JEE-Physics. A body in translation motion can move on either a straight line path or curvilinear path.

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1 J-hsics ricle Kinemics Kinemics In kinemics we sd hw bd mes wih knwing wh i mes. ll pricles f rigid bd in rnslin min me in idenicl fshin hence n f he pricles f rigid bd in rnslin min cn be sed represen rnslin min f he bd. This is wh, while nlzing is rnslin min, rigid bd is cnsidered pricle nd kinemics f rnslin min s pricle kinemics. ricle kinemics dels wih nre f min i.e. hw fs nd n wh ph n bjec mes nd reles he psiin, elci, ccelerin, nd ime wih n reference mss, frce nd energ. In her wrds, i is sd f gemer f min. Tpe s f Trnsl in M in bd in rnslin min cn me n eiher srigh line ph r criliner ph. Recili ner Mi n Trnslin min n srigh line ph is knwn s reciliner rnslin. I is ls knwn s ne dimensinl min. cr rnning n srigh rd, rin rnning n srigh rck nd bll hrwn ericll pwrds r drpped frm heigh ec re er cmmn emples f reciliner rnslin. Crili ner Mi n Trnslin min f bd n criliner ph is knwn s criliner rnslin. If he rjecr is in plne, he min is knwn s w dimensinl min. bll hrwn sme ngle wih he hriznl describes criliner rjecr in ericl plne; sne ied sring when whirled describes circlr ph nd n insec crwling n he flr r n wll re emples f w dimensinl min. If ph is n in plne nd reqires regin f spce r lme, he min is knwn s hree dimensinl min r min is spce. n insec fling rndml in rm, min f fbll in sccer gme er cnsiderble drin f ime ec re cmmn emples f hree dimensinl min. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Reference Frme Min f bd cn nl be bsered if i chnges is psiin wih respec sme her bd. Therefre, fr min be bsered here ms be bd, which is chnging is psiin wih respec her bd nd persn wh is bsering min. The persn bsering min is knwn s bserer. The bserer fr he prpse f inesigin ms he is wn clck mesre ime nd pin in he spce ched wih he her bd s rigin nd se f crdine es. These w hings he ime mesred b he clck nd he crdine ssem re clleciel knwn s reference frme. In his w, min f he ming bd is epressed in erms f is psiin crdines chnging wih ime. 1

2 J-hsics si in Vecr, Velci nd cceler in Vecr Fr nlzing rnslin min, we ssme he ming bd s pricle nd represen i s mhemicl pin. Cnsider pricle ming n criliner ph. si i n Vecr I describes psiin f pricle relie her pricle nd is ecr frm he ler wrds he firs. T sd min f pricle we he ssme reference frme fied wih sme her bd. The ecr drwn frm he rigin f he crdine ssem represening he reference frme he lcin f he pricle is knwn s psiin ecr f he pricle. Cnsider pricle ming in spce rces ph shwn in he figre. Is psiin cninsl chnges wih ime nd s des he psiin ecr. n insn f ime, is psiin ecr r is shwn in he fllwing figre. B r f r i r s z D is pl cem en Vecr & D is nce Tr eled Di splcemen nd disnce reled Displcemen is mesre f chnge in plce i.e. psiin f pricle. I is defined b ecr frm he iniil psiin he finl psiin. Le he pricle mes frm pin B n he criliner ph. The ecr B r is displcemen. Disnce reled is lengh f he ph rersed. We cn s i ph lengh. Here in he figre lengh f he cre s frm B is he disnce reled. Disnce reled beween w plces is greer hn he mgnide f displcemen ecr whereer pricle chnges is direcin dring is min. In nidirecinl min, bh f hem re eql. erge Velci nd erge Speed erge elci f pricle in ime inerl is h cnsn elci wih which pricle wld he cered he sme displcemen in he sme ime inerl s i cers in is cl min. I is defined s he ri f displcemen he cncerned ime inerl. If he pricle mes frm pin pin B in ime inerl i f, he erge elci in his ime inerl is gien b he fllwing eqin. r rf ri f i Similr erge elci, erge speed in ime inerl is h cnsn speed wih which pricle wld rel he sme disnce n he sme ph in he sme ime inerl s i rels in is cl min. I is defined s he ri f disnce reled he cncerned ime inerl. If in ming frm pin B, he pricle rels ph lengh i.e. disnce s in ime inerl i f, is erge speed c is gien b he fllwing eqin. c s h Lengh f i z r f B( f ) r i r s ( i ) nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

3 J-hsics erge speed in ime inerl is greer hn he mgnide f erge elci ecr whereer pricle chnges is direcin dring is min. In nidirecinl min, bh f hem re eql. Ins n nes Velci nd speed If we ssme he ime inerl be infiniesimll smll i.e. 0, he pin B pprches mking he chrd B cincide wih he ngen. Nw we cn epress he insnnes elci b he fllwing eqins. r dr lim 0 d The insnnes elci eqls he re f chnge in is psiin ecr r wih ime. Is direcin is lng he ngen he ph. Insnnes speed is defined s he ime re f disnce reled. s ds c lim 0 d Y cn esil cnceie h when 0, n nl he chrd B b ls he rc B bh pprch cincide wih ech her nd wih he ngen. Therefre ds dr. Nw we cn s h speed eqls mgnide f insnnes elci. Insnnes speed ells s hw fs pricle mes n insn nd insnnes elci ells s in wh direcin nd wih wh speed pricle mes n insn f ime. ccelerin Insnnes ccelerin is mesre f hw fs elci f bd chnges i.e. hw fs direcin f min nd speed chnge wih ime. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 n insn, i eqls he re f chnge in elci ecr wih ime. d d ecr qni chnges, when is mgnide r direcin r bh chnge. ccrdingl, ccelerin ecr m he w cmpnens, ne respnsible chnge nl speed nd he her respnsible chnge nl direcin f min. z B 3 n T

4 J-hsics Cmpnen f ccelerin respnsible chnge speed ms be in he direcin f min. I is knwn s ngenil cmpnen f ccelerin T. The cmpnen respnsible chnge direcin f min ms be perpendiclr he direcin f min. I is knwn s nrml cmpnen f ccelerin n. ccelerin ecr f pricle ming n criliner ph nd is ngenil nd Nrml cmpnens re shwn in he figre. Cr i li ner Trnsl in in Cr e si n cr dine ssem: Sperpsiin f hree reciliner Mins Cnsider pricle ming n hree dimensinl criliner ph B. n insn f ime i is pin (,, z) ming wih elci nd ccelerin. Is psiin ecr is defined b eqins r i ˆ j ˆ zkˆ Differeniing i wih respec ime, we ge elci ecr. dr d d dz ˆ i ˆ j k ˆ ˆ ˆ ˆ i j zk d d d d Here d d, d d nd z dz d re he cmpnens f elci ecrs in he, nd z direcins respeciel. Nw he ccelerin cn be bined b differeniing elci ecr wih respec ime. d d d i j k i j k d d d d d ˆ ˆ z ˆ ˆ ˆ ˆ z ccelerin ecr cn ls be bined b differeniing psiin ecr wice wih respec ime. i j k i j k d d d d d r d ˆ d ˆ d ˆ ˆ ˆ ˆ z In he be w eqins, d d d d, d d d d cmpnens f ccelerin ecrs in he, nd z direcins respeciel. nd d z d d d re he In he be eqins, we cn nlze ech f he cmpnens, nd z f min s hree indiidl reciliner mins ech lng ne f he es, nd z. z z lng he is d d nd d d lng he is d d nd d d m ple lng he z is z dz d nd 4 z d z d criliner min cn be nlzed s sperpsiin f hree simlnes reciliner mins ech lng ne f he crdine es. r i j kˆ, where r is in siin ecr r 1.5 f pricle ries wih ime ccrding he lw 1 ˆ 4 ˆ meers nd is in secnds. () 3 Find sible epressin fr is elci nd ccelerin s fncin f ime. (b) Find mgnide f is displcemen nd disnce reled in he ime inerl = 0 4 s. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

