Vectors & Motion. Monday, August 29, 11

Transcription

1 Vectors & Motion

2 Representing Motion

3 Four Types of Motion We ll Study

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5 What is a vector?

6 What is a vector? A quantity with magnitude & direction

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8 Quiz 1. What is the difference between speed and velocity? A. Speed is an average quantity while velocity is not. B. Velocity contains information about the direction of motion while speed does not. C. Speed is measured in mph, while velocity is measured in m/s. D. The concept of speed applies only to objects that are neither speeding up nor slowing down, while velocity applies to every kind of motion. E. Speed is used to measure how fast an object is moving in a straight line, while velocity is used for objects moving along curved paths.

9 Answer 1. What is the difference between speed and velocity? A. Speed is an average quantity while velocity is not. B. Velocity contains information about the direction of motion while speed does not. C. Speed is measured in mph, while velocity is measured in m/s. D. The concept of speed applies only to objects that are neither speeding up nor slowing down, while velocity applies to every kind of motion. E. Speed is used to measure how fast an object is moving in a straight line, while velocity is used for objects moving along curved paths.

10 Reading Quiz 2. The quantity 2.67 x 10 3 m/s has how many significant figures? A. 1 B. 2 C. 3 D. 4 E. 5

11 Answer 2. The quantity 2.67 x 10 3 m/s has how many significant figures? A. 1 B. 2 C. 3 D. 4 E. 5

12 Quiz 3. If Sam walks 100 m to the right, then 200 m to the left, his net displacement vector points A. to the right. B. to the left. C. has zero length. D. Cannot tell without more information.

13 Answer 3. If Sam walks 100 m to the right, then 200 m to the left, his net displacement vector points A. to the right. B. to the left. C. has zero length. D. Cannot tell without more information.

14 Quiz 4. Velocity vectors point A. in the same direction as displacement vectors. B. in the opposite direction as displacement vectors. C. perpendicular to displacement vectors. D. in the same direction as acceleration vectors. E. Velocity is not represented by a vector.

15 Answer 4. Velocity vectors point A. in the same direction as displacement vectors. B. in the opposite direction as displacement vectors. C. perpendicular to displacement vectors. D. in the same direction as acceleration vectors. E. Velocity is not represented by a vector.

16 What are the components of vector A?

17 What are the components of vector A?

18 Making a Motion Diagram

19 Examples of Motion Diagrams

20 The Particle Model A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.

21 Position and Time The position of an object is located along a coordinate system. At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0).

22 Displacement The change in the position of an object as it moves from initial position x i to final position x f is its displacement x = x f x i.

23 Checking Understanding Maria is at position x = 23 m. She then undergoes a displacement x = 50 m. What is her final position? A. 27 m B. 50 m C. 23 m D. 73 m

24 Answer Maria is at position x = 23 m. She then undergoes a displacement x = 50 m. What is her final position? A. 27 m B. 50 m C. 23 m D. 73 m

25 Checking Understanding Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

26 Answer Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster? A

27 Checking Understanding Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C. They are both moving at the same speed.

28 Answer Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C. They are both moving at the same speed.

29 Speed of a Moving Object 40 m m The car moves 40 m in 1 s. Its speed is = s s 20 m m The bike moves 20 m in 1 s. Its speed is 1 s = 20 s.

30 Velocity of a Moving Object

31 Example Problem At t =12 s, Frank is at x = 25 m. 5 s later, he s at x = 20 m. What is Frank s velocity?

32 Example Problem At t =12 s, Frank is at x = 25 m. 5 s later, he s at x = 20 m. What is Frank s velocity? v = Δx Δt = x 2 x 1 t 2 t 1 = (12 + 5) 12 = 5 5 = 1 m/s

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37 Vectors A quantity that requires both a magnitude (or size) and a direction can be represented by a vector. Graphically, we represent a vector by an arrow. The velocity of this car is 100 m/s (magnitude) to the left (direction). This boy pushes on his friend with a force of 25 N to the right.

