Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009

Transcription

1 Physics 111 Lectue 5 (Walke: 3.3-6) Vectos & Vecto Math Motion Vectos Sept. 11, 2009 Quiz Monday - Chap. 2 1 Resolving a vecto into x-component & y- component: Pola Coodinates Catesian Coodinates x y = cos 25.0 = (1.50 m)(0.906) = 1.36 m = sin 25.0 = (1.50 m)(0.423) = m 2

2 Signs of vecto components: 3 Magnitude & Angle fom Components Vecto magnitude and angle can be calculated fom the scala components using tigonomety and the Pythagoean theoem: A = A + A θ = 2 2 x y 1 tan Ay / A x 4

3 Magnitude & Angle fom Components Catesian Coodinates Pola Coodinates = + = (1.36 m) + (0.634 m) = 2.25 m = 1.50 m θ x y [ ] 1 1 = tan (0.634 m) / (1.36 m) = tan (0.466) = Popeties of Vectos Equality of Two Vectos Two vectos ae equal if they have the same magnitude and the same diection Movement of vectos in a diagam Any vecto can be moved paallel to itself without being affected 6

4 Moe Popeties of Vectos Negative Vectos A vecto is the negative of anothe if they have the same magnitude but opposite diections A A = -B Resultant Vecto B The esultant vecto is the sum of a given set of vectos 7 Adding Vectos When adding vectos, thei diections must be taken into account Units must be the same Gaphical Methods Use scale dawings & geomety Algebaic Method Use components Usually moe convenient 8

5 Adding Vectos Gaphically (Tiangle o Polygon Method) Choose a scale Daw the fist vecto A with the appopiate length and in the diection specified, with espect to a coodinate system Daw the next vecto B with the appopiate length and in the diection specified, stating at the end of vecto A 9 Gaphically Adding Vectos Continue dawing the vectos tip-to-tail The esultant is dawn fom the oigin of A to the end of the last vecto Measue the length of R and its angle Use the scale facto to convet length to actual magnitude 10

6 Adding Vectos 11 Gaphically Adding Vectos When you have many vectos, just keep epeating the pocess until all ae included The esultant is still dawn fom the oigin of the fist vecto to the end of the last vecto 12

7 Notes about Vecto Addition Vectos obey the Commutative Law of Addition The ode in which the vectos ae added doesn t affect the esult 13 Vecto Subtaction Special case of vecto addition To find A B, use A+(-B) Continue with standad vecto addition pocedue 14

8 Subtacting Vectos 15 Adding Vectos Algebaically Choose a coodinate system and sketch the vectos Find the x- and y-components of all the vectos Add all the x-components This gives total, o Resultant, x-component R x : R x = vx 16

9 Adding Vectos Algebaically Add all the y-components This gives R y : R y = v y Use the Pythagoean Theoem to find the magnitude of the Resultant: 2 2 R = R x + Ry Use the invese tangent function to find the diection of R: R 1 y θ = tan R x 17 Adding Vectos by Components 18

10 Multiplying o Dividing a Vecto by a Scala The esult of the multiplication o division is a vecto The magnitude of the vecto is multiplied o divided by the scala If the scala is positive, the diection of the esult is the same as of the oiginal vecto If the scala is negative, the diection of the esult is opposite that of the oiginal vecto 19 Multiplication of Vecto by Scala 20

11 2-Dim. Vecto Summay Scala: numbe, with appopiate units Vecto: quantity with magnitude and diection Vecto components: A x = A cos θ, A y = A sin θ Magnitude: A = (A x2 + A y2 ) 1/2 Diection: θ = tan -1 (A y / A x ) = actan (A y / A x ) Gaphical vecto addition: Place tail of second at head of fist; vecto sum points fom tail of fist to head of last 21 Unit Vectos A unit vecto is a symbol fo a cetain defined diection, such as Noth, o along the x axis. A unit vecto has magnitude equal to 1 and no units, but has a defined diection. Example: the unit vecto in the x diection is usually witten: xˆ Now, I can wite a vecto as a sum of scala components multiplied by the coesponding unit vecto: A = A xˆ + A x whee A x and A y ae scala components (with units). y yˆ Sept. 11, 2009 Physics Lectue 5 22

