Phscs 1 Lectue 4 ltoda: hapte 3 Lectue 4 v Intoduce scalas and vectos v Peom basc vecto aleba (addton and subtacton) v Inteconvet between atesan & Pola coodnates Stat n nteestn 1D moton poblem: ace 9.8 m 9.8 m l Two cas stat out at a speed o 9.8 m/s. One tavels alon the hoontal at constant veloct whle the second ollows a 9.8 m lon nclne anled below the hoontal down and then an dentcal nclne up to the nsh pont. The acceleaton o the ca on the nclne s sn (because o avt & lttle ) l t what anles, an, s the ace a te? Fnsh Phscs 1: Lectue 4, P 1 Phscs 1: Lectue 4, P ace l ccodn to ou dnamcal equatons v (t) + v t + ½ a t v v(t) v + a t l tval soluton s and the ace eques seconds. l Hoontal tavel v onstant veloct so t d / v 9.8 m (cos ) / 9.8 m/s v t 1 cos seconds l Inclne tavel v 9.8 m 9.8 m/s ( t /) + ½ sn (t /) v 1 ½ t + ½ sn (½ t ) sn (t ) + 4 t 8 4 ± t 16 + 3 sn + sn 4 + 8 sn sn Phscs 1: Lectue 4, P 3 Solvn analtcall s a challene + 4+ 8sn cos sn cos + sn cos Let cos and 1- sn Solve: 6 4 + 4 8 + 4 sth ode polnomnal, ( 1, 1-1+4-8+4 ) Phscs 1: Lectue 4, P 4 Solvn aphcall s ease + 4 + 8 sn t1 cos t sn Solve aphcall. + 4 + 8 sn 1.6 sn tme (sec.) 1..8.63 ad 36 cos scence poject l You dop a bus o the Wlls Towe (44 m above the sde walk). It so happens that Supeman les b at the same nstant ou elease the bus. Supeman s ln down at 35 m/s. l How ast s the bus on when t catches up to Supeman?.4...4.6.8 1. 1. 1.4 I < 36 then the nclne s aste nle (adans) I < 36 then the hoontal tack s aste Phscs 1: Lectue 4, P 5 Phscs 1: Lectue 4, P 6 Pae 1
Phscs 1 Lectue 4 scence poject scence poject l You dop a bus o the Wlls Towe (44 m above the sde walk). It so happens that Supeman les b at the same nstant ou elease the ca. Supeman s ln down at 35 m/s. l How ast s the bus on when t catches up to Supeman? l Daw a pctue Phscs 1: Lectue 4, P 7 t l Daw a pctue l uves ntesect at two ponts vsupeman v Supeman v bus vsupeman v v Supeman Supeman Phscs 1: Lectue 4, P 8 t Home Eecse: Welcome to Wsconsn Welcome to Wsconsn l You ae taveln on a two lane hhwa n a ca on a speed o m/s (45 mph). You ae notce that a dee that has jumped n ont o a ca n the opposte lane taveln at 4 m/s (9 mph) and that ca avods httn the dee but does so b movn nto ou lane! Thee s a head on collson and ou ca tavels a ull m beoe comn to est. ssumn that ou acceleaton n the cash s constant. What s ou acceleaton n tems o the numbe o s (assumn s 1 m/s )? l You ae taveln on a two lane hhwa n a ca on a speed o m/s (~45 mph). You ae notce that a dee that has jumped n ont o a ca taveln at 4 m/s and that ca avods httn the dee but does so b movn nto ou lane! Thee s a head on collson and ou ca tavels a ull m beoe comn to est. ssumn that ou acceleaton n the cash s constant. What s ou acceleaton n tems o the numbe o s (assumn s 1 m/s )? l Daw a Pctue l Ke acts (what s mpotant, what s not mpotant) l ttack the poblem Phscs 1: Lectue 4, P 9 Phscs 1: Lectue 4, P 1 Welcome to Wsconsn l You ae taveln on a two lane hhwa n a ca on a speed o m/s. You ae notce that a dee that has jumped n ont o a ca taveln at 4 m/s and that ca avods httn the dee but does so b movn nto ou lane! Thee s a head on collson and ou ca tavels a ull m beoe comn to est. ssumn that ou acceleaton n the cash s constant. What s the mantude o ou acceleaton n tems o the numbe o s (assumn s 1 m/s )? l Ke acts: v ntal m/s, ate m ou v. v ntal + v ntal + ½ a - ntal - m v ntal + ½ a v v v ntal + a -v ntal /a v - m v ntal (-v ntal /a ) + ½ a (-v ntal /a ) oodnate Sstems and Vectos l In 1 dmenson, onl 1 knd o sstem, v Lnea oodnates () +/- l In dmensons thee ae two commonl used sstems, v atesan oodnates (,) v cula oodnates (,) l In 3 dmensons thee ae thee commonl used sstems, v atesan oodnates (,,) v lndcal oodnates (,,) v Sphecal oodnates (,,φ) v - m -½v ntal / a a ( m/s) / m 1m/s Phscs 1: Lectue 4, P 11 Phscs 1: Lectue 4, P 1 Pae
