Lecture 4. Home Exercise: Welcome to Wisconsin

Transcription

1 Lectue 4 ltoday: h. 3 (all) & h. 4 (stat) v Pefom vecto algeba (addition and subtaction) v Inteconvet between atesian and Pola coodinates v Wok with D motion Deconstuct motion into x & y o paallel & pependicula Obtain velocities Obtain acceleations Deduce components paallel and pependicula to the tajectoy path Physics 07: Lectue 4, Pg 1 Home Execise: Welcome to Wisconsin l You ae taveling on a two lane highway in a ca going a speed of 0 m/s (45 mph). You ae notice that a dee that has jumped in font of a ca in the opposite lane taveling at 40 m/s (90 mph) and that ca avoids hitting the dee but does so by moving into you lane! Thee is a head on collision and you ca tavels a full m befoe coming to est. Assuming that you acceleation in the cash is constant. What is you acceleation in tems of the numbe of g s (assuming g is 10 m/s )? Physics 07: Lectue 4, Pg Page 1

2 Welcome to Wisconsin l You ae taveling on a two lane highway in a ca going a speed of 0 m/s (~45 mph). You ae notice that a dee that has jumped in font of a ca taveling at 40 m/s and that ca avoids hitting the dee but does so by moving into you lane! Thee is a head on collision and you ca tavels a full m befoe coming to est. Assuming that you acceleation in the cash is constant. What is you acceleation in tems of the numbe of g s (assuming g is 10 m/s )? l Daw a Pictue l Key facts (what is impotant, what is not impotant) l Attack the poblem Physics 07: Lectue 4, Pg 3 Welcome to Wisconsin l You ae taveling on a two lane highway in a ca going a speed of 0 m/s. You ae notice that a dee that has jumped in font of a ca taveling at 40 m/s and that ca avoids hitting the dee but does so by moving into you lane! Thee is a head on collision and you ca tavels a full m befoe coming to est. Assuming that you acceleation in the cash is constant. What is the magnitude of you acceleation in tems of the numbe of g s (assuming g is 10 m/s )? l Key facts: v initial 0 m/s, afte m you v 0. v x x initial + v initial t + ½ a t x - x initial - m v initial t + ½ a t v v v initial + a t -v initial /a t v - m v initial (-v initial /a ) + ½ a (-v initial /a ) v - m -½v initial / a a (0 m/s) / m 100m/s Physics 07: Lectue 4, Pg 4 Page

3 oodinate Systems and vectos l In 1 dimension, only 1 kind of system, v Linea oodinates (x) +/- l In dimensions thee ae two commonly used systems, v atesian oodinates (x,y) v icula oodinates (,θ) l In 3 dimensions thee ae thee commonly used systems, v atesian oodinates (x,y,z) v ylindical oodinates (,θ,z) v Spheical oodinates (,θ,φ) Physics 07: Lectue 4, Pg 5 Vectos l In 1 dimension, we can specify diection with a + o - sign. l In o 3 dimensions, we need moe than a sign to specify the diection of something: l To illustate this, conside the position vecto in dimensions. Example: Whee is oston? v hoose oigin at New Yok v hoose coodinate system oston is 1 miles notheast of New Yok [ in (,θ) ] OR oston is 150 miles noth and 150 miles east of New Yok [ in (x,y) ] oston New Yok Physics 07: Lectue 4, Pg 6 Page 3

4 Vectos look like... l Thee ae two common ways of indicating that something is a vecto quantity: v oldface notation: A A o A v Aow notation: A Physics 07: Lectue 4, Pg 7 Vectos act like l Vectos have both magnitude and a diection v Vectos: position, displacement, velocity, acceleation v Magnitude of a vecto A A l Fo vecto addition o subtaction we can shift vecto position at will (NO ROTATION) l Two vectos ae equal if thei diections, magnitudes & units match. A A A, Physics 07: Lectue 4, Pg 8 Page 4

5 Scalas l A scala is an odinay numbe. v A magnitude ( + o - ), but no diection v May have units (e.g. kg) but can be just a numbe v No bold face and no aow on top. l The poduct of a vecto and a scala is anothe vecto in the same diection but with modified magnitude A A Physics 07: Lectue 4, Pg 9 Execise Vectos and Scalas While I conduct my daily un, seveal quantities descibe my condition Which of the following is cannot be a vecto? A. my velocity (3 m/s). my acceleation downhill (30 m/s). my destination (the lab - 100,000 m east) D. my mass (150 kg) Physics 07: Lectue 4, Pg 10 Page 5

6 Vectos and D vecto addition l The sum of two vectos is anothe vecto. A + A Physics 07: Lectue 4, Pg 11 D Vecto subtaction l Vecto subtaction can be defined in tems of addition. - + (-1) Physics 07: Lectue 4, Pg 1 Page 6

7 D Vecto subtaction l Vecto subtaction can be defined in tems of addition. - + (-1) Diffeent diection and magnitude! Physics 07: Lectue 4, Pg 13 Unit Vectos l A Unit Vecto points : a length 1 and no units l Gives a diection. l Unit vecto u points in the diection of U v Often denoted with a hat : u û û U U û l Useful examples ae the catesian unit vectos [ i, j, k ] o v Point in the diection of the x, y and z axes. R x i + y j + z k o R x i + y j + z k [ xˆ, yˆ, zˆ] z k y j i x Physics 07: Lectue 4, Pg 14 Page 7

8 Vecto addition using components: l onside, in D, A +. (a) (A x i + A y j ) + ( x i + y j ) (A x + x )i + (A y + y ) (b) ( x i + y j ) l ompaing components of (a) and (b): v x A x + x y v y A y + y v [ ( x ) + ( y ) ] 1/ A A y x A x Physics 07: Lectue 4, Pg 15 l Vecto A {0,,1} l Vecto {3,0,} l Vecto {1,-4,} Example Vecto Addition What is the esultant vecto, D, fom adding A++? A. {3,-4,}. {4,-,5}. {5,-,4} D. None of the above Physics 07: Lectue 4, Pg 16 Page 8

