anyhng abu h drcn f h mn undr udy. Bu n ral lf, drcn uually vry mpran. Whhr yu r drvng 60 ml pr hur nrh r 60 ml pr hur uh, mak a gra dal f dffrnc a whr yu nd up. Scalar ar quan ha ar dfnd nly by hr magnud; h numrcal valu. Spd, m and danc ar all xampl f calar. Whn w pak f 40 m/, 20 mnu r 3 ml w r n gvng any nfrman abu drcn. Vcr ar quan ha ar dfnd by bh h magnud and drcn. S, nad f ayng ha I ravld a danc f 400m, I wuld ay ha I ravld 400m nrh; I am nw dfnng vcr. Th vcr ha r dfnd by cmbnng danc wh drcn calld dplacmn. Th ymbl fr dplacmn Δx. 3 Pag

4 Exampl 6: Yu drv 1500m nrh and hn 500m uh. Drmn bh h al danc yu ravld and yur al dplacmn frm whr yu ard. Th am dffrnc x bwn pd and vlcy. Th ymbl fr vlcy v and h ymbl fr avrag vlcy vavg. Th avrag vlcy drmnd by dvdng yur al dplacmn by h m k fr ha dplacmn. Th mlar hw w calculad avrag pd by dvdng h al danc ravld by h al m k ravl ha danc. = d/ and v avg = Δx/ Exampl 7: If h ravl n Exampl 6 wa dn a cnan pd and rqurd a al m f 500, drmn h avrag pd and h avrag vlcy. Gvn Frmula S Up Slun Crdna Sym Th dplacmn f an bjc ll u hw pn ha changd. In rdr br undrand wha ha man w nd a way f dfnng pn; w nd a crdna ym. Th rqurmn f any crdna ym ar an rgn and an rnan. In hr wrd, yu nd pck a zr frm whch yu ll b makng maurmn and yu nd knw h drcn n whch yu wll b maurng. Th mpl yp f crdna ym n-dmnnal, n whch ca h crdna ym bcm a numbr ln, a hwn blw. Th rgn lcad a zr, ngav pn ar h lf f h rgn and pv pn ar h rgh. W can dnfy dffrn lcan n h numbr ln a uch a x0, x1 and x2. In h dagram abv, x0 lcad a 0, x1 lcad a +5m and x2 lcad a -5m. W can nw rfn ur dfnn f dplacmn, h chang n h pn f an bjc, a bng h dffrnc bwn an bjc fnal pn, x, and nal pn, x. I nw bcm clar why h ymbl fr dplacmn Δx. Th Grk lr dla, Δ, man h chang n Δx can b rad a dla x r h chang n x. Symblcally h bcm; Δx x x Exampl 8: An bjc mv frm an nal pn f +5m a fnal pn f +10m n a m f 10. Wha dplacmn dd undrg? Wha wa avrag vlcy? Exampl 9: An bjc mv frm an nal pn f +5m a fnal pn f -10m n a m f Wha dplacmn dd undrg? Wha wa avrag vlcy? 4 Pag

5 Inananu Vlcy and Acclran Th m brng wrld wuld b n n whch h pn f all bjc wr cnan...nhng wuld mv: vlcy wuld hav n manng. Frunaly ur wrld a l mr nrng han ha. Objc ar changng hr pn all h m, vlcy an mpran cncp. Bu ur wrld vn mr nrng, bjc ar al changng hr vlcy all h m: hy ar pdng up, changng drcn and/r lwng dwn. Ju a chang n pn vr m lad h da f vlcy, chang n vlcy vr m lad h cncp f acclran. In h am mannr ha w dfnd nananu pd a h pd maurd durng a vry hr prd f m, w can nw dfn nananu vlcy a h vlcy maurd durng a vry hr prd f m. Th ymbl, v wll b ud fr nananu vlcy. In a wrld wh acclran, h da f nananu vlcy vry mpran nc an bjc vlcy may fn b changng frm mmn mmn. v Δx/ fr a vry hr prd f m...an nan. W can nw dfn acclran a h chang n vlcy vr m. a Δv/ r a v v / Un f Acclran Th un f acclran can b drvd frm frmula: a Δv/ Th SI un f vlcy mr/cnd (m/) and f m h cnd (). Thrfr, h un f acclran (m/)/ r m//. Th h am a (m/) x (1/) nc dvdng by h am a mulplyng by 1/. Th rul n m/ 2 whch, whl n havng any nuv manng, a l ar kp rack f han m//, mr pr cnd pr cnd, h alrnav way f wrng h un fr acclran. Exampl 11 An bjc ravlng a a vlcy f 20m/ nrh whn xprnc an acclran vr 12 ha ncra vlcy 40m/ n h am drcn. Wha wa h magnud and drcn f h acclran? L lv h algbracally by dfnng vlc ward h nrh a pv and ward h uh a ngav. Gvn Frmula S Up Slun 5 Pag

