Chapter 4: Motion in 2 and 3 dimensions

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Chapter 4: Motion in 2 and 3 dimensions

• To introduce the concepts and notation for vectors: quantities that, in a single package, convey magnitude

and direction motion• To generalize our 1d kinematics to higher numbers of

dimensions, using vector notation• To generalize Newtonian dynamics using vectors

• To appreciate the glory of the Free Body Diagram and how it enables one to utilize N2

• To discuss projectile motion, and motion on a curve

Chapter 4 Goals:


Polar coordinates in 2 dimensions

The position of a point • (x,y) are the cartesian coordinates• two numbers needed to specify full information• convention for is to start along x and swing counterclockwise

r hypotenusex adjacenty opposite

(only) distance 22 yxr


Polar coordinates in 2 dimensions

r hypotenusex adjacenty opposite

ho

a

22 yxr

0] if 180

Arctan

tan

sinsin

coscos

x

xya

o

ryho

rxha


The displacement vector: we move into a higher dimension at last!

• initial position is ri

• final position is rf

• displacement is r:= rf – ri

x

ri

rf

r

• Notice : the positions are origin-dependent, but the displacement is not!!

y

• If it behaves like the displacement vector, then it is a vector!! It is the ‘paradigmatic’ vector.• vector subtraction is a bit tricky


Obvious language for vectors in two dimensions: magnitude and direction

x

y

\A\

A

• magnitude of A: written |A| (some books say A)• it is just the length of A• |A| is always positive or 0• technically it is not a scalar but don’t worry about that• |A| carries the units of A

• direction of A: written A

• usually counterclockwise from x axis• units of A are degrees or radians


Pros of magnitude-direction language• we can write A = {|A|,A}• lends itself to obvious pictures• can easily be converted to compass (map) language: East x, and North y• addition and subtraction of vectors is done pictorially

• accuracy of pictorial addition and subtraction is limited to human ability

• laws of sines and cosines needed to calculate: messy• only convenient in 2d!! 3d requires the dreaded spherical trigonometry because there are 2 angles!!

Cons of magnitude-direction language


Triangle method for adding vectors

• A (and B) are two displacement vectors with B following A• magnitude : |A| or A• direction: an angle , just like in previous figure•A is followed by B, to give resultant R = A + B• ‘tip-to-tail’ triangle method of adding vectors• One can just keep going, tip to tail style


Rules for adding vectors (and to multiply by scalar)• A + B = B + A: vector addition is commutative• (A + B) + C = A + (B + C): vector addition is associative • the negative of a vector is – A: same length as A but opposite direction [additive inverse]• There is a zero vector (no length)• vectors carry units and when added units mus be the same• ‘tip-to-tail’ triangle method of addition• cA: also a vector (opposite direction) if c<0 but ‘rescaled’ in length: |cA| = |c| |A|


How to subtract one vector from another• A – B := A + (– B): to subtract, just add the additive inverse• put both A and B tail-to-tail: then C = A – B has its tail at the tip of B (the one subtracted) and has its tip at the tip of A (the one added)• it seems to ‘start’ at B and end at A• This procedure is extremely important when finding changes in vectors!!!• Example: the displacement vector is precisely the change in the position vector!!

{show Active Figure AF_0306


Addition Example : Taking a Hike

A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.

• Using ruler and protractor, lay out the two displacements A and B. Then, with the same tools, measure and then


A different and more flexible language: components

x

y

|A|

A

Ax

Ay

•in this sketch, A is larger than 90

but you can deal…• |A| is the hypotenuse; |A| ≥ 0• Ax is the adjacent and here < 0

• Ay is the opposite and here > 0

•“drop a perpendicular” from tip to either Cartesian axis• you have made a right triangle

• draw vector with tail at origin: standard position

• we write A = < Ax , Ay >


Converting from magnitude-angle to components

Converting from magnitude-angle to components

• This assumes that the angle is defined as ccw from x axis

Pros of component language• addition and subtraction of two (or more) vectors is simplicity itself: just add (subtract) the components like scalars and the resultant vector‘s (difference vector’s) components are known!