38 Displacement Vectors A displacement vector starts at an object s initial position and ends at its final position. It doesn t matter what the object did in between these two positions. In motion diagrams, the displacement vectors span successive particle positions.

39 Exercise Alice is sliding along a smooth, icy road on her sled when she suddenly runs headfirst into a large, very soft snowbank that gradually brings her to a halt. Draw a motion diagram for Alice. Show and label all displacement vectors.

40 Exercise Alice is sliding along a smooth, icy road on her sled when she suddenly runs headfirst into a large, very soft snowbank that gradually brings her to a halt. Draw a motion diagram for Alice. Show and label all displacement vectors.

41 Adding Displacement Vectors

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43 How do we add or subtract vectors?

44 How do we add or subtract vectors? Vectors may be added graphically, head to tail.

45 Vector Addition The sum of vectors is independent of the order of addition

46 What are the two ways we can use to specify a vector?

47 What are the two ways we can use to specify a vector? 1. Magnitude & direction

48 What are the two ways we can use to specify a vector? 1. Magnitude & direction 2. Components

49 Calculations using components Note: Components are not vectors

50 What are unit vectors?

51 What are unit vectors?

52 What happens when we multiplying a vector by a scalar?

53 What happens when we multiplying a vector by a scalar?

54 What vector products can be computed?

55 What vector products can be computed? There are two different types of vector products: The scalar or dot product of two vectors The vector or cross product of two vectors

56 The scalar or dot product of two vectors The scalar product of two vectors A and B is denoted by A B, and is a scalar. Because of this notation, the scalar product is also called the dot product.

57 Special Cases for Dot Product

58 Special Cases for Dot Product If A & B are parallel, ϕ=0, cosϕ=1, A B=AB

59 Special Cases for Dot Product If A & B are parallel, ϕ=0, cosϕ=1, A B=AB If A & B are in opposite directions, ϕ=180, cosϕ=-1, A B=-AB

60 Special Cases for Dot Product If A & B are parallel, ϕ=0, cosϕ=1, A B=AB If A & B are in opposite directions, ϕ=180, cosϕ=-1, A B=-AB If A & B are perpendicular, ϕ=90, cosϕ=0, A B=0

61 Special Cases for Dot Product If A & B are parallel, ϕ=0, cosϕ=1, A B=AB If A & B are in opposite directions, ϕ=180, cosϕ=-1, A B=-AB If A & B are perpendicular, ϕ=90, cosϕ=0, A B=0 For unit vectors

62 The vector or cross product of two vectors We measure the angle ϕ from A towards B and take it to be the smaller of the two possible angles, so ϕ ranges from 0 to 180. Then sin ϕ 0 and the magnitude is never negative, as has to be the case for a vector. magnitude of cross product is A B = ABsinφ

63 The vector or cross product of two vectors magnitude of cross product is A B = ABsinφ

64 The vector or cross product of two vectors magnitude of cross product is A B = ABsinφ The vector product of two vectors A and B is denoted by A B. Because of this notation, the vector product is also called the cross product.

65 The vector or cross product of two vectors magnitude of cross product is A B = ABsinφ The vector product of two vectors A and B is denoted by A B. Because of this notation, the vector product is also called the cross product. A and B are vectors, and A B is a vector.

66 The vector or cross product of two vectors magnitude of cross product is A B = ABsinφ The vector product of two vectors A and B is denoted by A B. Because of this notation, the vector product is also called the cross product. A and B are vectors, and A B is a vector. We will use this latter to describe torque, angular momentum, as well as magnetic fields and forces.

