Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe the motion, since the results will be independent of n prticulr coordinte sstem.
t time t the prticle is t position, which is locted b the position ector r mesured from the fied origin O. Both the mgnitude nd direction of r re known t time t. t time t+dt, the prticle is t ', locted b the position ector r Dr. Pth of prticle r Dr r t+dt ' Ds Dr t ' D O
O r Pth of prticle Dr r t+dt ' Ds Dr t ' D The displcement of the prticle during Dt is the ector Dr which represents the ector chnge of position nd is independent of the choice of origin. If nother point ws selected s the origin the position ectors would he chnged but Dr would remin the sme.
Pth of prticle r Dr r t+dt ' Ds Dr t ' D O The distnce ctull trelled b the prticle s it moes long the pth from to ' is the sclr length Ds mesured long the pth. It is importnt to distinguish between Ds nd D r.
Velocit The erge elocit of the prticle between nd ' is defined s Dr Dt which is ector whose direction is tht of. The mgnitude of is Dr. Dr The erge speed of the prticle between nd ' is Dt r Pth of prticle Dr r t+dt ' Ds Dr t ' Ds Dt Clerl, the mgnitude of the erge r elocit nd the speed pproch D Dt one nother s the interl Dt decreses nd nd ' become closer together. Ds Dt O
The instntneous elocit of the prticle is defined s the limiting lue of the erge elocit s the time Dt pproches zero. lim Dt0 We obsere tht the direction of pproches tht of the tngent to the pth s Dt pproches zero nd, thus, the elocit is lws ector tngent to the pth. The mgnitude of is clled the speed nd is the sclr ds dt s Dr Dt dr dt r O r Pth of prticle Dr r t+dt Dr ' t Ds '
The chnge in elocities, which re tngent to the pth nd re t nd t during time Dt is ector D. D D Here indictes both chnge in mgnitude nd direction of. Therefore, when the differentil of ector is to be tken, the chnges both in mgnitude nd direction must be tken into ccount.
ccelertion The erge ccelertion of the prticle between nd ' is defined s D Dt which is ector whose direction is tht of D. Its mgnitude is D Dt O The instntneous ccelertion of the prticle is defined s the limiting lue of the erge ccelertion s the time interl pproches zero. D d lim r Dt dt r Dt0 Pth of prticle Dr r t+dt Dr ' t Ds ' D
s Dt becomes smller nd pproches zero, the direction of pproches d. D The ccelertion includes the effects of both the chnges in mgnitude nd direction of. In generl, the direction of the ccelertion of prticle in curiliner motion is neither tngent to the pth nor norml to the pth. If the ccelertion Pth of prticle ws diided into two t+dt ' ' components one tngent nd the other r Dr Ds norml to the pth, it Dr D would be seen tht the t norml component r would lws be O directed towrds the center of curture.
If elocit ectors re plotted from some rbitrr point C, cure, clled the hodogrph, is formed. ccelertion ectors re tngent to the hodogrph.
Three different coordinte sstems re commonl used in describing the ector reltionships for plne curiliner motion of prticle. These re: Rectngulr (Crtesin) Coordintes (Krtezen e Dik Koordintlr) Norml nd Tngentil Coordintes (Doğl e Norml-Teğetsel Koordintlr) Polr Coordintes (Polr e Kutupsl Koordintlr) The selection of the pproprite reference sstem is prerequisite for the solution of problem. This selection is crried out b considering the description of the problem nd the mnner the dt re gien.
Crtesin Coordinte sstem is useful for describing motions where the - nd -components of ccelertion re independentl generted or determined. Position, elocit nd ccelertion ectors of the curiliner motion re indicted b their nd components. j j Pth of prticle q O r i i
Let us ssume tht t time t the prticle is t point. With the id of the unit ectors i nd j, we cn write the position, elocit nd ccelertion ectors in terms of - nd -components. r i j i j i j i j i j i j j j O Pth of prticle r i q i s we differentite with respect to time, we obsere tht the time derities of the unit ectors re zero becuse their mgnitudes nd directions remin constnt.
The mgnitudes of the components of nd re: In the figure it is seen tht the direction of is in direction. Therefore when writing in ector form - sign must be dded in front of. j j Pth of prticle q O r i i
q tn tn The direction of the elocit is lws tngent to the pth. No such thing cn be sid for ccelertion. O Pth of prticle q r i i j j
If the coordintes nd re known independentl s functions of time, =f 1 (t) nd =f (t), then for n lue of the time we cn obtin r. Similrl, we combine their first derities nd to obtin nd their second derities nd to obtin. Inersel, if nd re known, then we must tke integrls in order to obtin the components of elocit nd position. If time t is remoed between nd, the eqution of the pth cn be obtined s =f().
Projectile Motion (Eğik tış Hreketi) n importnt ppliction of two-dimensionl kinemtic theor is the problem of projectile motion. For first tretment, we neglect erodnmic drg nd the curture nd rottion of the erth, nd we ssume tht the ltitude chnge is smll enough so tht the ccelertion due to the grit cn be considered constnt. With these ssumptions, rectngulr coordintes re useful to emplo for projectile motion.
ccelertion components; =0 = -g pe; =0 o = o = o = o sinq o q o = o cosq o = g ' '
Horizontl Verticl ) ( 1 0 0 0 0 0 0 0 0 0 g t gt t gt constnt g - - - - - We cn see tht the - nd -motions re independent of ech other. Elimintion of the time t between - nd -displcement equtions shows the pth to be prbolic. If motion is emined seprtel in horizontl nd erticl directions,