Chapter 3 Vectors in Physics. Copyright 2010 Pearson Education, Inc.

Transcription

1 Chapter 3 Vectors in Physics

2 Units of Chapter 3 Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion

3 3-1 Scalars Versus Vectors Scalar: number with units Vector: quantity with magnitude and direction How to get to the library: need to know how far and which way

4 3-2 The Components of a Vector Even though you know how far and in which direction the library is, you may not be able to walk there in a straight line:

5 3-2 The Components of a Vector Can resolve vector into perpendicular components using a two-dimensional coordinate system:

6 3-2 The Components of a Vector Length, angle, and components can be calculated from each other using trigonometry:

7 3-2 The Components of a Vector Signs of vector components:

8 3-3 Adding and Subtracting Vectors Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.

9 3-3 Adding and Subtracting Vectors Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.

10 3-3 Adding and Subtracting Vectors

11 3-3 Adding and Subtracting Vectors Subtracting Vectors: The negative of a vector is a vector of the same magnitude r pointing r r in the opposite direction. Here, D A.! B

12 Example: Vector A has a length of 5.00 meters and points along the x-axis. Vector B has a length of 3.00 meters and points 120 from the +x-axis. Compute A+B (C). y B C 120 A x

13 Example continued: B y y B 60 B x 120 A x sin! cos! tan! opp hyp adj hyp sin! cos! opp adj sin 60 cos60 By " By Bsin 60 B! Bx " Bx! Bcos60! B ( 3.00m) sin m ( 3.00m) cos60! 1.50 m and A x 5.00 m and A y 0.00 m

14 Example continued: The components of C: C C x y A A x y + B + B x y 5.00 m + (! 1.50 m) 3.50 m 0.00 m m 2.60 m y The length of C is: θ C C x 3.50 m C y 2.60 m x C 2 2 C C x + Cy 2 ( 3.50 m) + ( 2.60 m) 4.36 m 2 The direction of C is: C tan" C " tan! 1 y x 2.60 m 3.50 m ( ) From the +x-axis

15 Example: Margaret walks to the store using the following path: miles west, miles north, miles east. What is her total displacement? Give the magnitude and direction. y Take north to be in the +y direction and east to be along +x. r 2 r 3 Δr r 1 x

16 Example continued: The displacement is Δr r f r i. The initial position is the origin; what is r f? The final position will be r f r 1 + r 2 + r 3. The components are r fx r 1 + r miles and r fy +r miles. Δr y y Δr θ Δr x x Using the figure, the magnitude and direction of the displacement are 2 2! r! r x +! ry miles! ry tan " 1 and " 45! r x N of W.

17 3-4 Unit Vectors Unit vectors are dimensionless vectors of unit length.

18 3-4 Unit Vectors Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

19 3-5 Position, Displacement, Velocity, and Acceleration Vectors Position vector r f points from the origin to the location in question. The displacement vector! r points from the original position to the final position.

20 3-5 Position, Displacement, Velocity, and Acceleration Vectors Average velocity vector: (3-3) So v r av is in the same direction as! r.

21 3-5 Position, Displacement, Velocity, and Acceleration Vectors Instantaneous velocity vector is tangent to the path:

22 3-5 Position, Displacement, Velocity, and Acceleration Vectors Average acceleration vector is in the direction of the change in velocity:

23 3-5 Position, Displacement, Velocity, and Acceleration Vectors Velocity vector is always in the direction of motion; acceleration vector can point anywhere:

24 Example (text problem 3.42): At the beginning of a 3 hour plane trip you are traveling due north at 192 km/hour. At the end, you are traveling 240 km/hour at 45 west of north. (a) Draw the initial and final velocity vectors. y (north) v f v i x (east)

25 Example continued: (b) Find Δv. The components are " v " v x y v v fx fy! v! v ix iy! v + v f f sin 45! 0! 170 km/hr cos 45! v The magnitude and direction are: i! 22.3 km/hr 2 2! v! v x +! vy 171 km/hr tan% " v " v y x ! $ tan # 1 ( ) 7.5 South of west

26 Example continued: (c) What is a av during the trip? a " v! 170 km/hr x 2! a v x,av " t 3 hr av!t " vy! 22.3 km/hr 2 ay,av! 7.43 km/hr " t 3 hr! 56.7 km/hr The magnitude and direction are: a av a 2 x,av + a 2 y,av 57.2 km/hr 2 tan# a a y,av x,av ! # tan " 1 (0.1310) 7.5 South of west

27 3-6 Relative Motion The speed of the passenger with respect to the ground depends on the relative directions of the passenger s and train s speeds:

28 3-6 Relative Motion This also works in two dimensions:

29 Example: You are traveling in a car (A) at 60 miles/hour east on a long straight road. The car (B) next to you is traveling at 65 miles/hour east. What is the speed of car B relative to car A?

30 Example continued: t0 t>0 +x A Δr AG A Δr BA B Δr BG B From the picture: Divide by Δt: " r " r v v BG BA BA BA " r " r v BG AG BG! + " r! " r v AG BA AG 65 miles/hr east! 5 miles/hour east 60 miles/hr east

31 Example: You are traveling in a car (A) at 60 miles/hour east on a long straight road. The car (B) next to you is traveling at 65 miles/hour west. What is the speed of car B relative to car A?

32 Example continued: t>0 t0 +x t>0 A Δr AG A B Δr BG B Δr BA From the picture: Divide by Δt: " r v BA BA " r v BG BG!! " r v AG AG 65 miles/hr west! 60 miles/hr east 125 miles/hr west

33 Summary of Chapter 3 Scalar: number, with appropriate units Vector: quantity with magnitude and direction Vector components: A x A cos θ, B y B sin θ Magnitude: A (A x 2 + A y2 ) 1/2 Direction: θ tan -1 (A y / A x ) Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

34 Summary of Chapter 3 Component method: add components of individual vectors, then find magnitude and direction Unit vectors are dimensionless and of unit length Position vector points from origin to location Displacement vector points from original position to final position Velocity vector points in direction of motion Acceleration vector points in direction of change of motion Relative motion: v r v r + v r