5 J-hsics Sl in () Velci is defined s he firs deriie f psiin ecr wih respec ime. dr i ˆ ˆj kˆ m/s d ccelerin is defined s he firs deriie f elci ecr wih respec ime. d ˆ 1 i ˆj m/s d (b) Displcemen r is defined s he chnge in plce f psiin ecr. r 8ˆi ˆj 8kˆ m 3 3 Mgnide f displcemen 3 r m 3 Disnce s is defined s he ph lengh nd cn be clcled b inegring speed er he cncerned ime inerl s d 4 4d d 16 m Reciliner Min Criliner min cn be cnceied s sperpsiin f hree reciliner mins ech lng ne f he Cresin es. Therefre, we firs sd reciliner min in deil. We cn clssif reciliner min prblems in fllwing cegries ccrding gien infrmin. Reciliner Min Unifrm Velci Min Unifrm ccelerin Min ccelered Min Vrible ccelerin Min I. ccelerin s fncin f ime. II. ccelerin s fncin f psiin. III. ccelerin s fncin f elci. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Unifr m Velci M i n In nifrm elci min, bd mes wih cnsn speed n srigh line ph wih chnge in direcin. If bd sring frm psiin = he insn = 0, mes wih nifrm elci in he psiie direcin, is eqin f min n ime is = + Velci ime ( ) grph fr his min is shwn in he fllwing figre. 5

6 J-hsics s we knw h, he re beween grph nd he ime es eqls chnge in psiin i.e displcemen, he psiin ime relinship r psiin n insn cn be bined. siin Velci re = = - Time siin-ime grph Time Unifrm cceleri n Mi n Min in which ccelerin remins cnsn in mgnide s well s direcin is clled nifrm ccelerin min. In he min digrm, is shwn pricle ming in psiie direcin wih nifrm ccelerin. I psses he psiin, ming wih elci he insn = 0 nd cqires elci ler insn. =0 d d d d 0...(i) Nw frm he be eqin, we he 0 d d d d 1...(ii) 1 limining ime, frm he be w eqins, we he...(iii) qins (i), (ii) nd (iii) re knwn s he firs, secnd nd hird eqins f min fr nifrml ccelered bdies. ccelerin ime ( ) grph fr his min is shwn in he fllwing figre. ccelerin ccelerin-ime grph Time s we knw h, he re beween grph nd he ime es eqls chnge in elci, elci ime relin r elci n insn cn be bined. ccelerin re = = Time 6 Velci Velci-ime grph The re beween grph nd he ime es eqls chnge in psiin. Therefre, psiin ime relin r psiin n insn cn be bined. Time nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

7 J-hsics Velci siin re= re = Slpe f his ngen eqls he iniil elci Time siin-ime grph Time m ple pricle ming wih nifrm ccelerin psses he pin = m wih elci 0 m/s he insn = 0. Sme ime ler i is bsered he pin = 3 m ming wih elci 10 m/s. () Sl in Wh is is ccelerin? (b) Find is psiin nd elci he insn = 8 s. (c) Wh is he disnce reled dring he inerl = 0 8 s? In he djining figre he gien nd reqired infrmin shwn re n scle. s min digrm is schemic represenin nl. () Using he hird eqin f nifrm ccelerin min, we he 0 m/s 10 m/s 10 0 ( ) 5 m/s ( ) (3 ) = = 3 (b) Using secnd eqin f nifrm ccelerin min, we he m Using he firs eqin f nifrm ccelerin min, we he m/s 8 (c) Where he pricle rerns, is elci ms be zer. Using he hird eqin f nifrm ccelerin min, we he ( 5) m 0 m/s 5 m/s This lcin is shwn in he djining mdified min digrm. The disnce-reled s is s 80 m = = 4 nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 m ple bll is drpped frm he p f bilding. The bll kes 0.50 s fll ps he 3 m lengh f windw, which is sme disnce belw he p f he bilding. () (b) Sl in Hw fs ws he bll ging s i pssed he p f he windw? Hw fr is he p f he windw frm he pin which he bll ws drpped? ssme ccelerin g in free fll de gri be 10 m/s dwnwrds. The bll is drpped, s i sr flling frm he p f he bilding wih zer iniil elci ( = 0). The min digrm is shwn wih he gien infrmin in he djining figre. Using he firs eqin f he cnsn ccelerin min, we he 7

8 J-hsics (i) ' 0 10( 0.5) (ii) Using les f nd in fllwing eqin, we he windw heigh () Frm eqin (i), we he m/s (p) Frm fllwing eqin, we he ' s h 3 m ' h 61.5 cm Vri ble cceler i n M i n Mre fen, prblems in reciliner min inle ccelerin h is n cnsn. In hese cses ccelerin is epressed s fncin f ne r mre f he ribles, nd. Le s cnsider hree cmmn cses. ccelerin gien s fncin f ime If ccelerin is gien fncin f ime s = f(), frm eqin = d/d we he d f()d d f()d The be eqin epresses s fncin f ime, s = g(). Nw sbsiing g() fr in eqin = d/d, we he d g d d g()d The be eqin ield psiin s fncin f ime. m ple Sl in The ccelerin f pricle ming lng he -direcin is gien b eqin = (3 ) m/s. he insns = 0 nd = 6 s, i ccpies he sme psiin. () Find he iniil elci. (b) Wh will be he elci = s? B sbsiing he gien eqin in eqin d d, we he...(i) 0 d 3 d d 3 d 3 B sbsiing eq. (i) in eqin = d/d, we he (ii) d 3 d d 3 d 0 () ppling he gien cndiin h he pricle ccpies he sme crdine he insns = 0 nd = 6 s in eq. (ii), we he m/s (b) Using in eq. (i), we he m/s 8 nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

9 ccelerin s fnci n f psii n J-hsics If ccelerin is gien fncin f psiin s = f(), we he se eqin = d/d. Rerrnging erm in his eqin we he d = d. Nw sbsiing f() fr, we he d f d d f()d The be eqin prides s wih elci s fncin f psiin. Le relin bined in his w is = g(). Nw sbsiing g() fr in eqin = d/d, we he d d d g() d g() The be eqin ields he desired relin beween nd. m ple Sl in ccelerin f pricle ming lng he -is is defined b he lw 4, where is in m/s nd is in meers. he insn = 0, he pricle psses he rigin wih elci f m/s ming in he psiie - direcin. () Find is elci s fncin f is psiin crdines. (b) Find is psiin s fncin f ime. (c) Find he mimm disnce i cn g w frm he rigin. () B sbsiing gien epressin in he eqin = d/d nd rerrnging, we he d 4d d 4 d Since he pricle psses he rigin wih psiie elci f m/s, s he mins sign in he eq. (i) hs been drpped. (b) B sbsiing be bined epressin f elci in he eqin = d/d nd rerrnging, we he d d 1 d d sin sin (c) The mimm disnce i cn g w frm he rigin is 1m becse mimm mgnide f sine fncin is ni. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 ccelerin s fncin f elci If ccelerin is gien s fncin f elci s =f(), b sing eqin = d/d we cn bin elci s fncin f ime. d d d f() d f() Nw sing eqin = d/d we cn bin psiin s fncin f ime In nher w if we se eqin = d/d, we bin elci s fncin f psiin. d d d f() d f() Nw sing eqin = d/d we cn bin psiin s fncin f ime 9