67 Special Cases for Cross Product

68 Special Cases for Cross Product If A & B are parallel or antiparallel, ϕ=0 or 180, sinϕ=0, A B=0

69 Special Cases for Cross Product If A & B are parallel or antiparallel, ϕ=0 or 180, sinϕ=0, A B=0 The vector product of any vector with itself is zero

70 Special Cases for Cross Product If A & B are parallel or antiparallel, ϕ=0 or 180, sinϕ=0, A B=0 The vector product of any vector with itself is zero

71 Vector product vs Scalar product magnitude of dot product is A B = AB cosφ magnitude of cross product is A B = ABsinφ Be careful not to confuse scalar and vector products as the expressions for their magnitude look similar.

72 Vector product vs Scalar product magnitude of dot product is A B = AB cosφ magnitude of cross product is A B = ABsinφ Be careful not to confuse scalar and vector products as the expressions for their magnitude look similar. When A & B are parallel: the magnitude of the scalar product is a maximum the magnitude of the vector product is zero

73 Vector product vs Scalar product magnitude of dot product is A B = AB cosφ magnitude of cross product is A B = ABsinφ Be careful not to confuse scalar and vector products as the expressions for their magnitude look similar. When A & B are parallel: the magnitude of the scalar product is a maximum the magnitude of the vector product is zero When A & B are perpendicular: the magnitude of the scalar product is zero the magnitude of the vector product is a maximum

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76 Example Problem: Adding Displacement Vectors Jenny runs 1 mi to the northeast, then 1 mi south. Graphically find her net displacement.

77 Velocity Vectors

78 Example: Velocity Vectors Jake throws a ball at a 60 angle, measured from the horizontal. The ball is caught by Jim. Draw a motion diagram of the ball with velocity vectors.

79 l2.m clear;clc;close all % Let us define two vectors a & b a = [1 2 3]; b = [4 5 6]; % Find the dot or scalar product c = dot(a,b) % Find the cross or vector product d = cross(a,b) whos %define origin o=[0 0 0]; % plot a circle at the origin plot3(0,0,0,'og','markersize',3) % ensure all subsequent plot commands overlay on same plot hold on drawvector(o,a) drawvector(o,b) drawvector(o,d) hold off grid on c = 32 d = Name Size Bytes Class Attributes a 1x3 24 double b 1x3 24 double c 1x1 8 double d 1x3 24 double

80 l2simpledrawvector.m % clear the memory, command window, and close all open figures clear;clc;close all % Let us define two vectors a & b a = [6 0]; b = [4*cosd(30) 4*sind(30)]; % Find the dot or scalar product c = dot(a,b) % get a list of all the current variables whos %define origin o=[0 0]; % % Plot vectors plot(0,0,'og') hold on drawvector(o,a) drawvector(o,b) hold off grid on xlim([-1 7]) ylim([-1 3]) xlabel('x','fontsize',25) ylabel('y','fontsize',25) c = Name Size Bytes Class Attributes a 1x2 16 double b 1x2 16 double c 1x1 8 double

81 l2simpledrawvector.m % % we can explicitly specify the size of the window we open and where it is % on the screen width=800; height=800; left=1; bottom=1; figure('position',[left, bottom, width, height]) % % Plot vectors % plot a circle at the origin plot(0,0,'og','markersize',20) hold on drawvector(o,a) drawvector(o,b,'r') hold off grid on % Annotate the angle between a & b text(1,0.3,'\theta',... 'HorizontalAlignment','left','FontSize',20) % Annotate vectors text(3,-0.25,'$\vec{a}$','interpreter','latex','fontsize',20) text(1.8,1.5,'$\vec{b}$','interpreter','latex','fontsize',20)

82 l2simpledrawvector.m % % we can explicitly specify the size of the window we open and where it is % on the screen width=800; height=800; left=1; bottom=1; figure('position',[left, bottom, width, height]) % % Plot vectors % plot a circle at the origin plot(0,0,'og','markersize',20) hold on drawvector(o,a) drawvector(o,b,'r') hold off grid on % Annotate the angle between a & b text(1,0.3,'\theta',... 'HorizontalAlignment','left','FontSize',20) % Annotate vectors text(3,-0.25,'$\vec{a}$','interpreter','latex','fontsize',20) text(1.8,1.5,'$\vec{b}$','interpreter','latex','fontsize',20) % Make a directory for the plots plot_dir='plots/'; mkdir(plot_dir)