Phscs 1 Lectue 4 Scalas and Vectos l scala s an odna numbe. v Has mantude ( + o - ), but no decton v Ma have unts (e.. k) but can be just a numbe v Repesented b an odna chaacte Eamples: mass (m, M) kloams dstance (d,s) metes spn constant (k) Newtons/mete Vectos act lke l Vectos have both mantude and a decton v Vectos: poston, dsplacement, veloct, acceleaton v Mantude o a vecto l Fo vecto addton o subtacton we can sht vecto poston at wll (NO ROTTION) l Two vectos ae equal the dectons, mantudes & unts match., Phscs 1: Lectue 4, P 13 Phscs 1: Lectue 4, P 14 Vectos look lke... l Thee ae two common was o ndcatn that somethn s a vecto quantt: Scalas and Vectos l scala can t be added to a vecto, even the have the same unts. v oldace notaton: o l The poduct o a vecto and a scala s anothe vecto n the same decton but wth moded mantude v ow notaton: -.75 Phscs 1: Lectue 4, P 15 Phscs 1: Lectue 4, P 16 Eecse Vectos and Scalas Whle I conduct m dal un, seveal quanttes descbe m condton Vectos and D vecto addton l The sum o two vectos s anothe vecto. + Whch o the ollown s cannot be a vecto?. m veloct (3 m/s). m acceleaton downhll (3 m/s). m destnaton (the lab - 1, m east) D. m mass (15 k) Phscs 1: Lectue 4, P 17 Phscs 1: Lectue 4, P 18 Pae 3
Phscs 1 Lectue 4 D Vecto subtacton l Vecto subtacton can be dened n tems o addton. - - + (-1) + - - Deent decton and mantude! Phscs 1: Lectue 4, P 19 Unt Vectos l Unt Vecto ponts : a lenth 1 and no unts l Gves a decton. l Unt vecto u ponts n the decton o U v Oten denoted wth a hat : u û l Useul eamples ae the catesan unt vectos [, j, k ] o v Pont n the decton o the [ ˆ, ˆ, ˆ], and aes. R + j + k k o R + j + k j û U U û Phscs 1: Lectue 4, P Vecto addton usn components: l onsde, n D, +. (a) ( + j ) + ( + j ) ( + ) + ( + ) (b) ( + j ) l ompan components o (a) and (b): v + v + v [ ( ) + ( ) ] 1/ l Vecto {,,1} l Vecto {3,,} l Vecto {1,-4,}. {3,-4,}. {4,-,5}. {5,-,4} D. None o the above Eample Vecto ddton What s the esultant vecto, D, om addn ++? Phscs 1: Lectue 4, P 1 Phscs 1: Lectue 4, P onvetn oodnate Sstems (Decomposn vectos) l In pola coodnates the vecto R (,) l In atesan the vecto R (, ) (,) l We can convet between the two as ollows: cos sn î + ĵ In 3D + tan -1 ( / ) + + (,) Phscs 1: Lectue 4, P 3 Moton n o 3 dmensons l Poston l Dsplacement l Veloct (av.) l cceleaton (av.), t and, t v av. v a av. Phscs 1: Lectue 4, P 4 Pae 4
Phscs 1 Lectue 4 Knematcs l In -dm. poston, veloct, and acceleaton o a patcle: + j v v + v j (, j unt vectos ) a a + a j ( ) ( ) wth, constant accel. : d v l ll ths complet s hdden awa n () v d / a d / ( ) + v d v d a d a 1 wth, constant accel. : ( ) + v + a + 1 a Phscs 1: Lectue 4, P 5 Knematcs l The poston, veloct, and acceleaton o a patcle n 3-dmensons can be epessed as: + j + k v v + v j + v k (, j, k unt vectos ) a a + a j + a k ( ) ( ) ( ) d v d a d d v v d d a a wth, constant accel., e.. ( ) + v l ll ths complet s hdden awa n () v d / a d / 1 + a Phscs 1: Lectue 4, P 6 Lectue 4 ssnment: Read hapte 4.1 to 4.3 Phscs 1: Lectue 4, P 7 Pae 5