{show Active Figure AF_0303}

Ayy

A

Axx

A

AA

ho

AAh

a

sinsin

coscos

AA

AA

x

yAyx A

AAA tan22A


Revisiting the vector addition example

Section 3.4

km 16.9 km, 37.7

km 9.16

km 6.34km 7.17

km 7.37

km 20km 7.17

R

yyy

xxx

BAR

BAR

km 7.17.707- km 25315in km 25sin

km 7.17.707 km 25315cos km 25cos

sA

A

Ay

Ax

A

A

km 6.34.866 km 4060in km 40sin

km 20.0 .500 km 4060cos km 40cos

sB

B

Ay

Ax

B

B


Finishing the Example

Section 3.4

• Now we would work back from component language to magnitude-angle language:

1.204

1.24.448Arctan448.

km 37.7km 9.16tan

R

x

yR R

R

• ‘net displacement is 41.3 km, at a bearing of 65.9° East of North’• R = {41.3 km, 24.1°}

km 3.41km 1706km 286km 1420

km 6.91km 7.37

21

221

22

2222

yx RRR


A third language that uses components: unit vectors

x

y

|A|

A

• i is a vector of unit length, with no units, that points along x• j and k are similar, along y (and z)• create the vectors i Ax & j Ayj Ay

i Ax

• A truly explicit way to write A• remember: |i| = |j| = |k| = 1• (no units, and but one unit long!!)

) as written (also

) ˆ( ˆˆ

A

kjiA

zyx AAA


What other operations can we do with vectors?• cannot divide by a vector; vectors are only ‘upstairs’• dot (scalar (inner)) product of two• cross (vector (outer (wedge))) product of two• we can take their derivative with respect to a scalar• we can integrate them but usually we integrate some kind of scalar product…

The scalar product of two vectors

A

B

• put them tail-to-tail, with the angle between (0° ≤ ≤ 180°)

cos: BABA

• the result is indeed a scalar• A∙B = B ∙A (commutative)


More about the scalar product• a measure of the parallelness of the vectors, as well as the magnitudes• A∙B = (|A|)(|B| cos ) = length of A times the length of B’s projection along the line of A• A∙B = (|B|)(|A| cos ) = length of B times the length of A’s projection along the line of B• A∙A = |A|2

• i∙i = j∙j = k∙k =1 i∙j = j∙k = 0 etc.• a vector’s component in a certain direction is the scalar product of that vector with a unit vector in that direction: Cn = C∙n • [to make a unit vector, just divide a vector by its magnitude: a = A/|A| ]


Kinematics in ‘higher dimensions’ I:position and displacement as vector

• position is now a vector: kjir ˆˆˆ: zyx

• the three components of the position: r:= <x, y, z>• if the arena is 2d, drop the z-stuff• displacement is change in position: r = rB – rA or

r = rf – ri or, best of all, r = r(t+t) – r(t)• it is a vector with tail at r(t) and tip at r(t+t)• displacement vector not usually drawn in standard position but may be, especially if you are adding a second displacement to the first and you don’t really care about ‘initial’ position, or you take that to be zero• the displacement vector is origin-independent!


Kinematics in ‘higher dimensions’ II: the (instantaneous) velocity vector

• here we used the fact that the unit vectors do not vary with time as the body moves around•the three components of the velocity v:= <vx, vy , vz >

where vx(t)=dx(t)/dt and similarly for y and z

• velocity is now a vector:

0ˆ0ˆ0ˆ

ˆˆ

ˆˆ

ˆˆ

argument] '' [drop lim:)()(0

kji

kk

jj

ii

rrv

dt

dz

dt

dy

dt

dxdt

dz

dt

dz

dt

dy

dt

dy

dt

dx

dt

dx

tt

tdt

dt

t


Kinematics in ‘higher dimensions’ III

x

y

r(t)

r(t+t)

r(t)

path of object

• how do we understand this?• start with average velocity: of course, <v> = r/t

• <v> is a vector: it’s a vector (r) times a number (1/t)• magnitude: |<v>| = |r |/|t|• direction: same as direction of r


Kinematics in ‘higher dimensions’ IV

x

y

r(t)r(t+t)

r(t)

path of object

• we now let t get really small (t 0) and call it dt• as that shrinks, so does r : call it dr

r(t+dt)

dr(t)

• direction: tangent to the path in space!!!

• magnitude: |v| = |dr |/dt| = speed 222zyx vvv v

Chapter 4: Motion in 2 and 3 dimensions

Documents

Transcript of Chapter 4: Motion in 2 and 3 dimensions