12 Unit Vectos ŷ xˆ Example of vecto witten in tems of scala components and unit vectos: = ( 2m) xˆ + (3m) yˆ Sept. 11, 2009 Physics Lectue 5 23/24 Motion in Two Dimensions Using + o signs fo diection is usually not sufficient to fully descibe motion in moe than one dimension Vectos ae used to fully descibe motion Inteested in displacement, velocity, and acceleation vectos Sept. 11, 2009 Physics Lectue 5 24

13 The position of an object is descibed by its position vecto, = xxˆ + Displacement yyˆ The displacement of the object is defined as the change in its position = f - i Sept. 11, 2009 Physics Lectue 5 25 = xxˆ+ yyˆ The Displacement Vecto = 2 1 = 2 1 = ( x ˆ ˆ ˆ ˆ 2x+ y2y) ( xx 1 + yy 1 ) = xxˆ+ y yˆ Sept. 11, 2009 Physics Lectue 5 26

14 Velocity The aveage velocity is the atio of the displacement to the time inteval fo the displacement v av = t The diection of v av is the diection of The instantaneous velocity is the limit of the aveage velocity as t appoaches zeo The diection of the instantaneous velocity is along a line tangent to the path of the paticle and in the diection of motion v = lim t 0 t Sept. 11, 2009 Physics Lectue 5 27 The Aveage Velocity Vecto Aveage velocity vecto: (3-3) So v av is in the same diection as. Sept. 11, 2009 Physics Lectue 5 28

15 Example: A Dagonfly A dagonfly is obseved initially at position: ˆ ˆ 1 = (2.00 m) x+ (3.50 m) y Thee seconds late, it is obseved at position: ˆ ˆ 2 = ( 3.00 m) x+ (5.50 m) y What was the dagonfly s aveage velocity duing this time? v av 2 1 = = t t [( 3.00 m) (2.00 m) ] xˆ+ [(5.50 m) (3.50 m) ] = (3.00 s) = ( 1.67 m/s) xˆ+ (0.667 m/s) yˆ yˆ Sept. 11, 2009 Physics Lectue 5 29 Example: Velocity of a Sailboat Sailboat has coodinates (130 m, 205 m) at t 1 =0.0 s. Two minutes late its position is (110 m, 218 m). v v ; (b) Find av av v = v xˆ+ v yˆ (a) Find av xav y av x 110 m 130 m vxav = = = m/s t 120 s y 218 m 205 m vyav = = = m/s t 120 s v = ( m/s) xˆ+ (0.108 m/s) yˆ av v = + = av 2 2 ( m/s) (0.108 m/s) m/s m/s θ = actan = m/s o v av Sept. 11, 2009 Physics Lectue 5 30/24 v av

16 Acceleation The aveage acceleation is defined as the ate at which the velocity changes a av The instantaneous acceleation is the limit of the aveage acceleation as t appoaches zeo a = v = t lim t 0 v t Sept. 11, 2009 Physics Lectue 5 31 Ways an Object Might Acceleate a = lim t 0 v t The magnitude of the velocity (the speed) can change The diection of the velocity can change Then thee is acceleation, even though the magnitude is constant -- example: motion in a cicle at constant speed Both the magnitude and the diection can change Sept. 11, 2009 Physics Lectue 5 32

17 Position, Displacement, Velocity, & Acceleation Vectos Velocity vecto v always points in the diection of motion. The acceleation vecto a can point anywhee. Sept. 11, 2009 Physics Lectue 5 33/24 Acceleation on a Cuve Fo vehicle moving on a cuve, magnitude of velocity may stay the same but diection changes. Fo example, conside ca that has initial velocity of 12 m/s east, and 10 seconds late its velocity is 12 m/s south. v ( 12 m/s) yˆ (12 m/s) xˆ 2 2 a ( 1.2 m/s ) ˆ ( 1.2 m/s ) ˆ av = = = x+ y t (10.0 s) Sept. 11, 2009 Physics Lectue 5 34/24

18 Vecto Motion with Constant Acceleation Aveage velocity: ( ) 2 1 vav = v0 + v Velocity as a function of time: vt () = v+ at 0 Position as a function of time: 1 t () = 0 + vt av = 0 + ( v vt ) 1 2 t () = + vt+ at Sept. 11, 2009 Physics Lectue 5 35 Befoe the next lectue, ead Walke 3.6,4.1-2 Quiz Monday - Chap. 2 Homewok Assignment #3a should be submitted using WebAssign by 11:00 PM on Tuesday, Sept