10 J-hsics m ple ccelerin f pricle ming lng he -is ries ccrding he lw =, where is in m/s nd is in m/s. he insn = 0, he pricle psses he rigin wih elci f m/s ming in he psiie - direcin. () Find is elci s fncin f ime. (b) Find is psiin s fncin f ime. (c) (d) (e) Sl in Find is elci s fncin f is psiin crdines. Find he mimm disnce i cn g w frm he rigin. Will i rech he be-menined mimm disnce? () B sbsiing he gien relin in eqin d d, we he d d...(i) 0 d d e (b) B sbsiing he be eqin in = d/d, we he...(ii) d e d d e d 1 e 0 0 (c) Sbsiing gien epressin in he eqin d d nd rerrnging, we he...(iii) d d d d 1 0 (d) (e) q. (iii) sggess h i will sp = 1 m. Therefre, he mimm disnce w frm he rigin i cn g is 1 m. q. (ii) sggess h cer 1 m i will ke ime whse le ends infini. Therefre, i cn neer cer his disnce. rjecile Min n bjec prjeced b n eernl frce when cnines me b is wn ineri is knwn s prjecile nd is min s prjecile min. fbll kicked b pler, n rrw sh b n rcher, wer sprinkling wer fnin, n hlee in lng jmp r high jmp, blle r n riller shell fired frm gn re sme emples f prjecile min. In simples cse when prjecile des rech gre heighs be he grnd s well s des n cer er lrge disnce n he grnd, ccelerin de gri cn be ssmed nifrm hrgh is min. Mreer, sch prjecile des n spend mch ime in ir n permiing he wind nd ir resisnce gher pprecible effecs. Therefre, while nlzing hem, we cn ssme gri be nifrm nd neglec effecs f wind s well s ir resisnce. Under hese circmsnces when n bjec is hrwn in direcin her hn he ericl, is rjecr ssmes shpe f prbl. In he figre, bll hrwn fllw prblic rjecr is shwn s n emple f prjecile min. 10 The bll is hrwn rblic Trjecr The bll lnds n he grnd hrwn nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

11 J-hsics presen, we sd prjeciles ming n prblic rjecries nd b he erm prjecile min; we sll refer his kind f min. Fr prjecile me n prblic rjecr, he fllwing cndiins ms be flfilled. ccelerin ecr ms be nifrm. Velci ecr neer cincides wih line f ccelerin ecr. nlzing rjeci le mi n Since prbl is plne cre, prjecile min n prblic rjecr becmes n emple f w dimensinl min. I cn be cnceied s sperpsiin f w simlnes reciliner mins in w mll perpendiclr direcins, which cn be nlzed seprel s w Cresin cmpnens f he prjecile min. rjecile Min ner he Hriznl r Fl Grnd sing Cresin cmpnens Cnsider min f bll hrwn frm grnd s shwn in he figre. The pin frm where i is prjeced is knwn s pin f prjecin, he pin where i flls n he grnd is knwn s pin f lnding r rge. The disnce beween hese w pins is knwn s hriznl rnge r rnge, he heigh frm he grnd f he highes pin i reches dring fligh is knwn s mimm heigh nd he drin fr which i remin in he ir is knwn s ir ime r ime f fligh. The elci wih which i is hrwn is knwn s elci f prjecin nd ngle which elci f prjecin mkes wih he hriznl is knwn s ngle f prjecin. Velci f rjecin Time f rjecin = 0 in f rjecin ngle f rjecin Mimm Heigh (H) Rnge (R) Time f Fligh = T crefl bserin f his min reels h when bll is hrwn is ericl cmpnen f elci decreses in is pwrd min, nishes he highes pin nd herefer increses in is dwnwrd min de gri similr min f bll hrwn ericll pwrds. he sme ime, he bll cnines me nifrml in hriznl direcin de ineri. The cl prjecile min n is prblic rjecr is sperpsiin f hese w simlnes reciliner mins. In he fllwing figre, he be ides re shwn represening he ericl b -is nd he hriznl b -is. Trge - C m pnen f M in = 0 = 0 = 0 - C m pnen f M in = T = R = H = H = H B nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 = 0 = T = 0 11 C = T = R rjecile m in res led in is w C resin cm pnen s. rjecile m in s s perpsiin f w recilin er m ins n e in ericl nd her in hrizn l d irecin.

12 J-hsics Ver icl r -cmpnen f m in. Cmpnen f iniil elci in he ericl direcin is. Since frces her hn griinl pll f he erh re negligible, ericl cmpnen f ccelerin f he bll is g ericll dwnwrds. This cmpnen f min is described b he fllwing hree eqins. Here denes -cmpnen f elci, denes psiin crdine n insn. g...(i) g 1...(ii) g...(iii) Hriznl r - cmpnen f min. Since effecs f wind nd ir resisnce re ssmed negligible s cmpred effec f gri, he hriznl cmpnen f ccelerin f he bll becmes zer nd he bll mes wih nifrm hriznl cmpnen f elci. This cmpnen f min is described b he fllwing eqin....(i) qi n f rjecr qin f he rjecr is relin beween he nd he crdines f he bll wih inlemen f ime. T elimine, we sbsie is epressin frm eqin (i) in eqin (ii). g n...() cs er prjecile min cn be nlzed sing he be fie eqins. In specil cse f ineres, if he prjecile lnds he grnd gin, is ime f fligh, he mimm heigh reched nd hriznl rnge re bined sing he be eqins. 1 Time f Fligh he highes pin f rjecr when T, he ericl cmpnen f Mi mm Heigh elci becmes zer. he insn =T, he bll srikes he grnd wih ericl cmpnen f elci. B sbsiing eiher f hese cndiins in eqin (i), we bin he ime f fligh. T g he highes pin f rjecr where = H, he ericl cmpnen f elci becmes zer. B sbsiing his infrmin in eqin (iii), we bin he mimm heigh. Hri znl Rnge Mi mm Rnge Trjecr qin lerne frm 1 H g The hriznl rnge r simpl he rnge f he prjecile min f he bll is disnce reled n he grnd in is whle ime f fligh. sin R T g I is he mimm disnce reled b prjecile in he hriznl direcin fr cerin elci f prjecin. The be epressin f rnge mkes bis h bin mimm rnge he bll ms be prjeced ngle 45. Sbsiing his cndiin in he epressin f rnge, we bin he mimm rnge R m. R m g If rnge is knwn in dnce, he eqin f rjecr cn be wrien in n lernie frm inling hriznl rnge. n 1 R g nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

13 J-hsics m ple bll is hrwn wih 5 m/s n ngle 53 be he hriznl. Find is ime f fligh, mimm heigh nd rnge. Sl in In he djining figre elci f prjecin = 5 m/s, ngle f prjecin = 53, he hriznl nd ericl cmpnens nd f elci f prjecin re shwn. Frm hese infrmin we he cs m/s nd sin 53 0 m/s Using eqins fr ime f fligh T, mimm heigh H nd rnge R, we he T 0 g 10 4 s 0 H 0 m g 10 R 15 0 g m m ple bll 4 s fer he insn i ws hrwn frm he grnd psses hrgh pin, nd srikes he grnd fer 5 s frm he insn i psses hrgh he pin. ssming ccelerin de gri be 9.8 m/s find heigh f he pin be he grnd. Sl in The bll prjeced wih elci ˆ ˆ i j frm reches he pin wih elci ˆ ˆ i j nd his he grnd pin Q he insn T = = 9 s s shwn in he djining min digrm. 5 m = 4 s = 0 = 9 Q nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Frm eqin f ime f fligh, we he is iniil -cmpnen f elci Sbsiing be in eq. (ii) nd rerrnging erms, we he he heigh f he pin. rjecile n inclined plne g g T m 13 1 T gt g riller pplicin fen finds rge eiher p hill r dwn hill. These siins cn pprimel be mdeled s prjecile min p r dwn n inclined plne. rjecile p n inclined plne rjecile dwn n inclined plne