6 Exampl 12 Wha wll an bjc vlcy b a h nd f 8.0 f nal vlcy +35m/ and ubjc an acclran f -2.5m/ 2? Gvn Frmula S Up Slun Fr Fall Yu nw knw nugh b abl undrand n f h gra dba ha markd h bgnnng f wha w nw call phyc. Th rm phyc wa bng ud by h ancn Grk mr han 2000 yar ag. Thr phlphy, much f dcrbd n h bk ld Phyc by Arl, ncludd m da ha d unl Gall mad m mpran argumn and maurmn ha hwd h ancn Grk phyc b f lmd valu. Th phyc f ancn Grc ncludd h da ha all bjc wr mad up f a cmbnan f fur lmn (h ffh lmn wa rrvd fr bjc ha wr bynd h arh). Th fur lmn b fund n ur wrld wr arh, war, ar and fr. Each f h lmn had hr naural plac. If yu rmvd an lmn frm naural plac, wuld, whn rlad, mmdaly mv back ha plac; and wuld d wh naural (cnan) vlcy. Thr vw f h wrld culd b hugh f a f cncnrc crcl wh ach f h lmn ccupyng a layr. Earh ccupd h cnr f h crcl, rck, whch ar prdmnanly mad f arh wuld naurally mv dwn, ward h cnr f ur wrld. Abv arh wa war, whch wuld fll h ara abv h rck, lk a lak r an can abv h land ha frm h lak r can bd. Abv war ar, whch n vrywhr n ur wrld, abv bh arh and war. Fnally, fr r up hrugh h ar, archng fr naural lcan abv vryhng l. All bjc wr cndrd b a mxur f h fur lmn. Rck wr prdmnanly arh: f yu drp a rck fall a r g back naural lcan a h cnr f h arh. In dng, wll pa hrugh war and ar: If yu drp a rck n a lak, nk h bm. Fr pa upward h hgh lcan, f yu mak a fr, alway pa upward hrugh h ar. On cnclun ha h ld ha bjc whch wr mad f a hghr prcnag f arh wuld fl a grar drv rach hr naural lcan. Snc arh h hav f h lmn, h wuld man ha havr bjc wuld fall far han lghr bjc. Al, hy wuld fall wh a naural cnan vlcy. Tha phlphy d fr mr han 2000 yar unl Gall, n h 1600 mad a r f argumn, and cnducd a r f xprmn, ha prvd ha nhr f h w cnclun wa accura. H hwd ha h naural ndncy f all unupprd bjc fall ward h cnr f h arh wh h am acclran: 9.8m/2. Tha numbr 9.8m/2 ud fn ha a gvn wn ymbl: g. In mdrn rm, h cnclun can b ad a fllw. All unupprd bjc fall ward h cnr f h arh wh an acclran f g: 9.8m/ 2. Th amn rqur m xplanan and m cava. 1. Unupprd man ha nhng hldng h bjc up. S f yu rla mhng and nhng ppng frm fallng, hn unupprd. In ha ca, all bjc wll xprnc h am acclran dwnward. I d n dpnd n hw havy h bjc : all bjc fall wh ha am acclran. 6 Pag

