Chapte 4 Mtin in Tw and Thee Dimensins In this chapte we will cntinue t stud the mtin f bjects withut the estictin we put in chapte t me aln a staiht line. Instead we will cnside mtin in a plane (tw dimensinal mtin) and mtin in space (thee dimensinal mtin) The fllwin ects will be defined f tw- and thee- dimensinal mtin: Displacement Aeae and instantaneus elcit Aeae and instantaneus acceleatin We will cnside in detail pjectile mtin and unifm cicula mtin as examples f mtin in tw dimensins Finall we will cnside elatie mtin, i.e. the tansfmatin f elcities between tw efeence sstems which me with espect t each the with cnstant elcit (4-1)
Psitin Vect The psitin ect f a paticle is defined as a ect whse tail is at a efeence pint (usuall the iin O) and its tip is at the paticle at pint P. Example : The psitin ect in the fiue is: xiˆ+ j ˆ+ zkˆ P i + j+ k m ( 3ˆ ˆ 5ˆ) (4 -)
Displacement Vect F a paticle that chanes pstin ect fm 1 t we define the displacement ect as fllws: 1 The psitin ects 1 and ae witten in tems f cmpnents as: xiˆ+ j ˆ + zk x iˆ+ ˆj+ zk 1 1 1 1ˆ ( ) ( ) ( ) 1 1 1 ˆ The displacement can then be witten as: 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)
Aeae and Instantaneus Velcit Fllwin the same appach as in chapte we define the aeae elcit as: displacement aeae elcit time inteal a xiˆ+ j ˆ+ zkˆ xiˆ j ˆ zkˆ + + t t t t t t t + t We define as the instantaneus elcit ( me simpl the elcit) as the limit: d lim t t 0 (4-4)
If we allw the time inteal t t shink t ze, the fllwin thins happen: 1. Vect mes twads ect and 0. The diectin f the ati (and thus a )appaches the diectin t f the tanent t the path at psitin 1 3. a d ( ˆ ˆ ˆ) dx ˆ d ˆ dz xi + j+ zk i + j+ kˆ ˆ ˆ ˆ xi + j+ zk The thee elcit cmpnents ae ien b the equatins: t t + t d x dx d (4-5) z dz
Aeae and Instantaneus Acceleatin The aeae acceleatin is defined as: chane in elcit aeae acceleatin time inteal a a t t 1 We define as the instantaneus acceleatin as the limit: d d lim ( ˆ ˆ ˆ) d d x ˆ ˆ dz a ˆ ˆ ˆ ˆ xi + j+ zk i + j+ k axi + a j+ azk t t 0 Nte: Unlike elcit, the acceleatin ect des nt hae an specific elatinship with the path. The thee acceleatin cmpnents ae ien b the equatins: a x d d x a a z d z a d (4-6)
Pjectile Mtin The mtin f an bject in a etical plane unde the influence f aitatinal fce is knwn as pjectile mtin The pjectile is launched with an initial elcit The hizntal and etical elcit cmpnents ae: (4-7) csθ sinθ x Pjectile mtin will be analzed in a hizntal and a etical mtin aln the x- and -axes, espectiel. These tw mtins ae independent f each the. Mtin aln the x-axis has ze acceleatin. Mtin aln the - axis has unifm acceleatin a - (4-7)
Hizntal Mtin: x 0 0 0 The elcit aln the x-axis des nt chane ( θ ) csθ (eqs.1) x x + cs Vetical Mtin: 0 a a x (eqs.) Aln the -axis the pjectile is in fee fall t sinθ0 t (eqs.3) + ( 0sinθ0) t (eqs.4) t ( θ ) ( ) If we eliminate t between equatins 3 and 4 we et : 0sin 0 (4-8) Hee x and ae the cdinates f the launchin pint. F man pblems the launchin pint is taken at the iin. In this case x 0 and 0 Nte: In this analsis f pjectile mtin we nelect the effects f ai esistance (4-8)
