Chper Lner M n
.1 Aerge Velcy The erge elcy prcle s dened s The erge elcy depends nly n he nl nd he nl psns he prcle. Ths mens h prcle srs rm pn nd reurn bck he sme pn, s dsplcemen, nd s s erge elcy s zer. Remrk: There s derence beween dsnce nd dsplcemen. Dsnce, sclr quny, s he cul lng he ph reled by prcle, bu dsplcemen, ecr quny, s he shres dsnce beween he nl nd he nl psns he prcle.
Speed s he mgnude he elcy. Ths mens h he speed cn neer be nege. The erge speed ders rm erge elcy n h cers he l dsnce rher hn he l dsplcemen, h s erge speed The SI un elcy s m/s. ldsnce.1 Insnneus Velcy The nsnneus elcy prcle s dened s lm d 0 d
In he psn-me grph sme nsn s he slpe he ngen h nsn. Le he psn prcle res wh me ccrdng he grph shwn The nsnneus elcy specc me A s he slpe he ngen h pn. he slpe hs ngen lne ges he nsnneus elcy A
Emple.1 The psn prcle res wh me ccrdng +3 wh n m nd n s. ) Fnd he erge elcy r he nerl 0 s b) Fnd he nsnneus elcy 1.5 s Slun nd nd, we he subsue r nd n he - reln, h s, nd ( ) + 3( ) 0 ( ) + 3( ) 4 + 6 10m 10 0 5m/ s 0 Nw we he r he erge elcy d And r he nsnneus elcy + 3 d The nsnneus elcy 1.5 s s bned by subsung r 1.5 s n he ls equn (1.5) + 3 6 m/ s
.3 Accelern When he elcy mng bdy chnges wh me, we sy h he bdy hs ccelern. The erge ccelern prcle s dened s The SI un elcy s m/s. The nsnneus ccelern prcle s dened s lm d 0 d d d
Emple. The elcy n bjec res wh me ccrdng V5-3 wh n m/s nd n s. ) Fnd he erge ccelern r he nerl 1 s s b) Fnd he nsnneus ccelern s Slun nd nd, we he subsue r nd n he - reln, h s, 5( ) 3 5 3 m/s nd 5( ) 3 10 3 7m/s Nw we he r he erge ccelern And r he nsnneus ccelern d d 5m/s 7 1 5m/s Ne h he ccelern s ndependen me (cnsn).
Emple.3 The psn-me grph prcle mng lng he -s s gen n he gure. Fnd ) Fnd he erge elcy r he nerl s 5 s. b) Fnd he erge ccelern r he nerl 0.5 s.5 s. Slun ) As s cler rm he grph, 3 m, nd 0. Nw we ge 0 5 3 1m/s b) Snce ny pn s he slpe he - grph h pn, we he 0.5 s 6m/s, nd.5 s 0, s (m) 6 3 1 3 4 5 (s) 0.5 6 0.5 3 m/s -3
.4 Lner mn wh cnsn ccelern The smples ype lner mn s he unrm mn n whch he ccelern s cnsn. In such cse here we dene by, by, nd by, nd ke 0. The be equn nw s + Als, becuse s cnsn, we cn wre + + + + 1 +
Nw s + 1 + + + ( ) +
Sregy r slng prblems wh cnsn ccelern: () Chse yur crdnes such h he prcle begns s mn rm he rgn ( 0). () Decde he sense he pse drecn. () Mke ls he knwn qunes. D n rge wre ny ecr quny (,,, ) h he drecn ppse yur pse sense s nege quny. () Mke sure h ll he qunes he he sme sysem uns. () () Accrdng wh s gen nd wh s requesed, yu cn esly decde whch equn r equns yu need sle r he unknwns.
