Chapers Knemacs Poson, Dsance, Dsplacemen Mechancs: Knemacs and Dynamcs. Knemacs deals wh moon, bu s no concerned wh he cause o moon. Dynamcs deals wh he relaonshp beween orce and moon. The word dsplacemen mples he esence o an nal poson (locaon) and a nal poson oen o he same objec bu a deren mes. The dsplacemen, r, (as a resul o an objec s moon n a perod o me) s he ecor ha pons rom he nal poson oward he nal poson, whn a suable rame o reerence. I s he change n poson. The magnude o a dsplacemen has he un o lengh. D Quanes: Implc Vecors A ecor s characerzed by a magnude and a drecon. (elocy, dsplacemen,...) A scalar quany has only a magnude bu no drecon. (mass, emperaure,...) I he moon o an objec s lmed along a sragh lne (s moon s onedmensonal), he dsplacemen o hs objec can hae only wo drecons (one beng oppose o he oher). I a coordnae sysem s chosen or he dsplacemen (as n denng whch drecon s pose), he dsplacemen may appear o be scalar-lke. Howeer, one should noe ha he sgn o he dsplacemen conans normaon on drecon. aerage elocy = a Fnal poson (coordnae) Inal poson (coordnae)
Aerage Velocy and Speed aerage speed = dsance raeled / elapsed me (scalar) aerage elocy = dsplacemen / elapsed me (ecor) Velocy and speed reer o he rae o change o an objec s poson (wh respec o some saonary reerence pon, e.g. he orgn). The use o he erms aerage elocy and aerage speed mples a dene range (n me, or n space beween wo markers) n whch he elocy or he speed (o an objec) has been aeraged. I an objec s aerage elocy (/speed) s zero or a perod o me, does s aerage speed (/elocy) hae o be zero or ha perod o me? No (/Yes) Ag. Velocy (D) Slope O Lne Through Two End Pons aerage elocy y slope y ( y alue) ( alue) a Noe: slopes may hae uns In he gure, wha s he aerage elocy beween s and 3 s? beween s and s? beween s and 4s?
Insananeous Velocy (D): Slope O Tangen aerage elocy nsananeous elocy lm = slope o (he angen lne o) he - plo nsananeous speed Insananeous Velocy Eercses lm Wha s he nsananeous elocy a. s? (-6) m / (.8 s.5 s) = 3 m/s A whch me s he nsananeous elocy he greaes: A, B, C or D? Wha s he nsananeous elocy a 3. s? ~ m/s Answer: A 3
Acceleraon: he rae wh whch he elocy changes Velocy s he rae o change o dsplacemen (a ecor). Acceleraon s he rae o change o elocy (also a ecor). The analyss o hese wo quanes s ery smlar. The rae o change o a quany s he slope o a plo o hs quany agans me. aerage acceleraon (ecor) a nsananeous acceleraon (ecor) a lm Acceleraon Eamples Noe ha he y-as here s he elocy and no dsance as beore. A negae acceleraon does no necessarly mply ha he speed o an objec s reducng. 4
Judgng Sgns o Acceleraon Deermnaon o he sgn o acceleraon rom - plos. Accel. car Dsplacemen Velocy Dsplacemen? The area under ersus cure (rom o ) s equal o he dsplacemen beween and. The relaonshp beween a and s he same as ha beween and. 5
Moon wh Consan Acceleraon Consan acceleraon means ha a s consan (ed). Frs, a ral case I he acceleraon s zero, he elocy does no change wh me, and he objec wll moe wh consan elocy,. I an objec s a o a me =, wll be ound a poson a me =. = o +, = - o = Equaons For Consan -D Acceleraon Non-zero consan acceleraon: a (= consan) a Inal elocy a = : o Velocy a me s = o + a The aerage elocy beween me zero and me s ( o ) o a The oal dsplacemen he objec beween me and me s he produc o he aerage elocy and he me elapsed (). I he objec s assumed o be a he orgn a =, s poson a me s o a o + a o 6
More on Dsplacemen n D o a a / a o o area o a rapezod s (/)*(op + base) * hegh = (/) * ( o + o + a) * = o + a / area = = o + a / Wha any o o,, or a were negae? Equaons For Consan -D Acceleraon o a ( o ) o a Add one more equaon by mulplyng he rs wo equaons ogeher o a No No a No No To ge he poson o he objec a me, we need o add he dsplacemen o he nal poson o he objec = o + 7
Whch Equaon o Use? In he me akes a car o accelerae along a sragh lne rom 3m/s o m/s wh a consan m/s acceleraon, how much dsance wll hae raeled? A rock s dropped wh an nal downward elocy o m/s rom he op o a 5m buldng, wha s s elocy jus beore hs he ground? (he consan acceleraon o any ree allng objec s 9.8m/s downward) A rock s dropped wh an nal downward elocy o m/s rom he op o a 5m buldng, how long does ake or o reach he ground? In 3 seconds, an objec has acceleraed rom o m/s wh a consan acceleraon. How much dsance has raeled durng hs me. o a ( o ) o a o a No always necessary o use he mos drec way. I you can see one way o ge o he soluon, ake. Vercal Moon wh Consan Graaonal Accel. Acceleraon due o gray a = g = 9.8m/s downward Or, g = - 9.8m/s (because up s usually dened as he pose y drecon.) Ths acceleraon s seen or all objecs on earh, bg or small, heay or lgh, n he absence o ar ressance. How o work on problems such as how hgh does go? or Velocy anshes a he hghes pon. how long s n he ar? Dsplacemen as speced n he queson s reached! Equaon A + B + C = has soluons: B B A 4 AC rock and eahers o g y ( o ) y o g o g y Can usually ell whch sgn (+ or - ) o use. 8
Eample Problems 57. A con s dropped rom a ho-ar balloon ha s 3 m aboe he ground and rsng a. m/s upward. For he con, nd (a) he mamum hegh reached, (b) s poson and elocy 4. s aer beng released, and (c) he me beore hs he ground. The poson-me graph or an objec mong along a sragh pah s shown. (a) Fnd he aerage elocy o hs objec durng he me nerals. s o 4. s. (b) Fnd he nsananeous elocy a =. s. (c) Fnd he aerage acceleraon beween =. s and =4. s. (d) Fnd he aerage speed beween =. s and =8. s. Reew o Chaper Dsplacemen noles drecon and dsance (ecor). Aerage elocy = dsplacemen/elapsed me Insananeous elocy = (aerage) elocy or an nnesmally small me perod. Aerage acceleraon = elocy araon / elapsed me Insananeous acceleraon = elocy araon / nnesmally small elapsed me Equaons or objec undergong consan acceleraon n -D. Free allng noles a consan acceleraon o 9.8m/s downward or all objecs. 9