2D Motion

Vectors

Lana Sheridan

De Anza College

Jan 14, 2020

Last time

• falling objects

• varying acceleration
Overview

• vectors

• addition and subtraction of vectors

1-D Kinematics with varying acceleration
An asteroid falls in a straight line toward the Sun, starting from
rest when it is 1.00 million km from the Sun. Its acceleration is
given by, a = − xk2 where x is the distance from the Sun to the
asteroid, and k = 1.33 × 1020 m3 /s2 is a constant.

After it has fallen through half-a-million km, what is its speed?

1-D Kinematics with varying acceleration
An asteroid falls in a straight line toward the Sun, starting from
rest when it is 1.00 million km from the Sun. Its acceleration is
given by, a = − xk2 where x is the distance from the Sun to the
asteroid, and k = 1.33 × 1020 m3 /s2 is a constant.

After it has fallen through half-a-million km, what is its speed?

0
Z xf
v2 2
vi7
− 
= a dx
2 xi
Z xf  
k
v2 = 2 − 2 dx
xi x
1-D Kinematics with varying acceleration
An asteroid falls in a straight line toward the Sun, starting from
rest when it is 1.00 million km from the Sun. Its acceleration is
given by, a = − xk2 where x is the distance from the Sun to the
asteroid, and k = 1.33 × 1020 m3 /s2 is a constant.

After it has fallen through half-a-million km, what is its speed?

0
Z xf
v2 2
vi7
− 
= a dx
2 xi
Z xf  
k
v2 = 2 − 2 dx
xi x
 
k k
v2 = 2 −
xf xi
v = 5.16 × 105 m/s
Math you will need for 2-Dimensions

Before going into motion in 2 dimensions, we will review some

things about vectors.
Vectors

scalar
A scalar quantity indicates an amount. It is represented by a real
number. (Assuming it is a physical quantity.)

vector
A vector quantity indicates both an amount and a direction. It is
represented more than one real number. (Assuming it is a physical
quantity.)

There are many ways to represent a vector.

• a magnitude and (an) angle(s)
• magnitudes in several perpendicular directions
Representing Vectors: Angles
Bearing angles
Example, a plane flies 750 km h−1 N
at a bearing of 70◦

Generic reference angles

A baseball is thrown at 10 m s−1 , 30◦ above the horizontal.
Representing Vectors: Unit Vectors

Magnitudes in several perpendicular directions: using unit vectors.

Unit vectors have a magnitude of one unit.

Representing Vectors: Unit Vectors

Magnitudes in several perpendicular directions: using unit vectors.

Unit vectors have a magnitude of one unit.

A set of perpendicular unit vectors defines a basis or decomposition

of a vector space.
Representing Vectors: Unit Vectors

Magnitudes in several perpendicular directions: using unit vectors.

Unit vectors have a magnitude of one unit.

A set of perpendicular unit vectors defines a basis or decomposition

of a vector space.

In two dimensions, a pair of perpendicular unit vectors are usually

denoted î and ĵ (or sometimes x̂, ŷ).
um of two other component vectors A x , which is parallel to the x axis, and A y , whic
parallel to the y axis. S
Components From SFigure S
3.12b, we see that the three vectors form
ight triangle
S
and that A 5 A x 1 A y . We shall often refer to the “component
f a vector A ,” written A x and A (without
y vector
#» the boldface notation). The compo
Consider the 2 dimensional S A = Ax î + Ay ĵ, where Ax and
ent Ax represents the projection
S
of A along the x axis, and the component A
A are numbers.
epresentsy the projection of A along the y axis. These components can S
be positiv
r negative. The component Ax is positiveSif the component vector A#»x points i
We then say that Ax is the i-component (or x-component) of A
he positive x direction and is negative if A x points in the #» negative x direction.
and Ay is the j-component (or y -component) of A.
milar statement is made for the component Ay.

y y

S S
S A A S
Ay Ay

u u
S
x S
x
O Ax O Ax
a b
Notice that Ax = A cos θ and Ay = A sin θ.
 S
r negative. The component Ax is positiveSif the component vector A x points i
he Components
positive x directionvs Magnitude-and-Angle
and is negative if A x points in theNotation
negative x direction.
milar statement is made for the component Ay.
Notice that Ax = A cos θ and Ay = A sin θ.
y y

S S
S A A S
Ay Ay

u u
S
x S
x
O Ax O Ax
a b
Also notice,
#» q
A = | A| = A2x + A2y
and  
Ay
θ = tan−1
Ax
if the angle is given as shown.
Why Vectors?