14 J-hsics In he be lef figre is shwn shell prjeced frm pin wih elci n ngle hi rge pin phill. This prjecile min is clled prjecile p hill r inclined plne. Similrl in he be righ figre is shwn prjecile dwn hill r inclined plne. nlzing prjecile mi n p n incli ned plne sing Cresi n cmpnens Cnsider he prjecile min p n inclined plne described erlier. ssme Cresin crdine ssem whse -is cincides wih he line f fire nd he rigin wih he pin f prjecin s shwn. The line is lng he line f he grees slpe. Velci f prjecin mkes ngle wih he psiie -is, herefre is nd -cmpnens nd re cs( ) sin( ) ccelerin de gri g being ericl mkes he ngle wih he negie -is, herefre nd -cmpnens f ccelerin ecr re g sin g cs g = T Hriznl rjecile m in p n in clin ed pl ne resled in is w C res in cm pnen s. Min cmpnen lng he - is The prjecile srs wih iniil -cmpnen f elci in he psiie -direcin nd hs nifrm -cmpnen f ccelerin = g cs in he negie -direcin. This cmpnen f min is described b he fllwing hree eqins. Here denes -cmpnen f elci, denes psiin crdine n insn....(i)...(ii) 1...(iii) Min cmpnen lng he - is The -cmpnen f min is ls nifrml ccelered min. The prjecile srs wih iniil -cmpnen f elci in he psiie -direcin nd hs nifrm -cmpnen f ccelerin in he negie -direcin. This cmpnen f min is described b he fllwing hree eqins. Here denes -cmpnen f elci, denes psiin crdine n insn. =...(i) = ½...() =...(i) er prjecile min p n incline cn be nlzed sing he be si eqins. Qniies f ineres in riller pplicins nd hence in prjecile n incline plne re ime f fligh, rnge n he incline plne nd he ngle which he shell his he rge. Time f fligh. Ming in ir fr ime inerl T he prjecile when his he rge, is -cmpnen f elci becmes in he negie -direcin. Using his infrmin in eqin (i), we bin he ime f fligh. sin( ) T g cs 14 When he prjecile his he rge, is cmpnen f displcemen ls becmes zer. This infrmin wih eqin (ii) ls ield he ime f fligh. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

15 Rnge n he plne. J-hsics The rnge f prjecile n n incline plne is he disnce beween he pin f prjecin nd he rge. I eqls displcemen in he -direcin dring whle fligh. B sbsiing ime f fligh in eqin (), we bin epressin fr he rnge R. sin( ) cs R g cs nlsis f prjecile n n incline plne sing qin f rjecr Smeimes he hill m be w frm he pin f prjecin r he hill m n he nifrm slpe s shwn in he fllwing w figres. Trjecr Trjecr Hill = m + c ( p, p ) Hill = f() ( p, p ) In hese cses, he shpe f he hill cn be epressed b sible eqin f he frm = m + c fr nifrm slpe hill r = f() fr nnnifrm slpe hill. The rge where he prjecile his he hill is he inersecin f rjecr f he prjecile nd he hill. Therefre, crdines ( p, p ) f he rge cn be bined b simlnesl sling eqin f he hill nd eqin f rjecr f he prjecile. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Time f fligh Since prjecile me wih nifrm hriznl cmpnen f he elci ( ), is ime f fligh T cn be clcled frm he fllwing eqin. T p p cs m ple pricle is prjeced wih elci f 30 m/s n ngle 60 be he hriznl n slpe f inclinin 30. Find is rnge, ime f fligh nd ngle f hi. S l. The crdine ssem, prjecin elci nd is cmpnen, nd ccelerin de gri nd is cmpnen re shwn in he djining figre. Sbsiing crrespnding les in fllwing eqin, we ge he ime f fligh T T 3 s Sbsiing le f ime f fligh in fllwing eqin, we ge he rnge R. R T T R ( 3 ) 60m 1 1 In he djining figre, cmpnens f elci when he prjecile his he slpe pin re shwn. The ngle which elci ecr mkes wih he -is is knwn s ngle f hi. The prjecile his he slpe wih sch elci, whse -cmpnen is eql in mgnide h f elci f prjecin. The -cmpnen f elci is clcled b sbsiing le f ime f fligh in fllwing eqin n = 15 = 5 60º = 30 30º = 10 30º = 53 = 5 3 = 15? 3 = 15 V Hriznl

16 J-hsics Relie Min Min f bd cn nl be bsered, when i chnges is psiin wih respec sme her bd. In his sense, min is relie cncep. T nlze min f bd s, herefre we he fi r reference frme sme her bd s B. The resl bined is min f bd relie bd B. Rel i e psi in, Rel ie Velci nd Rel ie cceler i n Le w bdies represened b pricles nd B psiins defined b psiin ecrs r nd r, ming wih elciies nd nd B B ccelerins nd wih respec reference frme S. Fr nlzing B min f erresril bdies he reference frme S is fied wih he grnd. The ecrs rb / denes psiin ecr f B relie. Fllwing ringle lw f ecr ddiin, we he r B r rb /...(i) Firs deriies f r nd r wih respec ime eqls elci f pricle nd elci f pricle B B relie frme S nd firs deriie f rb / wih respec ime defines elci f B relie....(ii) B B / Secnd deriies f r nd r B pricle B relie frme S nd secnd deriie f rb / relie. B B / wih respec ime eqls ccelerin f pricle nd ccelerin f wih respec ime defines ccelerin f B...(iii) In similr fshin min f pricle relie pricle B cn be nlzed wih he help f djining figre. Y cn bsere in he figre h psiin ecr f relie B is direced frm B nd herefre r r. B / / B, B / / B nd B / / B The be eqins elcide h hw bd ppers ming nher bd B is ppsie hw bd B ppers ming bd. m ple mn when sndsill bseres he rin flling ericll nd when he wlks 4 km/h he hs hld his mbrell n ngle f 53 frm he ericl. Find elci f he rindrps. Sl in ssigning sl smbls, m nd r r / m elci f mn, elci f rin nd elci f rin relie mn, we cn epress heir relinship b he fllwing eq. r m r / m The be eqin sggess h sndsill mn bseres elci r f rin relie he grnd nd while he is ming wih elci m, he bseres elci f rin relie himself r / m. I is cmmn iniie fc h mbrell ms be held gins r / m 16 fr pimm precin frm rin. ccrding hese fcs, direcins f he elci ecrs re shwn in he djining figre. r m m r The ddiin f elci ecrs is represened ccrding he be eqin is ls represened. Frm he figre we he n 37 3 km/h r m r / m r / m 53 Vericl nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