7 2. Suppr can al cm frm ar ranc. S a parachu prvd uppr by cachng ar ha lw dwn h ky dvr. In ha ca, h parachu n an unupprd bjc: h r h upprd by ar ranc. Bu h gnrally ru a lr xn. S a fahr r an uncrumpld pc f papr al rcv uppr frm h ar: hy dn fall wh a cnan acclran hr. Gall cnclun an dalzan, aum ha w can gnr ar ranc, whch nvr cmplly ru nar h arh (r arplan and parachu wuld hav a hard m f ) bu wll wrk fr h prblm w ll b dng. 3. H cnclun d n dpnd n h mn f h bjc. S baball hrwn ward hm pla, drppd, r hrwn ragh up all fall wh h am acclran ward h cnr f h arh. Th an ara f gra cnfun fr udn, yu ll b rmndd f fn. Whnvr nhng ppng an bjc frm fallng, wll acclra dwnward a 9.8m/2, rgardl f vrall mn. In h bk, w wll aum ha ar ranc can b gnrd unl pcfcally ad b a facr. Exampl 13 An bjc drppd nar h urfac f h arh. Wha wll vlcy b afr ha falln fr 6.0? Gvn Frmula S Up Slun 7 Pag

8 Inrprng Mn Graph Thr ar w yp f mn graph ha w ll b cndrng, Pn vru Tm and Vlcy vru Tm. In bh ca h hrznal ax, h x- ax, ud rcrd h m. In a Pn vru Tm graph, h vrcal ax, h y-ax, ud rcrd h pn f h bjc. In a Vlcy vru Tm graph, h vrcal ax, h y-ax, ud rcrd h bjc vlcy. In h cn, w r gng larn hw mak and nrpr h graph and hr rlanhp ach hr. Pn vru m graph fr Cnan Vlcy If whl yu wr mvng yu wr rcrd yur pn ach cnd, d b ay mak a pn vru m graph. L ak h ca ha yu r walkng away frm yur hu wh a cnan vlcy f +1m/ (n ha nc vlcy a vcr dfnng rqur a drcn, +, and a magnud, 1m/ ). If yu dfn yur hu a zr and dfn yur arng m a zr hn h fr fv cnd f yur walk wuld gv yu h fllwng daa. Tm (cnd) Pn (mr) T cra a pn vru m graph all yu nd d graph h pn and hn cnnc hm wh a ragh ln. Th graph allw yu drcly rad yur pn a any gvn m. In fac, allw yu drmn yur pn fr m a whch yu mad n maurmn, ha h manng f h ln cnncng h pn. Fr nanc, yur pn a 1.5 cnd can b n b a x = 1.5m. Nw h aum ha yu wr ravlng a a cnan vlcy, bu ha h aumpn ha wa mad n crang h graph. Yu can al ndrcly rad yur vlcy frm h char. P n ( m r ) Pn v. Tm y = x R² = Tm (cnd) Th vlcy f an bjc wll b h lp f h ln n Pn vru Tm graph. 8 Pag

9 Th dfnn f h lp f a ln, m, m Δv/Δx. Th man ha h lp f a ln drmnd by hw much h vrcal valu, h y-valu, f h ln chang fr a gvn chang n hrznal valu, x-valu. If dn chang a all, hn h ln hrznal, ha n l. If ha a pv valu lpng upward, nc ha man ha h y-crdna g largr a yu mv h rgh alng h x-ax. A ngav lp man ha h ln ld dwnward, nc y-valu dcra a yu mv h rgh. If yu mak a graph f h pn vru m fr a mvng bjc, wh pn hwn n h y-ax and m n h x-ax, hn h lp f ha ln gvn by m Δy/Δx, bu n h ca h y-valu ar h pn, x, and h x-valu ar m,. Th can b cnfung nc h x n h dfnn f lp dffrn han h x ud n knmac quan. In h dfnn f lp, x man h hrznal ax. In h dcun f mn, x man h pn f h bjc. Whn w graph h pn f an bjc vru m, w alway pu h pn n h vrcal, y-ax, and m,, n h x-ax. Th can lad cnfun nc h y- crdna n h pn vru m graph gv h pn, whch x n knmac quan. Exampl 14 Drmn h lcan and vlcy f h bjc n h blw graph a whn = 2.5. P n ( m r ) Pn v. Tm y = x R² = Tm (cnd) 9 Pag

