Physics 207 Lecture 5. Lecture 5
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1 Lectue 5 Goals: Addess sstems with multiple acceleations in 2- dimensions (including linea, pojectile and cicula motion) Discen diffeent efeence fames and undestand how the elate to paticle motion in stationa and moving fames Begin to ecognize diffeent tpes of foces and know how the act on an object in a paticle epesentation Assignment: HW2, (haptes 2 & 3, due Wednesda) Read though hapte 6, Sections 1-4 Phsics 207: Lectue 5, Pg 1 Acceleation The aveage acceleation of paticle motion eflects changes in the instantaneous velocit vecto a The aveage acceleation need NOT be along the path Phsics 207: Lectue 5, Pg 3 Page 1
2 Instantaneous Acceleation The instantaneous acceleation vecto: The instantaneous acceleation is a vecto with components paallel (tangential) and/o pependicula (adial) to the tangent of the path hanges in a paticle s path and speed eflect acceleation If acceleation is tangential. onl the magnitude of the velocit vecto changes If acceleation is pependicula, onl the diection of the velocit vecto changes Phsics 207: Lectue 5, Pg 4 a t a Motion with non-zeo acceleation: a a v a Two possible options: hange in the magnitude of v hange in the diection of Animation a with a = a at need both path & time a = a + v a a = 0 a = 0 Phsics 207: Lectue 5, Pg 5 Page 2
3 Kinematics in 2 D The position, velocit, and acceleation of a paticle moving in 2-dimensions can be epessed as: = i + j v = v i + v j a = a i + a j = ( t) = ( t) v = d dt d dt v = d dt d dt 2 2 a = a 2 = 2 Special ases: 1. a =0 a = -g 2. Unifom icula Motion Phsics 207: Lectue 5, Pg 6 Special ase 1: Feefall ( t) = 0 + v t v = const. ( t) = v ( t) = v 0 + v 0 0 t g t g t and motion ae sepaate and t is common to both Now: Let g act in the diection, v 0 = v 0 and v 0 = 0 vs t vs t t = 0 vs t 0 4 t Phsics 207: Lectue 5, Pg 7 Page 3
4 Tajecto with constant acceleation along the vetical What do the velocit and acceleation vectos look like? Velocit vecto is alwas tangent to the cuve! Acceleation ma o ma not be! t = 0 vs Eample Poblem Given & 0 v 0 How fa does the knife tavel (if no ai esistance)? 4 Phsics 207: Lectue 5, Pg 8 Anothe tajecto an ou identif the dnamics in this pictue? How man distinct egimes ae thee? Ae v o v = 0? Is v >,< o = v? t = 0 vs t =10 Phsics 207: Lectue 5, Pg 9 Page 4
5 t = 0 vs Anothe tajecto an ou identif the dnamics in this pictue? How man distinct egimes ae thee? 0 < t < 3 3 < t < 7 7 < t < 10 I. v = constant = v 0 ; v = 0 II. v = -v = v 0 III. v = 0 ; v = constant < v 0 What can ou sa about the acceleation? t =10 Phsics 207: Lectue 5, Pg 10 Eecise 1 & 2 Tajectoies with acceleation A ocket is difting sidewas (fom left to ight) in deep space, with its engine off, fom A to B. It is not nea an stas o planets o othe outside foces. Its constant thust engine (i.e., acceleation is constant) is fied at point B and left on fo 2 seconds in which time the ocket tavels fom point B to some point Sketch the shape of the path fom B to. At point the engine is tuned off. Sketch the shape of the path afte point Phsics 207: Lectue 5, Pg 11 Page 5
6 Eecise 1 Tajectoies with acceleation Fom B to? B B A. A B. B. D. D E. None of these A B B D B Phsics 207: Lectue 5, Pg 12 Eecise 2 Tajectoies with acceleation Afte? A. A B. B. D. D E. None of these A B D Phsics 207: Lectue 5, Pg 13 Page 6
7 Eecise 3 Relative Tajectoies: Monke and Hunte All fee objects, if acted on b gavit, acceleate similal. A hunte sees a monke in a tee, aims his gun at the monke and fies. At the same instant the monke lets go. Does the bullet A. go ove the monke. B. hit the monke.. go unde the monke. Phsics 207: Lectue 5, Pg 14 Schematic of the poblem B ( t) = d = v 0 cos θ t B ( t) = h f = v 0 sin θ t ½ g t 2 M ( t) = d M ( t) = h ½ g t 2 Does M ( t) = B ( t) = h f? (,) = (d,h) Monke Does anone want to change thei answe? What happens if g=0? How does intoducing g change things? Bullet v 0 θ g h f ( 0, 0 ) = (0,0) (v,v ) = (v 0 cos θ, v 0 sin θ) Phsics 207: Lectue 5, Pg 15 Page 7