17 m ple J-hsics b cn be rwed 5 m/s n sill wer. I is sed crss 00 m wide rier frm sh bnk he nrh bnk. The rier crren hs nifrm elci f 3 m/s de es. () (b) (c) Sl in () In which direcin ms i be seered crss he rier perpendiclr crren? Hw lng will i ke crss he rier in direcin perpendiclr he rier flw? In which direcin ms he b be seered crss he rier in minimm ime? Hw fr will i drif? Velci f b n sill wer is is cpci me n wer srfce nd eqls is elci relie wer. b / w = Velci f b relie wer = Velci f b n sill wer n flwing wer, he wer crries he b lng wih i. Ths elci b f he b relie he grnd eqls ecr sm f b / w nd w. The b crsses he rier wih he elci b. b b / w w (b) T crss he rier perpendiclr crren he b ms be seered in direcin s h ne f he cmpnens f is elci ( b / w ) relie wer becmes eql nd ppsie wer flw elci w nerlize is effec. I is pssible nl when elci f b relie wer is grer hn wer flw elci. In he djining figre i is shwn h he b srs frm he pin nd mes lng he line (-is) de nrh relie grnd wih elci b. T chiee his i is seered n ngle wih he -is. b / w w b / w b / w b s sin 5 sin 3 37 b / w w Nrh (c) The b will cer rier widh b wih elci w b b / w b / w sin 37 4 m/s in ime, which is gien b b b / w b nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 b / 50s b (d) T crss he rier in minimm ime, he cmpnen perpendiclr crren f is elci relie grnd ms be kep mimm le. I is chieed b seering he b lws perpendiclr crren s shwn in he djining figre. The b srs frm he sh bnk nd reches pin n he nrh bnk. Time ken b he b is gien b b / b / w 40s Drif is he displcemen lng he rier crren mesred frm he sring pin. Ths, i is gien b he fllwing eqin. We dene i b d. d b Sbsiing b w 3 m/s, frm he figre, we he d = 10 m Dependn Mi n r Cnsri n Mi n ffec f min f ne bd n nher, when he re inercnneced hrgh sme sr f phsicl link f definie prper is wh we sd in dependn min. The definie prper f he cnnecing link is cnsrin h decides hw min f ne bd depends n h f he her. Therefre, dependn min is ls knwn s cnsrin min. 17 s

18 J-hsics In ris phsicl siins, we fen encner inercnneced bdies ffecing min f ech her. The rie f cnnecing link m be sring, rd r direc cnc. sring hs definie lengh nd cn nl pll bd, i cnn psh; rd ls hs definie lengh nd cn pll r psh bd, bdies in direc smh cnc cn nl psh ech her. These prblems re nlzed b he fllwing mehds. Mehd f cnsri n eqi n In his mehd, prper f cnnecing link is epressed in erms f psiin crdines f he bdies. This eqin is knwn s cnsrin eqin. Differeniing he cnsrin eqin nce wih respec ime we ge relinship beween heir elciies nd gin differeniing he elci relin wih respec ime we ge relinship beween heir ccelerins. Me h d f Vir l Wrk In his mehd, we se cnceps f frce nd wrk. Wrk is defined s sclr prdc f frce nd displcemen f he pin f pplicin f frce. If w bdies re cnneced b ineensible links r links f cnsn lengh, he sm f sclr prdcs f frces pplied b cnnecing links nd displcemen f cnc pins he ends f he cnnecing links eqls zer in eer infiniesimll smll ime inerl. Le he frces pplied b he cnnecing links n cnneced bdies re F, 1 F,... F... i n f crrespnding cnc pins in n infiniesimll smll ime inerl d re dr, dr,... 1 The principle sgges h n i i1 F dr 0 i F nd displcemens dr,... dr. The relin beween speeds f he cnc pin cn direcl be bined b diiding he eqin wih ime inerl d. i n n i i i1 F 0 When ngle beween frce ecrs nd elci ecrs d n r wih ime, we cn differenie he be eqin bin relinship beween ccelerins. Hweer, cre ms be ken in deciding ccelerin relin, when ngle beween frce ecrs nd elci ecrs r wih ime. In hese circmsnces, we m ge n ddiinl erm inling he deriie f he ngle beween he frce nd he elci. Therefre, presen we resric rseles se his mehd when ngle beween frce nd elci ecrs remin cnsn. In hese siins, we he n i i1 F 0 i m ple Sl in In he ssem shwn, he blck is ming dwn wih elci 1 nd C is ming p wih elci 3. press elci f he blck B in erms f elciies f he blcks nd C. Mehd f Cnsri ned qi ns. The mehd reqires ssigning psiin crdine ech f he ming bdies nd mking cnsrin eqin fr ech sring. In he gien ssem, here re fr seprel ming bdies nd w srings. The ming bdies re he hree blcks nd ne plle. We ssign psiin crdines 1,, 3 nd p ll mesred frm he fied reference ceiling s shwn in he figre. The reqired cnsrin eqin fr sring cnnecing blck nd plle is 1 p 1...(i) nd he reqired cnsrin eqin fr he her sring is 3 p...(i) B C 3 p B C 3 3 nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

19 J-hsics Le he blck B is ming dwn wih elci. The elciies re defined s 1 1,, nd 3 3 Differeniing erms f eq. (i) nd (ii), elimining nd sbsiing be les f elciies, we he 3 1 p Meh d f Vir l Wrk. The ensin frces pplied b he srings n ech cnc pin nd displcemens f he blcks re shwn in he djcen figre. Le he ensin in he sring cnneced blck B is T. The ensin in he sring cnnecing he blck nd he plle ms be T in rder jsif Newn s Secnd Lw fr mssless plle. F i i 0 T1 T T Vi sl Inspecin wih Spperpsiin. Min f bd in n inercnneced ssem eqls sm f indiidl effecs f ll her bdies. S elci f blck B eqls ddiin f indiidl effecs f min f nd C. The indiidl effec f min f is elci f B de min f nl nd cn esil be prediced b isl inspecin f he ssem. Le his indiidl effec be dened b B....(i) B 1 Indiidl effec f min f C n min f B is BC 3...(ii) ccrding he principle f sperpsiin, elci f blck B eqls 3 1 T 1 T T B T T T T C T 3 nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 De scrib i ng Tr nsl in M in b nglr Vrible s siin f pricle cn cmpleel be specified b is psiin ecr r, if mgnide r f he psiin ecr nd is rienin relie sme fied reference direcin is knwn. In he gien figre is shwn pricle lcin shwn b psiin ecr r. Mgnide f he psiin ecr is disnce r = f he pricle frm he rigin nd rienin f he psiin ecr is he ngle mde b line wih he psiie -is. We nw specif psiin f pricle b hese ribles r nd, knwn s plr crdines. When he pricle mes, eiher r bh f hese crdines chnge wih ime. If pricle mes rdill w frm he rigin, mgnide r f is psiin ecr r increses wih n chnge in ngle. Similrl, if pricle mes rdill wrds he rigin, r decreses wih n chnge in ngle. If pricle mes n circlr ph wih cener he rigin, nl he ngle chnges wih ime. If he pricle mes n n ph her hn rdil srigh line r circle cenered he rigin, bh f he crdines r nd chnge wih ime. nglr Mi n : Chnge in direcin f psiin ecr r is knwn s nglr min. I hppens when pricle mes n criliner ph r srigh-line ph n cnining he rigin s shwn in he fllwing figres. () 19 ng l r M in () r (r, )