10 Nw pbl ha h vlcy f an bjc wll chang durng h m bng brvd. Fr nanc, f yu wr walk away frm yur hu a a vlcy f 1m/ fr 6 cnd, p fr 3 cnd and hn run back yur hu n 3 cnd, h Pn vru Tm graph fr yur rp wuld lk lk h. Yu can rad h graph drmn yur pn a any m durng h rp. Yu can al frm h ha durng yur rp yu had hr dffrn vlc. Wha yur nal vlcy vr h fr 6 cnd? Wha yur vlcy bwn 6 & 9 cnd? Wha yur vlcy frm 9 12 cnd? Pn v. Tm P n ( m r ) Tm (cnd) Cndr h w pn-vru-m graph blw fr bjc A and B. a) Hw d h mn f h bjc A n graph 1 cmpar ha f A n graph 2? b) Hw d h mn f bjc B n graph 1 cmpar h mn f bjc B n graph 2? c) Whch bjc ha h mallr pd n graph 2? Explan. d) Dcrb wha happnng a h pn whr h funcn f bjc A nrc ha f bjc B. ) Whch bjc ravld a grar danc durng h fr 6 cnd n graph 1? Explan. 10 Pag

11 A udn ar walkng a 5 f/ n a crrdr A and 20 f away frm h nrcn f crrdr A and B. A cnd udn ar a h am m runnng a 8 f/ n crrdr B and 32 f away frm h nrcn. Vlcy vru Tm graph Th fgur n h rgh hw a vlcy-vru-clck radng graph ha rprn h mn f a bcycl mvng alng a ragh bk pah. Th pv drcn f h pn ax ward h a. 11 Pag

12 a) Dcrb h mn f h bk n wrd. b) Hw h graph dffrn frm h pn vru m graph fr h am mn? c) Thnk f hw yu can u h graph ma h bk dplacmn frm a clck radng f 10 a clck radng f 15. Explan. (Hn: Thnk f h ara f a rcangl.) d) U h graph ma h bk dplacmn frm a clck radng f ) Frmula a gnral rul fr ung a vlcy vru clck radng graph drmn an bjc dplacmn durng m m nrval f h bjc mvng a cnan vlcy. 12 Pag

13 Any mn ha can b rcrdd ung a Pn vru Tm graph can al b rcrdd ung a Vlcy vru Tm graph. Yur chc f graph wll hav dffrn bnf, bu mpran ha yu b abl hw hy rla n anhr. L ak h fr Pn vru Tm graph ha w dd abv and rca a a Vlcy vru Tm graph. In h ca, h vrcal ax rcrd vlcy; h hrznal ax cnnu ndca h m. In h fr graph, yu manand a cnan vlcy f 1 m/ fr 6 cnd h bcm: V l c y ( m r / c n d ) Vlcy v. Tm y = 1 R² = #N/A Tm (cnd) In h ca, h bjc vlcy can b rad drcly ff h graph, bu dplacmn and h danc ha ravld cann. Hwvr, w can drmn h bjc dplacmn, hw far frm whr ard, and h danc ha ravld by maurng h ara undr h curv (n h ca h hrznal ln a v = 1 m/). (If h vlcy alway pv, h danc ravld and dplacmn wll b h am.) A rcangl ha w par f ppng d. In h ca, n d wll b h hrznal ln ndcang h vlcy ha h bjc ravlng and h d ppng ha h hrznal ax. Th cnd par f d a vrcal ln drawn ragh up frm h m w ar maurng and h d ppng ha a vrcal ln drawn ragh up frm h m w p maurng. S fr an bjc mvng a cnan vlcy w can dfn a rcangular hap wh hgh vlcy and wh lngh h m nrval ha w r udyng. Th ara f a rcangl gvn by hgh mulpld by lngh, h ara f h rcangl vlcy, v, mulpld by h lapd m,. Ara = (hgh)(lngh) A = (vlcy)(m) A = v Exampl 15 Drmn h dplacmn f h fllwng bjc, and h danc ha ravld, durng fr 3 cnd f ravl. 13 Pag

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