8 Relative motion and fames of efeence Refeence fame S is stationa Refeence fame S is moving at v o This also means that S moves at v o elative to S Define time t = 0 as that time when the oigins coincide Phsics 207: Lectue 5, Pg 16 Relative Velocit The positions, and, as seen fom the two efeence fames ae elated though the velocit, v o, whee v o is velocit of the efeence fame elative to = v o t The deivative of the position equation will give the velocit equation v = v v o These ae called the Galilean tansfomation equations Refeence fames that move with constant velocit (i.e., at constant speed in a staight line) ae defined to be inetial efeence fames (IRF); anone in an IRF sees the same acceleation of a paticle moving along a tajecto. a = a (dv o / dt = 0) Phsics 207: Lectue 5, Pg 17 Page 8
9 ental concept fo poblem solving: and components of motion teated independentl. Eample: Man on cat tosses a ball staight up in the ai. You can view the tajecto fom two efeence fames: Refeence fame on the moving cat. (t) motion govened b 1) a = -g 2) v = v 0 g t 3) = 0 + v 0 g t 2 /2 motion: = v t Refeence fame on the gound. Net motion: R = (t) i + (t) j (vecto) Phsics 207: Lectue 5, Pg 18 Eample (with fames of efeence) Vecto addition An epeimental aicaft can fl at full thottle in still ai at 200 m/s. The pilot has the nose of the plane pointed west (at full thottle) but, unknown to the pilot, the plane is actuall fling though a stong wind blowing fom the nothwest at 140 m/s. Just then the engine fails and the plane stats to fall at 5 m/s 2. What is the magnitude and diections of the esulting B velocit (elative to the gound) B the instant the engine fails? B alculate: A + B A A + B = and A + B = Phsics 207: Lectue 5, Pg 19 Page 9
10 Eecise, Relative Motion You ae swimming acoss a 50 m wide ive in which the cuent moves at 1 m/s with espect to the shoe. You swimming speed is 2 m/s with espect to the wate. You swim acoss in such a wa that ou path is a staight pependicula line acoss the ive. How man seconds does it take ou to get acoss? a) 50 2 = 25 s b) c) 50 1 = 50 s 50 3 = 29 s 50m 2m/s 1m/s d) 50 2 = 35 s Phsics 207: Lectue 5, Pg 20 Eecise hoose ais along ivebank and ais acoss ive The time taken to swim staight acoss is (distance acoss) / (v ) Since ou swim staight acoss, ou must be tilted in the wate so that ou component of velocit with espect to the wate eactl cancels the velocit of the wate in the diection: 1m/s 2m/s = 3 m/s 1m/s Phsics 207: Lectue 5, Pg 21 Page 10
11 Genealized motion with onl adial acceleation Unifom icula Motion a t a v v a a = + a a a a hanges onl in the diection of v a = 0 A paticle doesn t speed up o slow down! Phsics 207: Lectue 5, Pg 22 Unifom icula Motion (UM) is common so we have specialized tems Ac tavesed s = θ Tangential velocit v t Peiod, T, and fequenc, f s Angula position, θ Angula velocit, ω v t θ Peiod (T): The time equied to do one full evolution, 360 o 2π adians Fequenc (f): 1/T, numbe of ccles pe unit time Angula velocit o speed ω = 2πf = 2π/T, numbe of adians taced out pe unit time (in UM aveage and instantaneous will be the same) Phsics 207: Lectue 5, Pg 23 Page 11
12 Eecise A Ladbug sits at the oute edge of a me-go-ound, and a June bug sits halfwa between the oute one and the ais of otation. The me-go-ound makes a complete evolution once each second. What is the June bug s angula velocit? A. half the Ladbug s. B. the same as the Ladbug s.. twice the Ladbug s. D. impossible to detemine. J L Phsics 207: Lectue 5, Pg 24 icula Motion UM enables high acceleations (g s) in a small space omment: In automobile accidents involving otation sevee inju o death can occu even at modest speeds. [In phsics speed doesn t kill.acceleation does (i.e., the sudden change in velocit).] Phsics 207: Lectue 5, Pg 25 Page 12
13 Recap Assignment: HW2, (haptes 2 & 3, due Wednesda) Read though hapte 6, Sections 1-4 Phsics 207: Lectue 5, Pg 26 Page 13
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