physics 218, lecture v1 physics 218 lecture 5 dr. david toback

Physics 218, Lecture V 1

Physics 218Lecture 5

Dr. David Toback


Chapter 3

• Kinematics in Two Dimensions

• Vectors


Overview

• Motion in multiple dimensions• Vectors: (Tools to solve problems)

– Why we care about them– Addition, Subtraction and Multiplication

• Graphical and Component– Unit Vectors

• Projectile Motion (Next lecture)– Problem Solving


Why do we care about Vectors?• Last week we worked in one dimension• However, as you may have noticed, the world is

not one-dimensional. • Three dimensions: X, Y and Z. Example:

1. Up from us2. Straight in front of us3. To the side from us

– All at 90 degrees from each other. Three dimensional axis.

– Need a way of saying how much in each direction

For this we use VECTORS


Another reason to care about vectors

• It turns out that nature has decided that the directions don’t really care about each other.

• Example: You have a position in X, Y and Z. If you have a non-zero velocity in only the Y direction, then only your Y position changes. The X and Z directions could care less. (I.e, they don’t change).

Represent these ideas with Vectors


• Vector notation: – In the book,

variables which are vectors are in bold

– On the overheads, I’ll use an arrow over it

• Vectors are REALLY important

• Kinda like calculus: These are the tools!

First the Math: Vector Notation

v

Some motion represented by vectors. What do these vectors represent

physically?


Vector Addition• Why might I need to add vectors?

– If I travel East for 10 km and then North for 4 km, it would be good to know where I am. What is my new position?

• Can think of this graphically or via components

– Graphically:• Lay down first vector (the first

part of my trip)• Lay down second vector (the

second part of the trip) with its tail at the head of the first vector

• The “Sum” is the vector from the tail of the first to the head of the second

First

Second

Sum

Adding vectors is a skill Use this in far more

than just physics


Examples without an axis


Multiplication

• Multiplication of a vector by a scalar

• Let’s say I travel 1 km east. What if I had gone 4 times as far in the same direction? – Just stretch it out, multiply the magnitudes

• Negatives: – Multiplying by a negative number turns the

vector around


Subtraction

• Subtraction is easy: – It’s the same as addition but turning around one

of the vectors. I.e., making a negative vector is the equivalent of making the head the tail and vice versa. Then add:

)V(- V V V 1212


The tricky part• We saw that if you travel

East for 10 km and then North for 4 km, you end up with the same displacement as if you traveled in a straight line NorthEast.

• Could think of this the other way: If I had gone NorthEast, it’s the equivalent of having gone both North and East.

My single vector in some funny direction, can be thought of as

two vectors in nice simple directions (like X and Y). This

makes things much easier.


Components• This is the tricky part that separates the good

students from the poor students• Break a vector into x and y components (I.e, a

right triangle) THEN add them• This is the sine and cosine game• Can use the Pythagorean Theorem: A2 + B2 = C2

Again, this is a skill. Get good at this!!!


Adding Vectors by Components

How do you do it?• First RESOLVE the vector by its

components! Turn one vector into two• V = Vx + Vy

Vx = VcosVy = Vsin

• Careful when using the sin and cos


Specifying a Vector

• Two equivalent ways: – Components Vx and Vy

– Magnitude V and angle

• Switch back and forth– Magnitude of V

|V| = (vx2 + vy

2)½

(Pythagorean Theorem)

– Tan = vy /vx


Example

What is the magnitude and angle of the displacement in this example?


Adding vectors in funny directions• Let’s say I walk in some random direction, then in another different

direction. How do I find my total displacement? • We can draw it

• It would be good to have a better way…


Addition using Components

This is the first half of how pros do it: To add two vectors, break both up into their X and Y components, then add separately

2y1yy

2x1xx

21

VVV

VVV

V V V

Magnitudes


Drawing the components


Unit Vectors

This is how the pros do it!

kV jV iV V

direction z in the means ˆ

directiony in the means ˆ

direction x in the means ˆ

zyx k

j

i


Simple Example

What is the displacement using Unit Vectors in this example?

k km 0 j km 5 i km 10 D



DD D

R

2

1

21R


Example: Adding Unit Vectors

12

21

2222

1111

RR RRR R:Calculate

kZ jY iX RkZ jY iX R


Mail Carrier and Unit Vectors

A rural mail carrier leaves leaves the post office and drives D1 miles in a Northerly direction to the next town. She then drives in a direction South of East for D2 miles to another town. Using unit vector notation, what is her displacement from the post office?


Vector Kinematics Continued

/dt kd /dt jd 0 /dt id used have We

:Note

kV jV iV

kdZ/dt)( jdY/dt)( i(dX/dt)

)/dtkZ jY id(X

/dtRd V

zyx


Constant Acceleration

ta v v

ta v v

and

tat v y y

tat v x x

as same theis

ta v v

tat v r R

y0yy

x0xx

2y2

10y0

2x2

10x0

0

221

00


Projectile Motion

• This is what all the setup has been for!

• Motion in two dimensions–For now we’ll ignore air friction


Projectile Motion

The physics of the universe:

The horizontal and The horizontal and vertical parts of the vertical parts of the

motion behave motion behave independentlyindependently

This is why we use vectors in the first place


Ball Dropping

• Analyze Vertical and Horizontal separately!!!

• Ay = g (downwards)

• Ax = 0

– Constant for Both cases!!!

Vx = 0 Vx>0


A weird consequence

An object projected horizontally will reach the ground at the same time as an object dropped vertically.

Proof:


Rest of this week

• Reading: Finish Chapter 3 if you haven’t already

• Homework: – Finish HW2 and be working on HW3

• Web quiz: If you don’t have ten 100’s yet, I recommend you do so before the exam (coming up!)

• Labs and Recitations: Both meet this week.• Next time: More on kinematics in two

dimensions and vectors


A Mail Carrier

A rural mail carrier leaves leaves the post office and drives D1 miles in a Northerly direction to the next town. She then drives in a direction degrees South of East for a distance D2 to another town.

What is the magnitude and angle of her displacement from the post office?


Vector stuff

1. Pythagorean theorem: We’ll use this a lot– For a right triangle (90 degrees)

– Length C is the hypotenuse

– A2 + B2 = C2

2. Vector equations

321321

1221

V )V V( )V V( V

V V V V


Using all this stuff

k)Z-Z( j)Y-Y( i)X-(X R

RR R

k)ZZ( j)YY( i)X(X R

RR R

kZ jY iX R

kZ jY iX R

121212

12

121212

21

2222

1111

physics 218, lecture v1 physics 218 lecture 5 dr. david toback

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