Chpte 4 Mtin in Tw nd Thee Dimensins In this chpte we will cntinue t stud the mtin f bjects withut the estictin we put in chpte t me ln stiht line. Insted we will cnside mtin in plne (tw dimensinl mtin) nd mtin in spce (thee dimensinl mtin). The fllwin ects will be defined f tw- nd thee- dimensinl mtin: Displcement Aee nd instntneus elcit Aee nd instntneus cceletin We will cnside in detil pjectile mtin nd unifm cicul mtin s exmples f mtin in tw dimensins Finll we will cnside eltie mtin, i.e. the tnsfmtin f elcities between tw efeence sstems which me with espect t ech the with cnstnt elcit. (4-1)
Psitin Vect The psitin ect f pticle is defined s ect whse til is t efeence pint (usull t the iin O) nd its tip is t the pticle t pint P. xiˆ + j ˆ + zkˆ P i + j + k m ( 3ˆ ˆ 5 ˆ) (4 -)
Displcement Vect F pticle tht chnes pstin ect fm 1 t we define the displcement ect s fllws: 1 The psitin ects 1 nd e witten in tems f cmpnents s: x iˆ ˆ + j + z k x iˆ + ˆj + z k 1 1 1 1 ˆ ( ) ( ) ( ) 1 1 1 ˆ The displcement cn then be witten s: x x iˆ + ˆj + z z kˆ xiˆ + j ˆ + zkˆ x x x 1 1 z z z 1 t 1 t (4-3)
Aee nd Instntneus Velcit Fllwin the sme ppch s in chpte we define the ee elcit s: ee elcit displcement time intel xiˆ + j ˆ + zkˆ xiˆ j ˆ zkˆ + + t t t t t The (instntneus) elcit is the limit: t t + Δt lim t t 0 d dt (4-4)
Aee nd Instntneus Acceletin The ee cceletin is defined s: ee cceletin chne in elcit time intel t t 1 The (instntneus) cceletin is the limit: d d lim ( ˆ ˆ ˆ ) d d x ˆ ˆ dz ˆ ˆ ˆ ˆ xi + j + zk i + j + k xi + j + zk t dt dt dt dt dt t 0 Nte: Unlike elcit, the cceletin ect des nt he n specific eltinship with the pth. The thee cceletin cmpnents e ien b the equtins: x d dt d x dt z d dt z d dt (4-6)
Pjectile Mtin The mtin f n bject in eticl plne unde the influence f ittinl fce is knwn s pjectile mtin The pjectile is lunched with n initil elcit The hizntl nd eticl initil elcit cmpnents e: (4-7) csθ sinθ x Pjectile mtin will be nlzed in hizntl nd eticl mtin ln the x- nd -xes, espectiel. These tw mtins e independent f ech the. Mtin ln the x-xis hs ze cceletin. Mtin ln the - xis hs unifm cceletin - (4-7)
Hizntl Mtin: x 0 0 0 The elcit ln the x-xis des nt chne ( θ ) csθ (eqs.1) x x + cs Veticl Mtin: 0 sinθ If we eliminte x (eqs.) Aln the -xis the pjectile is in fee fll ( sinθ ) t (eqs.3) + t 0 0 0 t (eqs.4) ( θ ) ( ) t between equtins 3 nd 4 we et: 0 sin 0 t (4-8) Hee x nd e the cdintes f the lunchin pint. F mn pblems the lunchin pint is tken t the iin. In this cse x 0 nd 0 Nte: In this nlsis f pjectile mtin we nelect the effects f i esistnce (4-8)
The equtin f the pth: t x ( cs θ ) t (eqs.) ( 0 sin θ0 ) t (eqs.4) If we eliminte t between equtins nd 4 we et: ( tn θ ) x ( θ ) cs The pth equtins hs the fm: x This equtin x + bx Nte: descibes the pth f the mtin This is the equtin f pbl The equtin f the pth seems t cmplicted t be useful. Appences cn deceie: Cmplicted s it is, this equtin cn be used s sht cut in mn pjectile mtin pblems (4-9)
x 0 0 ( θ ) cs θ (eqs.1) x cs t (eqs.) t 0 sin θ0 t (eqs.3) ( 0 sin θ0 ) t (eqs.4) O Hizntl Rne: The distnce OA is defined s the hizntl ne At pint A we he: 0 Fm equtin 4 we he: t t ( 0 sinθ0 ) t 0 t 0 sinθ0 0 This equtin hs tw slutins: Slutin 1. t 0 This slutin cespnd t pint O nd is f n inteest Slutin. sinθ 0 0 0 0 Fm slutin we et: t If we substitute t in eqs. we et: R R R mx t 0 This slutin cespnd t pint A sinθ sinθ csθ sin θ hs its mximum lue when θ 45 sin ϕ π/ R 3π/ Tw diffeent nles ie sme ne! (4-10) ϕ
t A H Mximum heiht H H sin θ The -cmpnent f the pjectile elcit is: At pint A: 0 sinθ t t H 0 0 0 0 0 0 t 0 sinθ0 0 sinθ0 ( ) ( 0 sinθ0 ) ( 0 sinθ0 ) sin θ sinθ t sinθ H t t (4-11)
t A H Mximum heiht H (ence) H sin θ We cn clculte the mximum heiht usin the thid equtin f kinemtics ( ) f mtin ln the -xis: In u pblem: 0, H, sin θ, 0, nd H H sin θ (4-1)
Unifm cicul Mtin: A pticle is in unifm cicul mtin if it mes n cicul pth f dius with cnstnt speed. Een thuh the speed is cnstnt, the elcit is nt. The esn is tht the diectin f the elcit ect chnes fm pint t pint ln the pth. The fct tht the elcit chnes mens tht the cceletin is nt ze. The cceletin in unifm cicul mtin hs the fllwin chcteistics: 1. Its ect pints twds the cente C f the cicul pth, thus the nme centipetl. Its mnitude is ien b the equtin: Q C P The time T it tkes t cmplete full elutin is knwn s the peid. It is ien b the equtin: R T π (4-13)
Reltie Mtin in One Dimensin: The elcit f pticle P detemined b tw diffeent bsees A nd B ies fm bsee t bsee. Belw we deie wht is knwn s the tnsfmtin equtin f elcities. This equtin ies us the exct eltinship between the elcities ech bsee peceies. Hee we ssume tht bsee B mes with knwn cnstnt elcit BA with espect t bsee A. Obsee A nd B detemine the cdintes f pticle P t be x PA nd x PB, espectiel. xpa xpb + xba Hee xba is the cdinte f B with espect t A d d d We tke deities f the be equtin: ( xpa ) ( xpb ) + ( xba ) dt dt dt PA PB + BA If we tke deities f the lst equtin nd tke d BA int ccunt tht 0 dt Nte: PA PB Een thuh bsees A nd B mesue diffeent elcities f P, the mesue the sme cceletin (4-15)
Reltie Mtin in Tw Dimensins: Hee we ssume tht bsee B mes with knwn cnstnt elcit BA with espect t bsee A in the x-plne. Obsees A nd B detemine the psitin ect f pticle P t be nd, espectiel. PA PB + PA PB BA We tke the time deitie f bth sides f the equtin d PA d d PB + BA PA PB + BA PA PB + BA dt dt dt If we tke the time deitie f bth sides f the lst equtin we he: d d d dba P B + B If we tke int ccunt tht 0 PA dt dt dt dt PA A PB (4-16) Nte: As in the ne dimensinl cse, een thuh bsees A nd B mesue diffeent elcities f P, the mesue the sme cceletin