Emple.4 A cr srs rm res nd mes wh cnsn ccelern. Aer 1 s s elcy becmes 10 m/s. Fnd, ) he ccelern he cr b) he dsnce he cr rels n he 1 s Slun Le he drecn mn be lng he pse -s, where he cr srs rm he rgn 0 ( 0). Nw + Bu 0, 10 m/s nd 1 s. 10 0 1 T nd he dsnce we use 10 m/s 1 ( 10)( 1) 70 m + 0 + 1
.5 Free Fllng Bdes A reely llng bdy s ny bdy mng reely under he nluence gry regrdless s nl mn. Assumng h he grnl ccelern, dened by g, s cnsn, we cn cnsder he mn ree llng bdy s lner mn wh cnsn ccelern. Snce he mn s ercl we ke ur s be he y-s wh he +e snce s upwrd. S we replce by y nd he ccelern by -g The e sgn s becuse he grnl ccelern s lwys dwnwrd. g9.8 m/s 3 /s
y g 1 g gy Remrk: Remember h he grnl ccelern, g, s cnsn n mgnude nd n drecn nd hs mens h he nege sgn g n he ls equns wll n be chnged unless yu chnge yur pse sense, regrdless he drecn mn.
Emple.4 An bjec s hrwn erclly upwrd wh n nl speed 5 m/s. ) Hw lng des ke rech s mmum hegh? b) Wh s he mmum hegh? c) Hw lng des ke reurn he grund? d) Wh s s elcy jus bere srkng he grund? 5m/s Slun ) A he mmum pn 0. usng he equn h g 5 0.55s g 9.8
b) The mmum hgh s he ercl dsnce wh 0. usng he equn gy y ( 5) 0 h 31.9m g 9.8 ( ) c) When reurnng he grund, he l dsplcemen he bjec s zer (y 0). usng he equn y 1 g 5 0 5 1 ( 9. 8) 5.1s 4.9 d) T nd he nl elcy he mn we use he equn g ( 9.8)( 5.1) 5m/s 5 The mnus sgn ndces h he drecn he elcy s dwnwrd Ne h we use he eq. sme resul. gy wh y 0 we ge he
Emple.6 Asuden, snds he edge he r buldng, hrws bll erclly upwrd wh n nl speed 0 m/s. The buldng s 50 mhgh, nd he bll jus mssed he edge he r n s wy dwn. Fnd, ) he me needed r he bll reurn he leel he r, b) he elcy nd he psn he bll 5 s, c) he elcy he bll jus bere hng he grund. 50 m 0m/s Slun ) When he bll reurns he leel he r, s dsplcemen, y, s zer. 1 y g 0 0 ( 9. ) 1 8 0 4.9 4.08s
b) Usng he equn g ( 9.8)( 5) 9 m/s And usng he equn y 1 g y ( 0)( 5) ( 9.8)( 5).5m c) Frm he equn 1 gy 37.15 m/s ( 0) ( 9.8)( ) 50 The pse slun s rejeced becuse he bll hs he grund whle llng.
.5 RELATIVE MOTION Suppse h w persns wn bsere he mn prcle P. The rs persn Omer s snry wh respec he erh, whle he secnd persn, Ahmed s mng wh cnsn speed rele he erh. Le he reerence rme Omer be reerred by rme E (rme he erh) nd he reerence rme Ahmed by rme A. The dsplcemen he prcle w.r.. he erh s dened by PE, he dsplcemen he prcle w.r.. Ahmed s dened by pa whle AE s s he dsplcemen he A rele E. Frm he gure we cn wre + PE pa E AE AE A PA PE P Derene w.r.. we ge + PE pa AE
Emple.7 A mn, n cr, s drng n srgh hghwy cnsn speed 70 km/h rele he erh. Suddenly, he sps ruck relng n he sme drecn wh cnsn speed 60 km/h rele he erh. ) Wh s he elcy he ruck rele he cr? b) Wh s he elcy he cr rele he ruck? c) I he cr sps he ruck when hey re 1.5 km pr, hw lng des ke he cr erke he ruck? Slun ) Le TC denes he elcy he ruck rele he cr, nd TE nd CE dene, respecely, he elces he ruck nd he cr rele he erh. TC TE TC + CE TE CE 60 70 10 km/h The mnus sgn ndces h he cr s mng n he ppse drecn rele he cr.
b) Agn he elcy he cr rele he ruck CT s CE CT + TE CT CE TE Ne h s ne shuld epec. 70 60 10km/h c) The me needed r he cr erpss he ruck s 1.5 CT 1.5 10 0.15h 9mnues