Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon Componens addon Noe ha a eco has a magnude and a decon, bu doesn hae a xed locaon n space. I can be edawn anwhee n space and sll emans he same eco. Veco Addon angula mehod eco subacon
Componens o a Veco A wo dmensonal eco can be expessed as he sum o s componens along wo pependcula axes. A A x + A A x A cos θ A A sn θ A ( A x + A ) 1/ Veco Addon b Componens When wo o moe ecos ae boken up no componens n he same coodnae ssem (pependcula axes), he esulan eco om he addon o hese ecos has componens whch ae he smple sums o he nddual eco componens. A A x + A A + B Ax A + A + Bx + B + B ) + ( A + B ) ( x x B B x + B x-componen -componen A+B [ (A x + B x ) + ( A + B ) ] 1/ θ θ an A + B 1 Ax + B x
Example: Veco Addon A hke begns a p b s walkng 5. km 45. souh o eas om he base camp. On he second da she walks 4. km n a decon 6. noh o eas, a whch pon she dscoes a oes ange s owe. (a) Deemne he componens o he hke s oal dsplacemen o he p. (b) Fnd he magnude and decon o he dsplacemen om he base camp. dsplacemen aeage eloc Dsplacemen, Vel., and Accel. n D Δ Δ nsananeous eloc Δ lm aeage acceleaon a nsananeous acceleaon Δ Δ a lm
Poson, Veloc, and Acceleaon: 1D o D o 3D Fo 1-D moon, we used o know ha.. he slope a a specc me o a plo o he poson as a uncon o me s he nsananeous eloc a ha specc me; he slope a a specc me o a plo o he eloc as a uncon o me s he nsananeous acceleaon a ha specc me, ec. Ae hese sll coec n D? Δ x + ( x x + + ) ( x x ) + ( x ) + Δ + x Δ Poson, Veloc, and Acceleaon: 1D o D o 3D Fo 1-D moon, we used o know ha.. he slope a a specc me o a plo o he poson as a uncon o me s he nsananeous eloc a ha specc me; he slope a a specc me o a plo o he eloc as a uncon o me s he nsananeous acceleaon a ha specc me, ec. Ae hese sll coec n D? The ae sll ald o moons n D o 3D, as a as he componens (o poson, eloc, acceleaon) along a specc decon ae concened. Fo example: The slope o he -componen o he eloc o an objec, ploed agans me, s he -componen o he nsananeous acceleaon. The slope o x-componen o he dsplacemen o an objec, ploed agans me, s he x-componen o he nsananeous eloc, ec.
Two-Dmensonal Moon The x-componen o he anslaonal (non-oaonal) moon o an objec s ndependen o s moon n he -decon. Equaons deed o moon n one dmenson can be used o descbe he x- o -componen o he moon n wo dmensons. x x + ax 1 ( x + x ) 1 x + ax x Moe x + aoen x han no: a x x cons. x x + a Δ 1 + ) ( Δ + + 1 a a Δ Pojecle Moon In he absence o a essance, he hozonal o x componen o he acceleaon s zeo, and he ecal o componen o he acceleaon s he acceleaon due o ga. These wo moons ae ndependen o each ohe. x 1 Δ + ( g) g9.8 m/s
How a (hgh, much me) s he kcko wh hs nal eloc? ox / Δ ( 4.9m / s ) How much me s n he a? / g o snθ / g How hgh n a does go? H ( / ) o sn g θ How a does go? R Wha s he ahes a ball wll go when hown wh he same o? x o snθ cos / g θ Range Relae Veloc The eloc o A elae o B, V AB s he eloc o A as ewed om he anage pon o B. (B s unconcened abou possble moon o sel.) To nd V AB : (1) Fnd a coodnae (C) common o boh A and B. The eloces o A and B on hs common coodnae ae V AC and V BC, especel. Then, V AB V AC V BC () In cases whee C s known o moe wh espec o A whle B s known o moe espec o C, an addon o ecos ma be moe anspaen: V AB V AC + V CB Mnemonc dece: Thnk o A, B, C as hee pons on a pece o pape and V AB as he eco om A o B, and ce esa.
Chape 3 Example Poblems Ca A s mong n he noheas decon wh a speed o 4. m/s. Ca B s mong due eas wh a speed o 5. m/s. Wha s he eloc o ca A as measued b an obsee on ca B? 55. A home un s h n such a wa ha he baseball jus cleas a wall 1 m hgh, locaed 13 m om home plae. The ball s h a an angle o 35 o he hozonal, and a essance s neglgble. Fnd (a) he nal speed o he ball, (b) he me akes he ball o each he wall, and (c) he eloc componens and he speed o he ball when eaches he wall. (Assume ha he ball s h a a hegh o 1. m aboe he gound.) Reew o Chape 3 Veco componens and addon. Two dmensonal knemacs can be analzed b 1D mehods usng he x and componens o s dsplacemen, eloc, and acceleaon. The x and analses ae done ndependen o each ohe. A an specc me, he x- and -componens can be pu ogehe o eld he eloc (o dsplacemen, o acceleaon) eco. Pojecle moon noles consan acceleaon n decon and no acceleaon n x decon. Smme n pojecle moon: same speed a same hegh, he - componen o he eloc s eesed on wa down om on wa up. Relae eloc.
Equaon Shee