83 l2simpledrawvector.m % % we can explicitly specify the size of the window we open and where it is % on the screen width=800; height=800; left=1; bottom=1; figure('position',[left, bottom, width, height]) % % Plot vectors % plot a circle at the origin plot(0,0,'og','markersize',20) hold on drawvector(o,a) drawvector(o,b,'r') hold off grid on % Annotate the angle between a & b text(1,0.3,'\theta',... 'HorizontalAlignment','left','FontSize',20) % Annotate vectors text(3,-0.25,'$\vec{a}$','interpreter','latex','fontsize',20) text(1.8,1.5,'$\vec{b}$','interpreter','latex','fontsize',20) % Make a directory for the plots plot_dir='plots/'; mkdir(plot_dir) % Send plots to files fn=[plot_dir '2DVectorDotProduct']; % Print to png file fnpng=[fn,'.png']; print('-dpng',fnpng); % Print to ps file fnps=[fn,'.eps'];print('-depsc2',fnps);

84 % Let us define two vectors a & b a = [6 0] b = [4*cosd(30) 4*sind(30)] r=a+b l2vectoraddition.m %define origin o=[0 0]; % % Plot vectors plot(0,0,'og') hold on drawvector(o,a,'g') drawvector(a,a+b,'b') drawvector(o,r,'r') hold off grid on see LatexSymbols.pdf xlim([-1 11]) ylim([-1 3]) xlabel('x','fontsize',25) ylabel('y','fontsize',25) % Annotate vectors text(3,-0.25,'$\vec{a}$','interpreter','latex','fontsize',20) text(7.8,.75,'$\vec{b}$','interpreter','latex','fontsize',20) text(3,1.25,'$\vec{a}+\vec{b}$','interpreter','latex','fontsize',20) title(['vector addition, $\vec{a}+\vec{b}$'],'interpreter','latex','fontsize',25) % Set default line width set(gca,'linewidth',1) % set axis tick mark direction set(gca,'tickdir','out') % set default font size, primarily for tick mark labels set(gca,'fontsize',16) % Make a directory for the plots plot_dir='plots/'; mkdir(plot_dir) % Send plots to files fn=[plot_dir '2DVectorAddition']; % Print to png file fnpng=[fn,'.png']; print('-dpng',fnpng); % Print to ps file fnps=[fn,'.eps'];print('-depsc2',fnps); a = 6 0 b = r = Warning: Directory already exists. Y Vector addition, A + B A + B B A X

85 title(['$\vec{c} = \vec{a} \times \vec{b}$'],'interpreter','latex','fontsize',25) Operators SymbolCommand SymbolCommandSymbolCommand \pm \mp \times \div \cdot \ast \star \dagger \ddagger \amalg \cap \cup \uplus \sqcap \sqcup \vee \wedge \oplus \ominus \otimes \circ \bullet \diamond \lhd \rhd \unlhd \unrhd \oslash \odot \bigcirc \triangleleft \Diamond \bigtriangleup \bigtriangledown \Box \triangleright \setminus \wr Relations SymbolCommand SymbolCommandSymbolCommand \le \ge \neq \sim \ll \gg \doteq \simeq \subset \supset \approx \asymp \subseteq \supseteq \cong \smile \sqsubset \sqsupset \equiv \frown \sqsubseteq \sqsupseteq \propto \bowtie \in \ni \prec \succ \vdash \dashv \preceq \succeq \models \perp \parallel \ \mid Latex Symbols Lowercase Letters SymbolCommandSymbolCommand SymbolCommandSymbolCommand \alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta \theta \vartheta \iota \kappa \lambda \mu \nu \xi \pi \varpi \rho \varrho \sigma \varsigma \tau \upsilon \phi \varphi \chi \psi \omega

86 publish('l2dotcross.m','pdf')