LECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1.


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1 LECTURE 5 ] DESCRIPTION OF PARTICLE MOTION IN SPACE The displcemen, veloci nd cceleion in D moion evel hei veco nue (diecion) houh he cuion h one mus p o hei sin. Thei full veco menin ppes when he picle is movin in D nd 3D spce. Hee one hs o del wih hei componens lon he seleced es. Le's conside he moion of picle fom poin P o P in 3D spce nd ssume h one hs seleced he fme (O) wih uni () The componens of displcemnen veco i j (3) e found b usin ( ) nd () s ; _ ; _ (4) Noe: Ofen, one uses he smbol S fo displcemen. In his cse "S" is no he ph lenh.  The vee veloci veco v (5) is defined houh he displcemen veco nd he coespondin inevl of ime. If he inevl of ime becomes smll, diecion ens o he nen of ph sin poin (P ). So, one es he insnneous veloci iniil poin defined s j P d lim P P d (6) Fiue Noe h he veco of veloci is lws nen o he ph he consideed poin bu eep in mind h is mniude is no equl o he slope of he ph h poin becuse his is no ph of posiion s funcion of ime. Bein veco, he insnneous veloci cn be epessed b is hee componens (υ, υ, υ ) in O; i Fiue P P P vecos i ; j; shown in fi.. The posiion of poins P, P in his fme is defined b he posiion vecos,. The coodines of P (,, ) nd P (,, ) e he componens of nd in fme O, oo: i j nd i j (7) d d( i j ) d d d d d d As i j ; _ ; _ ; (8) d d d d d d d d  In simil w, one would define he vee cceleion s v houh he diffeence of veloci vecos poins P nd P (see fiue 3). Ne, b decesin he inevl of ime one would e i j () The displcemen veco (fom P o P ) is defined houh nd s I is ph h eles diffeen coodines (,,).
2 P P P * Fiue 3 v  Fo picle movin in plne (D spce) wih consn cceleion, P P nd fom (9) one m e d d which, fo smll bu mesuble (phsicl) chne, is wien. One es = when he picle is P nd when i is P Δ = ; so, _ nd _. Wih hese noions one es he epession  P P o he insnneous cceleion veco poin P : d P lim P (9) d Bein veco, he cceleion hs is componens (,, ) P _ P o (3) whee _ nd _ e consn vecos. B pojecin epession (3) ove wo es O, O one es _ nd (4) _ Then, one pplies he e echnique (s fo D moion) fo υ = υ (), υ = υ () phs nd finds ou h nd (5) The epession (4) nd (5) hve pcicll he sme fom s in he cse of D inemics. So, fo picle movin wih consn cceleion in plne (D moion), he followin elions ppl: d d d d i j d d d d () So, d d d ; _ ; _ ; d d d () nd b usin (8) d ; d d _ ; _ ; d d d () ( ) ( ) (6) Noes ] The moion in plne cn be decomposed ino wo moions lon wo diecions pependicul o ech ohe. Ech of hese moions cn be sudied independenl; he ime is he sme fo boh. ] Anohe se of equions fo O e should be dded in (6) if he objec moves in 3D spce. ] PROJECTILE MOTION  The em pojecile is used fo n objec h, fe bein iven n iniil veloci, moves in spce onl unde he cion of viion foce. This is fis sep ppoimion model h nelecs ll ohe effecs (i esisnce, eh oion ), nd consides h he pojecile (sho bulle, missile, olf bll,..) moves ll ime in veicl plne. So, one cn chose wo es O, O nd ppl he elions (6) o descibe is moion. Boh ses hve he sme mhemicl fom s hose of D inemics.
3  In enel, one selecs O e lon he veicl poinin up, O e lon he hoionl nd he oiin of coodine ssem O on ound level o he poin whee he pojecile ss is moion. The se of elions (6) shows h one cn m conside he pojecile moion s consiued b wo independen moions: ) Hoionl one wih consn veloci υ = υ ; hee is no hoionl cceleion o modif i ( = ). b)veicl one wih cceleion =  ; vi cceleion veco dieced opposie o e O diecion. Remembe: The vi is he onl foce consideed in his model. Thee is no moion lon O e becuse he iniil veloci lon O is eo (υ = ) nd i is no modified in ime. Ne, fo =, =  he equions (6) become ( The wo followin emples cove he essenil siuions in pojecile poblems E_ : A bll dopped fom he 6m hih window of in movin hoionll 5m/s ouches ound. Find: ) Is flih ime inevl; b) Is hoionl displcemen; c) Flih ime if he in ws es. d) The speed he momen he bll ouches he ound. One ss counin he ime fom he momen he bll is dopped ( = s). As is moion hppens in plne one needs onl wo es (O, O). Ne, one selecs he oiin of he coodinive ssem on he eh sufce; so, =, he bll coodines e ( = m, = 6 m). One hs o sud he moion of pojecile wih = 6m, υ = ; =, υ = 5 m/s. (,) ) (7) ) The flih ime is defined b he veicl moion of pojecile fom o = 6m o = m. As υ = nd  = = 6m (8) one es 6m =  * / nd = m / 9.8m/s Fom he wo soluions =. nd =.s onl he fis one hs phsicl menin. So he flih ime of he bll is.s. b) The hoionl displcemen wih consn veloci 5m s follows fo.s. / O Fiue 4 R v (.s) (s lon s he bll hs no ouched he ound) So, R = 5*.= 55.5m (9) c) The sme condiion (8) defines he flih ime when he in is es. So, one es he sme esul =.s. Bu in his cse hee is no chne, i.e.no need fo O e. d) The insnneous speed when he bll ouches he eh sufce is equl o he mniude of veloci his momen nd fo his one hs o find is wo componens. The hoionl componen is υ = υ o = 5m/s. The veicl componen cn be found fom he ls condiion (7). As, = nd m / s, i comes ou h *9.8*6 7.6m / s. Ne, he fomul fo he mniude of veco ives _ nd _ 5.m / s. Noe h when ouchin ound is veloci veco is 5i. 84 j [m/s] 3