9 onveting oodinate Systems (Decomposing vectos) l In pola coodinates the vecto R (,θ) l In atesian the vecto R ( x, y ) (x,y) l We can convet between the two as follows: x x cos θ y y sin θ x î + y ĵ x + y θ tan -1 ( y / x ) y y θ x (x,y) x In 3D x + y + z Physics 07: Lectue 4, Pg 17 Decomposing vectos into components A mass on a fictionless inclined plane l A block of mass m slides down a fictionless amp that makes angle θ with espect to hoizontal. What is its acceleation a? m a θ Physics 07: Lectue 4, Pg 18 Page 9

10 Decomposing vectos into components A mass on a fictionless inclined plane l A block of mass m slides down a fictionless amp that makes angle θ with espect to hoizontal. What is its acceleation a? m g sin θ yˆ g - g j θ θ -g cos θ xˆ Physics 07: Lectue 4, Pg 19 Motion in o 3 dimensions l Position l Displacement l Velocity (avg.) l Acceleation (avg.), t i i v avg. and f, t f i t v a avg. t f Physics 07: Lectue 4, Pg 0 Page 10

11 Dynamics II: Motion along a line but with a twist (D dimensional motion, magnitude and diections) l Paticle motions involve a path o tajectoy l In -dimensions position, of a paticle x i + y j (i, j unit vectos ) Physics 07: Lectue 4, Pg 1 Instantaneous Velocity l ut how we think about equies knowledge of the path. l The diection of the instantaneous velocity is along a line that is tangent to the path of the paticle s diection of motion. v Physics 07: Lectue 4, Pg Page 11

12 Aveage Acceleation l The aveage acceleation of paticle motion eflects changes in the instantaneous velocity vecto (divided by the time inteval duing which that change occus). l Instantaneous acceleation a Physics 07: Lectue 4, Pg 3 Instantaneous Acceleation l The instantaneous acceleation is the limit of the aveage acceleation as v/ t appoaches zeo l The instantaneous acceleation is a vecto with components paallel (tangential) and/o pependicula (adial) to the tangent of the path l hanges in a paticle s path may poduce an acceleation v The magnitude of the velocity vecto may change v The diection of the velocity vecto may change (Even if the magnitude emains constant) v oth may change simultaneously (depends: path vs time) Physics 07: Lectue 4, Pg 4 Page 1

13 a T a Tangential a a v a & adial acceleation a + a a + a a a T Acceleation outcomes: If paallel changes in the magnitude of (speeding up/ slowing down) v v Pependicula changes in the diection of (tun left o ight) Physics 07: Lectue 4, Pg 5 Kinematics l In -dim. position, velocity, and acceleation of a paticle: with, with, if if x i + y j v v x i + v y j (i, j unit vectos ) a a x i + a y j x x( t) y y( t) constant x constant dx v x d x a x accel. : x( t) x y accel. : y( t) y l All this complexity is hidden away in ( t) v d / a d / v x + v 0 y dy v y d y a y t + a t 0 t a t Physics 07: Lectue 4, Pg 6 x y Page 13

14 Kinematics l The position, velocity, and acceleation of a paticle in 3-dimensions can be expessed as: x i + y j + z k dx v x d x v v x i + v y j + v z k (i, j, k unit vectos ) a a x i + a y j + a z k x x( t) y y( t) z z( t) a x dy dz v y v z d y d z a y a z l All this complexity is hidden away in ( t) v d / a d / Physics 07: Lectue 4, Pg 7 x Special ase Thowing an object with x along the hoizontal and y along the vetical. x and y motion both coexist and t is common to both Let g act in the y diection, v 0x v 0 and v 0y 0 x vs t y vs t t 0 x vs y y y t 0 4 t x Physics 07: Lectue 4, Pg 8 Page 14

15 t 0 Anothe tajectoy an you identify the dynamics in this pictue? How many distinct egimes ae thee? Ae v x o v y 0? Is v x >,< o v y? x vs y y x t 10 Physics 07: Lectue 4, Pg 9 Anothe tajectoy an you identify the dynamics in this pictue? How many distinct egimes ae thee? 0 < t < 3 3 < t < 7 7 < t < 10 t 0 v I. v x constant v 0 ; v y 0 v II. v x v y v 0 v III. v x 0 ; v y constant < v 0 x vs y y What can you say about the acceleation? x t 10 Physics 07: Lectue 4, Pg 30 Page 15

16 Execises 1 & Tajectoies with acceleation l A ocket is difting sideways (fom left to ight) in deep space, with its engine off, fom A to. It is not nea any stas o planets o othe outside foces. l Its constant thust engine (i.e., acceleation is constant) is fied at point and left on fo seconds in which time the ocket tavels fom point to some point v Sketch the shape of the path fom to. l At point the engine is tuned off. v Sketch the shape of the path afte point (Note: a 0) Physics 07: Lectue 4, Pg 31 Execise 1 Tajectoies with acceleation Fom to? A. A. A. D. D E. None of these D Physics 07: Lectue 4, Pg 3 Page 16

17 Execise Tajectoies with acceleation Afte? A. A.. D. D E. None of these A D Physics 07: Lectue 4, Pg 33 Execise Tajectoies with acceleation Afte? A. A.. D. D E. None of these A D Physics 07: Lectue 4, Pg 34 Page 17

18 Lectue 4 Assignment: Read all of hapte 4, h Physics 07: Lectue 4, Pg 35 Page 18