20 J-hsics nglr psiin : The crdine ngle n insn is knwn s nglr psiin f he pricle. nglr Displcemen : chnge in nglr psiin in ime inerl is knwn s nglr displcemen. nglr Velci : The insnnes re f chnge in nglr psiin wih respec ime is knwn s nglr elci. We dene nglr elci b smbl. nglr ccelerin : The insnnes re f chnge in nglr elci wih respec ime is knwn s nglr ccelerin. We dene nglr ccelerin b smbl. d d d d d d d d If pricle mes in plne, he psiin ecr rns eiher in clckwise r niclckwise sense. ssming ne f hese direcin psiie nd her negie, prblems f nglr min inling nglr psiin, nglr elci nd nglr ccelerin cn be sled in fshin similr prblems f reciliner min inling psiin, elci, nd ccelerin. Kinemics f Circlr Min bd in circlr min mes n circlr ph. presen, we discss nl rnslin min f bd n circlr ph nd disregrd n rin; herefre, we represen he bd s pricle. In he gien figre, pricle is shwn ming n circlr ph f rdis r. Here nl fr simplici cener f he circlr ph is ssmed he rigin f crdine ssem. In generl, i is n necessr ssme cener he rigin. siin ecr f he pricle is shwn b direced rdis r. Therefre, i is ls knwn s rdis ecr. The rdis ecr is lws nrml he ph nd hs cnsn mgnide nd s he pricle mes, i is he nglr psiin, which ries wih ime. nglr Vr ible s i n Ci rclr M i n nglr psiin, nglr elci nd nglr ccelerin knwn s nglr ribles r in differen mnner depending n hw he pricle mes. Mi n wi h nifrm nglr elci If pricle mes wih cnsn nglr elci, is nglr ccelerin is zer nd psiin ecr rns cnsn re. I is nlgs nifrm elci min n srigh line. The nglr psiin n insn f ime is epressed b he fllwing eqin. Mi n wi h nifrm nglr cceleri n = 0 + If pricle mes wih cnsn nglr ccelerin, is nglr elci chnges wih ime cnsn re. The nglr psiin, nglr elci nd he nglr ccelerin ber relins described b he fllwing eqins, which he frms similr crrespnding eqins h describe nifrm ccelerin min nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

21 Mi n wi h ri b le nglr cceleri n J-hsics Vrible nglr ccelerin f pricle is generll specified s fncin f ime, nglr psiin r nglr elci. rblems inling rible nglr ccelerin cn ls be sled in w nlgs crrespnding reciliner min prblems in which ccelerin is specified s fncin f ime, psiin r elci. Li ner Velci nd cceler in i n circlr M in The insnnes elci nd he insnnes ccelerin re ls knwn s liner elci nd liner ccelerin. In he figre is shwn pricle ming n circlr ph. s i mes i cers disnce s (rc lengh). s =r Liner elci is lws lng he ph. Is mgnide knwn s liner speed is bined b differeniing s wih respec ime. d r r d If speed f he pricle is nifrm, he circlr min is knwn s nifrm circlr min. In his kind f min s he pricle precedes frher, nl direcin f elci chnges. Therefre, insnnes ccelerin r liner ccelerin ccns fr nl chnge in direcin f min. Cnsider pricle in nifrm circlr min. I is shwn w infiniel clse insns nd + d, where is elci ecrs re nd d. These w elci ecrs re eql in mgnide nd shwn in djcen figre. Frm his figre, i is bis h he chnge d in elci ecr is perpendiclr elci ecr i.e wrds he cener. I cn be pprimed s rc f rdis eql mgnide f. Therefre we cn wrie d d. Hence ccelerin f his pricle is wrds he cener. I is knwn s nrml cmpnen f ccelerin r mre cmmnl cenripel ccelerin. Diiding d b ime inerl d we ge mgnide f cenripel ccelerin c. r s nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 d d r c r ccelerin nd elci f pricle in nifrm circlr min re shwn in he fllwing figre. T keep he pricle in nifrm circlr min ne frce cing n i ms be wrds he cener, herefre i is knwn s cenripel frce. If pricle mes wih ring speed, he ne frce n i ms he cmpnen lng he direcin f elci ecr in ddiin he cenripel frce. This cmpnen frce is lng he ngen he ph nd prdces cmpnen f ccelerin in he ngenil direcin. This cmpnen knwn s ngenil cmpnen f ccelerin T, ccns fr chnge in speed. d d T r r d d 1 C c s

22 J-hsics m ple Sl in nglr psiin f pricle ming n criliner ph ries ccrding he eqin 3 3 4, where is in rdins nd ime is in secnds. Wh is is erge nglr ccelerin in he ime inerl = s = 4s? Like erge liner ccelerin, he erge nglr ccelerin eqls ri f chnge in nglr elci he cncerned ime inerl. finl finl iniil iniil...(i) The nglr elci being re f chnge in nglr psiin cn be bined b eqin d d Sbsiing he gien epressin f he nglr psiin, we he (ii) Frm he be eq. (ii), nglr elciies nd 4 he gien insns s nd 4s re 4 = 4 rd/s nd 4 =8 rd/s. Sbsiing he be les in eq. (1), we he =1 rd/s m ple Sl in pricle srs frm res nd mes n cre wih cnsn nglr ccelerin f 3.0 rd/s. n bserer srs his spwch cerin insn nd recrd h he pricle cers n nglr spn f 10 rd he end f 4 h secnd. Hw lng he pricle hd med when he bserer sred his spwch? Le he insns when he pricle srs ming nd he bserer srs his spwch, re 0 =0 = 1. Dening nglr psiins nd nglr elci he insn 1 b 1 nd 1 nd he nglr psiin he insn 1 4 s b, we cn epress he nglr spn cered dring he inerl frm eq Sbsiing les 1,, 1 nd, we he 1 4 rd/s m ple Sl in Frm eq. = 0 +, we he 1 = Nw sbsiing 0 =0, 1 = 4 nd =3 rd/s, we he 1 = 8.0 s pricle mes n circlr ph f rdis 8 m. Disnce reled b he pricle in ime is gien b he 3 eqin s. Find is speed when ngenil nd nrml ccelerins he eql mgnide. 3 The speed, ngenil ccelerin nd he nrml ccelerin n re epressed b he fllwing eqins. ds d Sbsiing he gien epressin fr s, we he d s d...(i) nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

23 J-hsics Sbsiing he gien epressin fr s, we he 4 n r...(ii) 4 Sbsiing frm eq. (i) nd r =8m, we he 1 n The insn when he ngenil nd he nrml ccelerins he eql mgnide, cn be bined b eqing heir epressins gien in eq. (ii) nd (iii). n = T = s Sbsiing he be le f in eq. (i), we bin = 8 m/s...(iii) m ple Sl in pricle is ming n circlr ph f rdis 1.5 m cnsn nglr ccelerin f rd/s. he insn = 0, nglr speed is 60/ rpm. Wh re is nglr speed, nglr displcemen, liner elci, ngenil ccelerin nd nrml ccelerin he insn = s. Iniil nglr speed is gien in rpm (relin per mine). I is epressed in rd/s s rd 1 rpm 60 s 60 rd rd/s 60 s he insn = s, nglr speed nd nglr displcemen re clcled b sing eq. Sbsiing les rd/s, rd/s, s, we he 6 rd/s 1 Sbsiing les 0 rd, rd/s, 6 rd/s nd = s, we he = 8 rd Liner elci = s, cn be clcled b sing eq. = r nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Sbsiing r 1.5 m nd 6 rd/s, we he = 9 m/s Tngenil ccelerin nd nrml ccelerin = r Sbsiing r 1.5 m nd rd/s, we he = 3 m/s n = r Sbsiing = 6 rd/s nd r = 1.5 m, we he n = 54 m/s 3 n cn be clcled b sing eq. nd respeciel.

