Chapter 3: Two Chapter 3: Two Dimensional Motion and Dimensional Motion and

VectorsVectors

(Now the fun really starts)(Now the fun really starts)

Opening QuestionOpening Question

I want to go to the library. How do I I want to go to the library. How do I get there?get there?

Things I need to know:Things I need to know:– How far away is it?How far away is it?– In what direction(s) do I need to go? In what direction(s) do I need to go?

One dimensional motion vs two One dimensional motion vs two dimensional motiondimensional motion

One dimensional motion: Limited to One dimensional motion: Limited to moving in one dimension (i.e. back moving in one dimension (i.e. back and forth or up and down)and forth or up and down)

Two dimensional motion: Able to Two dimensional motion: Able to move in two dimensions (i.e. forward move in two dimensions (i.e. forward then left then back)then left then back)

Scalars and VectorsScalars and Vectors

Scalar: A physical quantity that has Scalar: A physical quantity that has magnitude but no directionmagnitude but no direction– Examples:Examples:– Speed, Distance, Weight, VolumeSpeed, Distance, Weight, Volume

Vector: A physical quantity that has Vector: A physical quantity that has both magnitude and directionboth magnitude and direction– Examples:Examples:– Velocity, Displacement, AccelerationVelocity, Displacement, Acceleration

Vectors are represented by Vectors are represented by symbolssymbols

Book uses Book uses boldfaceboldface type to indicate type to indicate vectorsvectors

Scalars are designated with Scalars are designated with italicsitalics

Use arrows to draw vectorsUse arrows to draw vectors

Vectors can be added graphicallyVectors can be added graphically

When adding vectors make sure that When adding vectors make sure that the units are the samethe units are the same

Resultant vector: A vector Resultant vector: A vector representing the sum of two or more representing the sum of two or more vectorsvectors

Adding Vectors GraphicallyAdding Vectors Graphically

Draw situation using a reasonable Draw situation using a reasonable scale (i.e. 50 m = 1 cm)scale (i.e. 50 m = 1 cm)

Draw each vector head to tail using Draw each vector head to tail using the right scalethe right scale

Use a ruler and protractor to find the Use a ruler and protractor to find the resultant vectorresultant vector

Example: p. 85 in textbookExample: p. 85 in textbookA student walks from his house to his friend’s house (a) then from his friend’s house to school (b). The resultant displacement (c) can be found using a ruler and protractor

Properties of vectorsProperties of vectors

Vectors can be added in any orderVectors can be added in any order

To subtract a vector add its oppositeTo subtract a vector add its opposite

Coordinate SystemsCoordinate Systems

To perform vector To perform vector operations operations algebraically we algebraically we must use must use trigonometrytrigonometry

SOH CAH TOASOH CAH TOA

Pythagorean Pythagorean Theorem:Theorem:

h

o)sin(

h

a)cos(

a

o)tan(

222 bahyp

Vectors have directionsVectors have directions

South

West

North

East of N

orth

North of East East

West of North

South of W

est

South of East

East of South

West

of South

North of West

Examples p. 91 #2Examples p. 91 #2

While following directions on a While following directions on a treasure map, a pirate walks 45.0 m treasure map, a pirate walks 45.0 m north then turns around and walks north then turns around and walks 7.5 m east. What single straight-line 7.5 m east. What single straight-line displacement could the pirate have displacement could the pirate have taken to reach the treasure?taken to reach the treasure?

Solving the problemSolving the problem Use the Pythagorean Use the Pythagorean

theoremtheorem

RR22 = (7.5 m) = (7.5 m)22 + (45m) + (45m) 22

R= 45.6 m…R= 45.6 m…

What are we missing??What are we missing??

45 mN

7.5 m east

Resultant=?

Find the directionFind the direction Can’t say it’s just Can’t say it’s just

NE because we NE because we don’t know the don’t know the value of the anglevalue of the angle

Find the angle Find the angle using trigusing trig

45 mN

7.5 m east

Resultant=?

a

o)tan(

What is the angle? What is the angle? (Make sure your calculator is in Deg (Make sure your calculator is in Deg

not Rad)not Rad) Use inverse tangentUse inverse tangent

Final Answer:Final Answer:

46.5 m at 9.46° East of 46.5 m at 9.46° East of North North oror 46.5 m at 46.5 m at 80.54 ° North of East 80.54 ° North of East

46.9

45

5.7tantan 11

a

o

45 mN

7.5 m east

Resultant=46.5 m

Θ=80.54 °°

Θ=9.46°°