The equatin f the path: t x ( cs θ ) t (eqs.) ( 0sin θ0) t (eqs.4) If we eliminate t between equatins and 4 we et: ( tan θ ) x ( θ ) cs x This equatin descibes the path f the mtin The path equatins has the fm: ax+ bx This is the equatin f a paabla Nte: The equatin f the path seems t cmplicated t be useful. Appeaances can deceie: Cmplicated as it is, this equatin can be used as a sht cut in man pjectile mtin pblems (4-9)
O x 0 0 R sin Acs A sina ( θ ) cs θ (eqs.1) x cs t (eqs.) t 0sin θ0 t (eqs.3) ( 0sin θ0) t (eqs.4) O π/ Hizntal Rane: The distance OA is defined as the hizantal ane R At pint A we hae: 0 Fm equatin 4 we hae: t t ( 0sinθ0) t 0 t 0sinθ0 0 This equatin has tw slutins: Slutin 1. t 0 This slutin cespnd t pint O and is f n inteest Slutin. 0 0 0 0 Fm slutin we et: t If we substitute t in eqs. we et: t sinθ 0 This slutin cespnd t pint A sinθ R sinθcsθ sin θ A t R has its maximum alue when θ 45 R max sinϕ (4-10) 3π/ ϕ
t A H Maximum 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 0sinθ0 0sinθ0 () ( 0sinθ0) ( 0sinθ0) sin θ sinθ t sinθ H t t (4-11)
t A H Maximum heiht H (ence) H sin θ We can calculate the maximum heiht usin the thid equatin f kinematics ( ) f mtin aln the -axis: a In u pblem: 0, H, sin θ, 0, and a H H sin θ (4-1)
Unifm cicula Mtin: A paticles is in unifm cicula mtin it mes n a cicula path f adius with cnstant speed. Een thuh the speed is cnstant, the elcit is nt. The easn is that the diectin f the elcit ect chanes fm pint t pint aln the path. The fact that the elcit chanes means that the acceleatin is nt ze. The acceleatin in unifm cicula mtin has the fllwin chaacteistics: 1. Its ect pints twads the cente C f the cicula path, thus the name centipetal. Its manitude a is ien b the equatin: a Q C P The time T it takes t cmplete a full elutin is knwn as the peid. It is ien b the equatin: R T π (4-13)
ˆ ˆ ( sin ) ˆ ( cs ) ˆ P xp xi + j θ i + θ j sin θ csθ Hee xp and P ae the cdinates f the tatin paticle P ˆ xp ˆ d dp Acceleatin ˆ dxp i + j a i + ˆj dp dxp We nte that: cs θ and x sinθ a i + j a a + a + csθ ˆ sin θ ˆ x ( csθ) ( sinθ) ( ) ( ) a / sinθ tanφ tan θ φ θ a pints twads C a / csθ x sinθ cs θ x P C A C (4-14) ( θ) ( θ) cs + sin 1
Relatie Mtin in One Dimensin: The elcit f a paticle P detemined b tw diffeent bsees A and B aies fm bsee t bsee. Belw we deie what is knwn as the tansfmatin equatin f elcities. This equatin ies us the exact elatinship between the elcities each bsee peceies. Hee we assume that bsee B mes with a knwn cnstant elcit BA with espect t bsee A. Obsee A and B detemine the cdinates f paticle P t be x PA and x PB, espectiel. xpa xpb + xba Hee xba is the cdinate f B with espect t A d d d We take deiaties f the abe equatin: ( xpa ) ( xpb ) + ( xba ) PA PB + BA If we take deiaties f the last equatin and take d BA int accunt that 0 apa apb Nte: Een thuh bsees A and B measue diffeent elcities f P, the measue the same acceleatin (4-15)
Relatie Mtin in Tw Dimensins: Hee we assume that bsee B mes with a knwn cnstant elcit BA with espect t bsee A in the x-plane. Obsees A and B detemine the psitin ect f paticle P t be and, espectiel. PA PB + PA PB BA We take the time deiatie f bth sides f the equatin d PA d d PB + BA PA PB + BA PA PB + BA If we take the time deiatie f bth sides f the last equatin we hae: d d d dba P B + B If we take int accunt that 0 a PA a PA A PB (4-16) Nte: As in the ne dimensinal case, een thuh bsees A and B measue diffeent elcities f P, the measue the same acceleatin