4 E_ : A bulle is fied fom he ound wih iniil veloci he nle θ wih hoionl. Find: ) The flih ime; b) The hoionl disnce o he poin i ouches he eh(rne). b) The shpe of is ph; d) The mimum heih i ives. ) We selec oiin on he ound. Fom he fiue 5, one finds ou he wo componens of iniil veloci s So, Fiue 5 The flih ime coesponds o he momen when he bulle ouches he eh sufce: =. sin sin * sin * flih () cos _ nd _ sin () Ne, usin hese componens eq.(7) fo = nd, = one es sin cos sin * cos * ( ) b) The ne R is defined fom he flih ime nd hoionl veloci cos. So, one es: sin * sin cos sin R * flih cos * () c) The jeco shpe cn be founds b he elion = (). To find his epession one isoles he ime vible  epession nd subsiues his epession. So, one es cos sin * n * * (3) cos cos cos This ls epession hs he fom of he pbol equion =A +B. So, he ph is pbol. d) The mimum heih coesponds o he momen when. One cn find his momen fom he equion The heih his momen is equl o m. So sin sin m (4) sin M _ Heih sin (5). m sin sin sin * nd Homewo: Do numeicl clculions fo m / s nd θ = 55. 4
5 3] UNIFORM (consn speed) CIRCULAR MOTION OF A PARTICLE This is cse of picle moion duin which he mniude of he veloci veco emins consn ll ime bu is diecion chnes followin cicul ph. As he veco of veloci chnes, his is moion wih cceleion. A fis, we will find he diecion of cceleion veco nd hen, is mniude.  Conside c movin consn speed on sih secion of hihw followed b 9 ih un. The velociies hve he sme mniude bu diffeen diecions. If he ime inevl is,, one m find he vee cceleion s Ave ( ) (6) This epession shows h he diecion of cceleion is he sme s h of veco ( ). The fiue 6. shows h he diecion of his veco is inside nd psses b he cene of he un. Fiue 6. 6.b 6.c The fiues 6.b,c show he siuions whee he 9 un is elied b wo 45 nd b fou.5 successive uns. All he vee cceleion vecos e dieced new inside nd pss b he cene of he uns. When he numbe of uns inceses, he ph oes close o cicul shpe nd, he limi, one dels wih he insnneous cceleion poin on cicul ph. c lim (7) Conclusion: The cceleion veco is ceneseein (cenipel) ech poin on cicul ph. Le s find is mniude. Conside n picle movin on cicle consn speed υ. Duin he ime inevl i is shifed fom posiion o h ; his is cull smll oion b on cicle. As he mniudes of posiion vecos e equl nd he shif is smll, one cn find he lenh(mniude) of he displcemen veco s * _ nd _ (8) Bein nen o ph, he veloci vecos e pependicul Fiue 7 o he vecos,. As he nle beween, is comes ou h he nle beween, is, oo., i Fom he inle of velociies, in ino ccoun h velociies hve he sme mniude υ, one finds 5
6 ou h he mniude of veco veloci chne is B compin he equions (8,9) i comes ou h Fom (7 ) nd (3) one finds ou h c lim * _ nd _ (9) lim lim (3)  So, in n unifom cicul moion, he cceleion is ll ime cenl seein nd is mniude is c (3) Fo poin movin consn speed on cicle fo lon ime one inoduces he concep of peiod T, i.e. he ime i es fo full evoluion. As he cicumfeence of cicle is π nd he picle moves consn speed on i, i comes ou h he ime fo full evoluion is * * T o. Then, b subsiuin his epession (3) one m find ou h T 4 c T * (3) 4] NONUNIFORM (chnin speed) CIRCULAR MOTION c Conside picle speedin up ound cicle. Is veloci momen becomes he ne momen = +Δ. Fom he definiion of cceleion ( ) i comes ou h is diecion is he sme s. As is no lined on he dil Fiue 8 diecion, i comes ou h lso is no lined on dil diecion. So, one decomposes in wo componens (fiue 8); one dil c nd one nenil. We now h he dil (cenipel) cceleion is due onl o he chne of diecion of veloci nd is mniude is c whee υ is he insnneous speed nd he dius of cicle (o ph cuvue he iven poin). The mniude of nenil cceleion is equl o he deivive of veloci mniude = speed When speedin up his deivive is posiive;, vecos hve he sme diecion in spce.if he picle is slowin down, his deivive is neive nd he veco hs opposie diecion vesus. 6 d d (33)
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