P221 Lecture 6

Vectors

11 May 2006Reading for today: Giancoli chapter 2, sections 1-3

Mark MessierDepartment of PhysicsIndiana University

My bottleneck: Vectors

• A few definitions I would expect my students to know:

I. Vector quantities have magnitude and direction

II. Position, velocity, and acceleration are vector quantities which are related. Velocity generates a change in position, acceleration generates a change in velocity.

• My bottleneck:

• In Physics I several students retain patterns of thought that work for motion in one dimension that do not work when applied to multi-dimensions.

I. Common misconception: vector quantity ⇔ representation

II. Common misconception: vector quantity ⇔ magnitude.

Review: Vectors in one page. I would expect students to have some familiarity with the ideas summarized here.

• Vectors are mathematical objects that express a magnitude and a direction. We need them to work with motion in more than one dimension

• Notation:

• They can be represented graphically as an arrow and by components with respect to a chosen coordinate system

• Operations using scalar s and vectors a, b, and c

‣ Addition/subtraction:

‣ Multiplication/division:

a vector “v”

the magnitude of the vector “v”

a unit vector defining the “x” direction

#2#1

Conceptual example: What do you think of these clips?

Hate it,Wicked boring,Awful noise ...

Love it!

Take it or leave it

-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5

On your first index card, please write your first name and your reaction to the two samples, in order, according to this scale:

French suite #2 in C minor, BMV 813, J.S. BachPerformed by Joanna MacGregor

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

“American Idiot” from American Idiot,Performed by Green Day

Questions and analysis (1): What have I done?

1) I have imagined that there exists a “musical taste space” and that your tastes in music are a point in that space

2) I have made a (crude) measurement of the position of your taste in that space

3) I can represent taste in music in several ways:

(I) Symbolically:

(II)Graphically:

(III)By components:

Q2: Of the three representations which would change if I used different clips?Take away: Vectors exist independent of their

representation; the same vector can have many representations. The results of any calculations using vectors must be independent of the representation used.

Q1: By doing this exercise, did I create your taste in music?

Questions and analysis (2):What can we do?

4)What meaning can we assign to the direction of ? What meaning can we assign to its magnitude?

5)What meaning does multiplication by a scalar s have for s>1? 0<s<1? s<0?

6)What is represented by ?

7)Suppose your tastes change. What is represented by ?

8)What does it mean if is large and positive? Large and negative?

9)What does it mean if is large? zero?

Connecting to the physics of motion...On your second index card, can you draw position, velocity, and acceleration vectors corresponding to this situation:

“I’m a big Green Day fan, although recently I find myself listening to them less and less. Bach’s music is a little too tame for me, but I’m taking this course in classical music which I think is going to make me appreciate his music more.”

Please be careful to label which three vectors in your picture represent the position, velocity, and acceleration.

Next time: Projectile motion