24 J-hsics m ple pricle is ming in circlr rbi wih cnsn ngenil ccelerin. fer s frm he beginning f min, ngle beween he l ccelerin ecr nd he rdis R becmes 45. Wh is he nglr ccelerin f he pricle? 45 n cener Sl in In he djining figre re shwn he l ccelerin ecr nd is cmpnens he ngenil ccelerins nd nrml ccelerins n re shwn. These w cmpnens re lws mll perpendiclr ech her nd c lng he ngen he circle nd rdis respeciel. Therefre, if he l ccelerin ecr mkes n ngle f 45 wih he rdis, bh he ngenil nd he nrml cmpnens ms be eql in mgnide. Nw frm eq. nd, we he n R R...(i) Since nglr ccelerin is nifrm, frm eq., we he Sbsiing 0 =0 nd = s, we he =...(ii) Frm eq. (i) nd (ii), we he = 0.5 rd/s 4 nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

25 J-hsics SM WRKD UT XMLS mp le#1 n n pen grnd mris fllws rck h rns his lef b n ngle f 60 fer eer 500 m. Sring frm gien rn, specif he displcemen f he mris he hird, sih nd eighh rn. Cmpre he mgnide f displcemen wih he l ph lengh cered b he mris in ech cse. Sl in 60 V 60 D IV VI III C 500m 60 II B VIII 500m m I VII III rn Displcemen = B BC C = 500 cs cs 60 = = 1000 m frm C Displcemen 1000 Disnce = = 1500 m. S Disnce VI rn : iniil nd finl psiins re sme s displcemen Displcemen 0 = 0 nd disnce = = 3000 m Disnce VIII rn : Displcemen = 60 (500) cs = 1000 cs 30 = m nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 Disnce = = 4000 m Displcemen Disn ce mp le# drnkrd wlking in nrrw lne kes 5 seps frwrd nd 3 seps bckwrd, fllwed gin b 5 seps frwrd nd 3 seps bckwrd, nd s n. ch sep is 1m lng nd reqires 1s. l he grph f his min. Deermine grphicll r herwise hw lng he drnkrd kes fll in pi 9m w frm he sr. (m) (sec) 5 i

26 J-hsics Sl in frm grph ime ken = 1 s R (5m 3m) + (5m 3m) + 5m = 9m l seps = 1 ime = 1 s m p l e # 3 Sl in mn wlks n srigh rd frm his hme mrke.5 km w wih speed f 5 km/h. n reching he mrke he insnl rns nd wlks bck wih speed f 7.5 km/h. Wh is he () mgnide f erge elci nd (b) erge speed f he mn, er he inerl f ime (i) 0 30 min. (ii) 0 50 min (iii) 0 40 min. Time ken b mn g frm his hme mrke, 1 dis n ce.5 1 speed 5 h Time ken b mn g frm mrke his hme, h Tl ime ken = 1 + h = 50 min. 3 6 (i) 0 30 min erge elci = displcemen ime inerl.5 =5 km/h wrds mrke (ii) (iii) erge speed = 0 50 min dis n ce ime inerl.5 = 5 km/h Tl displcemen = zer s erge elci = 0 S, erge speed = 5 50 / 60 = 6 km/h Tl disnce relled = = 5 km min Disnce cered in 30 min (frm hme mrke) =.5 km. Disnce cered in 10 min (frm mrke hme) wih speed 7.5 km/h = S, displcemen = = 1.5 km (wrds mrke) Disnce relled = = 3.75 km 1.5 erge elci = km/h. (wrds mrke) erge speed = 5.65 km/h. 6 = 1.5 km Ne : Ming bd wih nifrm speed m he rible elci. e.g. in nifrm circlr min speed is cnsn b elci is nn nifrm. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

27 J-hsics mp le#4 drier kes 0.0 s ppl he brkes fer he sees need fr i. This is clled he recin ime f he drier. If he is driing cr speed f 54 km/h nd he brkes cse decelerin f 6.0m/s, find he disnce relled b he cr fer he sees he need p he brkes n? Sl in Disnce cered b he cr dring he pplicin f brkes b drier 5 s 1 = = (0.) = = 3.0 m fer ppling he brkes; = 0, = 15 m/s, = 6 m/s s =? 5 Using = s 0 = (15) 6 s 1 s = 5 s m 1 Disnce relled b he cr fer drier sees he need fr i s = s 1 + s = = 1.75 m. mp le#5 pssenger is snding d disnce w frm bs. The bs begins me wih cnsn ccelerin. T cch he bs, he pssenger rns cnsn speed wrds he bs. Wh ms be he minimm speed f he pssenger s h he m cch he bs? S l. Le he pssenger cch he bs fer ime. The disnce relled b he bs, s 1 = (i) nd he disnce relled b he pssenger s = (ii) Nw he pssenger will cch he bs if d + s 1 = s...(iii) d + 1 = 1 + d = 0 [ d ] S he pssenger will cch he bs if is rel, i.e., d d S he minimm speed f pssenger fr cching he bs is d. mp le#6 If bd rels hlf is l ph in he ls secnd f is fll frm res, find : () The ime nd (b) heigh f is fll. plin he phsicll nccepble slin f he qdric ime eqin. (g = 9.8 m/s ) Sl in If he bd flls heigh h in ime, hen h= 1 g [ = 0 s he bd srs frm res]... (i) Nw, s he disnce cered in ( 1) secnd is h'= 1 g( 1)... (ii) S frm qins (i) nd (ii) disnce relled in he ls secnd. nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 h h'= 1 g 1 g( 1) i.e., h h'= 1 g( 1) B ccrding gien prblem s (h h ) = h 1 1 h g = 0 r 1 g g 1 r [s frm eqin (i) 7 1 h g [4 (4 4 )] / = 0.59 s r 3.41 s 0.59 s is phsicll nccepble s i gies he l ime ken b he bd rech grnd lesser hn ne sec while ccrding he gien prblem ime f min ms be greer hn 1s. s =3.41s nd h= 1 (9.8) (3.41) =57 m ]

28 J-hsics mp le#7 cr cceleres frm res cnsn re fr sme ime, fer which i deceleres cnsn re, cme res. If he l ime elpsed is ele () he mimm elci ined nd (b) he l disnce relled. Sl in () Le he cr cceleres fr ime 1 nd deceleres fr ime hen = (i) nd crrespnding elci ime grph will be s shwn in. fig. Frm he grph = slpe f line = m 1 = 1 m m nd slpe f line B = m = m 1 B m m + = m m = (b) Tl disnce = re nder grph = 1 = 1 m = 1 Ne: This prblem cn ls be sled b sing eqins f min ( = +, ec.). mp le#8 Drw displcemen ime nd ccelerin ime grph fr he gien elci ime grph -1 (ms ) 10 (s) Sl in Fr 0 5 s nd 1 =cnsn 10 5 = ms fr whle inerl s 1 = re nder he cre = = 5 m Fr 5 10, = 10ms 1 =0 fr whle inerl s = re nder he cre = 5 10 = 50 m Fr 10 1 linerl decreses wih ime 3 = 10 = 5 ms fr whle inerl s 3 = re nder he cre = 1 10 = 10 m s(m) (ms ) nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

29 J-hsics mp le#9 rcke is fired pwrds ericll wih ne ccelerin f 4 m/s nd iniil elci zer. fer 5 secnds is fel is finished nd i deceleres wih g. he highes pin is elci becmes zer. Then i cceleres dwnwrds wih ccelerin g nd rern bck grnd. l elci ime nd displcemen ime grphs fr he cmplee jrne. Tke g = 10 m/s. Sl in (m/s) 0 B C (s) s(m) 70 B 50 C (s) In he grphs, = = (4) (5) = 0 m/s B = 0 = g B 0 B s g 10 B = (5+)s = 7s Nw, s B = re nder grph beween 0 7 s = 1 (7) (0) = 70 m 1 Nw, s s g B BC BC 70 = 1 (10) BC s BC BC = = 10.7s ls s = re nder grph beween = 1 (5) (0) = 50 m mple#10 he heigh f 500m, pricle is hrwn p wih = 75 ms 1 nd pricle B is relesed frm res. Drw, ccelerin ime, elci ime, speed ime nd displcemen ime grph f ech pricle. Fr pricle : Time f fligh Fr ricle B B 1 =0 500 = Time f fligh 1 500m nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 500m = 0 500= 1 (10) = 10 s 3 = 0 s Velci B Time ken fr 1 = 0 (10) (10) = 100 ms 1 = 0 = = 7.5 s Velci 3, = = 15 ms 1 Heigh 1 = (10) (7.5) = 81.5 m 9 B

