PHYS 1443 Secon 001 Lecure #4 Monda, June 5, 006 Moon n Two Dmensons Moon under consan acceleraon Projecle Moon Mamum ranges and heghs Reerence Frames and relae moon Newon s Laws o Moon Force Newon s Law o Inera & Mass Newon s second law o moon Newon s hrd law o moon 1
Announcemens All o ou hae regsered or he homework Good job!!! Quz resul Class aerage: 10.5/14 Equalen o: 75/100 Top score: 14/14 Mal dsrbuon ls Problem has been ed. Please go ahead and subscrbe o he ls Phs1443-001-summer06 Era cred 5 pons done b Tomorrow, June 6 3 pons done b Thursda, June 8
Un Vecors Un ecors are he ones ha ells us he drecons o he componens Dmensonless Magnudes are eacl 1 Un ecors are usuall epressed n, j, k or, j, k So he ecor A can be re-wren as A A + A j A cos θ + A snθ j 3
Dsplacemen, Veloc, and Acceleraon n -dm Dsplacemen: Aerage Veloc: Insananeous Veloc: Aerage Acceleraon Insananeous Acceleraon: r a a r r lm lm 0 0 r r r r dr d d d d d How s each o hese quanes dened n 1-D? dr dr d d 4
Knemac Quanes n 1d and d Quanes Dsplacemen Aerage Veloc Ins. Veloc Aerage Acc. Ins. Acc. a 1 Dmenson a Dmenson r r r r r r lm 0 d d d d lm 0 d d Wha s he derence beween 1D and D quanes? r dr lm 0 d a d d r a lm 0 d d 5
-dm Moon Under Consan Acceleraon Poson ecors n - plane: r + j r Veloc ecors n - plane: + j Veloc ecors n erms o acceleraon ecor X-comp ( ) ( ) + j + j + a Y-comp + a + + a j + a + ( ) ( ) a + j + a+ a j 6
-dm Moon Under Consan Acceleraon How are he poson ecors wren n acceleraon ecors? Poson ecor componens Pung hem ogeher n a ecor orm Regroupng he aboe r 1 + + a + j 1 + + a 1 1 a + + + + + ( ) + j + ( ) + j r + + 1 a a j + a + aj 1 ( ) 7
Eample or -D Knemac Equaons A parcle sars a orgn when 0 wh an nal eloc (0-15j)m/s. The parcle moes n he plane wh a 4.0m/s. Deermne he componens o eloc ecor a an me,. Veloc ecor () ( ) Compue he eloc and speed o he parcle a 5.0 s. 0 + 4.0 5.0 15 j 40 15 j m/ s 5 + a 0, 5 +, 5j speed + + 4.0 m/ s ( ) + j ( ) ( ) ( ) ( ) + a 15 0 ( ) + 15 ( m/ s) 0 + 4.0 15 j( m/ s) ( ) ( ) ( ) 40 + 15 43 m/ s 8
Eample or -D Knemac Eq. Cn d Angle o he Veloc ecor θ an 1 an 1 15 40 1 3 an 1 8 Deermne he and componens o he parcle a 5.0 s. Can ou wre down he poson ecor a 5.0s? r 1 + a 0 5 15 5 75 ( m) + j 150 1 4 5 + 150( m) 75 j m ( ) 9
Projecle Moon A -dm moon o an objec under he graaonal acceleraon wh he ollowng assumpons Free all acceleraon, -g, s consan oer he range o he moon Ar ressance and oher eecs are neglgble A moon under consan acceleraon!!!! Superposon o wo moons Horzonal moon wh consan eloc ( no acceleraon ) Vercal moon under consan acceleraon ( g ) 10
a 0 Show ha a projecle moon s a parabola!!! -componen a Plug no he aboe a cos θ ι 1 + g + a j an ( ) θ gj snθ cosθ cosθ -componen g cos θ cos θ 1 snθ g 1 g cosθ sn θ In a projecle moon, he onl acceleraon s graaonal one whose drecon s alwas oward he cener o he earh (downward). Wha knd o parabola s hs? 11
Projecle Moon The onl acceleraon n hs moon. I s a consan!! 1
Eample or Projecle Moon A ball s hrown wh an nal eloc (0+40j)m/s. Esmae he me o lgh and he dsance he ball s rom he orgnal poson when landed. Whch componen deermnes he lgh me and he dsance? Flgh me s deermned b componen, because he ball sops mong when s on he ground aer he lgh. Dsance s deermned b componen n -dm, because he ball s a 0 poson when compleed s lgh. 1 40 + ( g) 0m 80 0 ( g) So he possble soluons are 80 0 or 8sec g 8sec ( ) Wh sn 0 he soluon? 0 8 160 m 13
Horzonal Range and Ma Hegh Based on wha we hae learned n he preous pages, one can analze a projecle moon n more deal Mamum hegh an objec can reach Mamum range Wha happens a he mamum hegh? A he mamum hegh he objec s ercal moon sops o urn around!! + a θ h sn g 0 Sole or A θ ι A A sn θ ι g 14
Horzonal Range and Ma Hegh Snce no acceleraon s n drecon, sll les een 0. R Range R ( ) A 1 + g h ( ) Hegh sn g cosθ θ snθ snθ g sn θ h g snθ g 1 g snθ g 15
Mamum Range and Hegh Wha are he condons ha ge mamum hegh and range o a projecle moon? h sn g θ Ths ormula ells us ha he mamum hegh can be acheed when θ 90 o!!! R snθ g Ths ormula ells us ha he mamum range can be acheed when θ 90 o,.e., θ 45 o!!! 16
Eample or a Projecle Moon A sone was hrown upward rom he op o a cl a an angle o 37 o o horzonal wh nal speed o 65.0m/s. I he hegh o he cl s 15.0m, how long s beore he sone hs he ground? cosθ 65.0 cos37 51.9 m/ s ι g snθ 65.0 sn 37 39.1 m/ s 78. 50 15.0 1 g Becomes 9.80 78. 50 0 78. ± ( 78.) 4 9.80 9.80.43 s or 10. 4 s 10. 4s ( 50 ) Snce negae me does no es. 17
Eample con d Wha s he speed o he sone jus beore hs he ground? cosθ ι 65.0 cos37 51.9 m / s g snθ g 39.1 9.80 10.4 6.8 m/ s + 51.9 + ( 6.8) 81.5 m/ s Wha are he mamum hegh and he mamum range o he sone? Do hese ourseles a home or un!!! 18
Obseraons n Deren Reerence Frames Resuls o Phscal measuremens n deren reerence rames could be deren Obseraons o he same moon n a saonar rame would be deren han he ones made n he rame mong ogeher wh he mong objec. Consder ha ou are drng a car. To ou, he objecs n he car do no moe whle o he person ousde he car he are mong n he same speed and drecon as our car s. O Frame S r 0 0 O Frame S r The poson ecor r s sll r n he mong rame S.no maer how much me has passed!! The poson ecor r s no longer r n he saonar rame S when me has passed. How are hese poson ecors relaed o each oher? r() r + 0 0 19
Relae Veloc and Acceleraon The eloc and acceleraon n wo deren rames o reerences can be denoed, usng he ormula n he preous slde: Frame S r 0 Frame S r Gallean ransormaon equaon r r 0 dr dr d d 0 0 O 0 O Wha does hs ell ou? d d d d 0 d d a a, when s consan 0 The acceleraons measured n wo rames are he same when he rames moe a a consan eloc wh respec o each oher!!! The earh s graaonal acceleraon s he same n a rame mong a a consan eloc wr he earh. 0