30 J-hsics - (ms ) =0s (s) - (ms ) =10s (s) (ms ) (s) (ms ) (s) speed -1 (ms ) speed -1 (ms ) (s) (s) displcemen(m) 81.5 (s) (s) displcemen(m) (s) mple#11 Tw ships nd B re 10 km pr n line rnning sh nrh. Ship frher nrh is sreming wes 0 km/h nd ship B is sreming nrh 0 km/h. Wh is heir disnce f clses pprch nd hw lng d he ke rech i? Sl in Ships nd B re ming wih sme speed 0 km/h in he direcins shwn in figre. I is w dimensinl, w bd prblem wih zer ccelerin. Le s find B = B B Here, 0 0 = 0 km/h B B N i.e., B is 0 km/h n ngle f 45 frm es wrds nrh. Ths, he gien prblem cn be simplified s : B B=10km B =0km/h B =0 km/h 45 0 is res nd B is ming wih B =0km/h Therefre, he minimm disnce beween he w is s min = C = B sin 45 0 = 10 in he direcin shwn in figre. 1 km = 5 km BC nd he desired ime is = 5 = B 0 = 1 h = 15 min 4 30 B 45 0 C B nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

31 mple#1 J-hsics In he figre shwn, he w prjecile re fired simlnesl. Find he minimm disnce beween hem dring heir fligh. 0 3 ms 0ms m 30 0 B Sl in Tking rigin nd is lng B Velci f w.r.. B = 0 3 cs 60i ˆ sin 60 ˆ j 0 cs150i ˆ sin150 ˆ j = 0 3 ˆ i ˆj 0 ˆi ˆj 0 3ˆi 0ˆj m d min B B d n 30 min 1 s sin sin 30 d min 10m mple #13 pricle is drpped frm he p f high bilding f heigh 360 m. The disnce relled b he pricle in ninh secnd is (g = 10 m/s ) S l i n () 85 m (B) 60 m (C) 40 m (D) cn' be deermined ns. (C) Tl ime ken b pricle rech he grnd T = H s g 10 Disnce relled in 8 secnds = 1 g = 1 (10) (8) = 30 m Therefre disnce relled in ninh secnd = = 40 m mple #14 bll is hrwn frm he grnd cler wll 3 m high disnce f 6 m nd flls 18 m w frm he wll, he ngle f prjecin f bll is nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 () n 3 1 (B) n 1 3 (C) n 1 1 Sl in Frm eqin f rjecr = n 1 R 3 = 6 n n = 3 3m 6m 18m 31 (D) n ns. (B)

32 J-hsics mple #15 pricle mes in XY plne sch h is psiin, elci nd ccelerin re gien b r i ˆ j ˆ ; ˆi ˆ j ; ˆi ˆj which f he fllwing cndiin is crrec if he pricle is speeding dwn? () + < 0 (B) + > 0 (C) + < 0 (D) + > 0 Sl in Fr speeding dwn. 0 + <0 ns. (C) mple #16 pricle is hrwn ericll pwrds frm he srfce f he erh. Le T be he ime ken b he pricle rel frm pin be he erh is highes pin nd bck he pin. Similrl, le T Q be he ime ken b he pricle rel frm nher pin Q be he erh is highes pin nd bck he sme pin Q. If he disnce beween he pins nd Q is H, he epressin fr ccelerin de gri in erms f T, T Q nd H, is :- () Sl in T Highes pin 6H T Q Q h H (B) 8H H H T T (C) T T (D) T T Q Time ken frm pin pin Time ken frm pin Q pin Q T 8(h H) nd g T Q 8h g Q T T Q h H g h g 8H 8H T T g g T T Q Q Q ns. (B) mple #17 n erplne is relling hriznll heigh f 000 m frm he grnd. The erplne, when pin, drps bmb hi sinr rge Q n he grnd. In rder h he bmb his he rge, wh ngle ms he line Q mke wih he ericl? [g = 10ms ] Q () 15 (B) 30 (C) 90 (D) 45 Slin : Le be he ime ken b bmb hi he rge. h = 000 = 1 g = 0 sec R = = (100) (0) = 000 m R 000 n 1 45 h ns. (D) nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65

33 J-hsics mple #18 Sme infrmins re gien fr bd ming in srigh line. The bd srs is min =0. Infrmin I : The elci f bd he end f 4s is 16 m/s Infrmin II : The elci f bd he end f 1s is 48 m/s Infrmin III : The elci f bd he end f s is 88 m/s The bd is cerinl ming wih () Unifrm elci (B) Unifrm speed (C) Unifrm ccelerin (D) D insfficien fr generlizin Sl in ns. (D) Here erge ccelerin = B we cn' s cerinl h bd he nifrm ccelerin. mple #19 lrge nmber f pricles re ming ech wih speed hing direcins f min rndml disribed. Wh is he erge relie elci beween n w pricles erged er ll he pirs? () (B) (/4) (C) (4/) (D) Zer Slin : Relie elci, where 1 = = r 1 ns. (C) If ngle beween hem be, hen r cs (1 cs ) sin Hence, erge relie elci 0 r sin d 4 d 0 mple #0 bll is prjeced s shwn in figre. The bll will rern pin : (ericl) wind gc gri nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65 (hriznl) () (B) lef pin (C) righ pin (D) nne f hese Slin : Here g c 1 g n Bll will rern pin. Iniil elci & ccelerin re ppsie ech her. 33 ns. ()

34 J-hsics mple #1 Thrgh ime inerl, while he speed f pricle increses s i mes lng he -is, is elci nd ccelerin migh be () psiie nd psiie respeciel. (C) negie nd negie respeciel. (B) psiie nd negie respeciel. (D) negie nd psiie respeciel. Slin : ns. (, C) Speed increses if bh elci & ccelerin he sme signs. mple # Three pin pricles, B nd C re prjeced frm sme pin wih sme speed =0 s shwn in figre. Fr his siin selec crrec semen(s). 53 B 37 () ll f hem rech he grnd sme ime. H C g (B) ll f hem rech he grnd differen ime. (C) ll f hem rech he grnd wih sme speed. (D) ll f hem he sme hriznl displcemen when he rech he grnd. Slin : ns. (B, C) Vericl cmpnen f iniil elciies re differen rech he grnd differen ime. mple #3 Slin : prjecile is hrwn wih speed in ir frm pin n he hriznl grnd n ngle wih hriznl. If he ir eers cnsn hriznl resisie frce n he prjecile hen selec crrec lernie(s). () he frhes pin, he elci is hriznl. (C) The ph f he prjecile m be prblic. (B) The ime fr scen eqls he ime fr descen. (D) The ph f he prjecile m be srigh line. ns. (C,D) Here l ccelerin = g = cnsn, s ph m be prblic r srigh line. mple # 4 6 rier f widh 'd' wih srigh prllel bnks flws de Nrh wih speed. b, whse speed is relie wer, srs frm nd crsses he rier. If he b is seered de Wes nd ries wih s = hen nswer he fllwing qesins. 4. The ime ken b b crss he rier is () d (B) d B 34 (C) d S W N (D) d d d nde6\_nd6 ()\D\014\K\J-dnced\SM\h\Kinmeics\nglish\her.p65