Of course, there is no reason to limit this to two dimensions.

Why Vectors?

Of course, there is no reason to limit this to two dimensions.

With three dimensions we introduce another unit vector k̂ = ẑ.

And we can have as many dimensions as we need by adding more
perpendicular unit vectors.
Why Vectors?

Of course, there is no reason to limit this to two dimensions.

With three dimensions we introduce another unit vector k̂ = ẑ.

And we can have as many dimensions as we need by adding more
perpendicular unit vectors.

Vectors are the right tool for working in higher dimensions.

Why Vectors?

Of course, there is no reason to limit this to two dimensions.

With three dimensions we introduce another unit vector k̂ = ẑ.

And we can have as many dimensions as we need by adding more
perpendicular unit vectors.

Vectors are the right tool for working in higher dimensions.

They have a property that correctly reflects what it means for

there to be more than one dimension: that each perpendicular
direction is independent of the others.
Why Vectors?

Of course, there is no reason to limit this to two dimensions.

With three dimensions we introduce another unit vector k̂ = ẑ.

And we can have as many dimensions as we need by adding more
perpendicular unit vectors.

Vectors are the right tool for working in higher dimensions.

They have a property that correctly reflects what it means for

there to be more than one dimension: that each perpendicular
direction is independent of the others.

This makes life much easier: we will be able to solve for motion in
the x direction separately from motion in the y direction.
nstant. If the position vector is known, the velocity of the particle
from Equations 4.3
Visualizing and 4.6, in
Motion which give
2 Dimensions
S dSr airdx dy
Imagine
v 5 an 5 i^ 1 puck
hockey j^ 5 moving
v x ^i 1 v ywith
^j (4.7)
horizontally constant
velocity:dt dt dt

y
d vectors,

velocity, x
a Figure
h in
across a
gure,
table at
es that
directio
ensions y in the y
two
puck, th
ons in
x ponent
ections.
ponent
b in the p

If it experiences a momentary upward (in the diagram)

acceleration, it will have a component of velocity upwards.
The horizontal motion remains unchanged!
 When two vectors are added, the sum is indepe
Vectors Properties and Operations
tion. (This fact may seem trivial, but as you will s
important when vectors are multiplied. Procedures
cussed in Chapters 7 and 11.) This property, which c
Equality construction in Figure 3.8, is known as the commut
#» #» S S S S
Vectors A=B
Commutative lawifofand only if the magnitudes and directions
addition A 1areB the
5 B 1 A
same. (Each component is the same.)

Addition
#» #»
A+B
S
D

S
D
C "
S
B "
S
S C

S
S B
A"

A "
S
R! S

S
B

R !
S S
B
S S
A A
S
Figure 3.6 When vector B is
S S
Figure 3.7 Geometric construc-
added to vector A , the resultant R is tion for summing four vectors. The
S
the vector that runs from the tail of resultant vector R is by definition
S S
 construction in Figure 3.8, is known as the commutative law of addition:
S S S S
Vectors Properties and
f addition A 1 BOperations
5 B 1 A (3.5)

Properties of Addition Draw B ,

S
S

then add A . S
A

#» #» #» #»
• A + B = B + A (commutative)

S
B
S

A"
D

S
A!
S

S
D
S B
B

B"
C "

R! S
S

S
B "
S
S C
S

B
A "

S
S

B
R !

S
S

S A
B S
Draw A ,
S
S then add B .
#» #» #»A #» #» #»
S
vector B is • ( A + B) +3.7CGeometric
Figure =A+ ( B + C) (associative)
construc- Figure 3.8S ThisSconstruction
shows that A 1 B 5 B 3.3 1 A Some Properties of Vectors
S S S
he resultant R is tion for summing four vectors. The or, in
S
from the tail of resultant vector R is by definition other words, that vector addition is
the one that completes the polygon. commutative.
S S S S Figure 3.9 Geo
Add B and C ; Add A and B;
S tions for verifyin
then add the then add C to
S law of addition.
result to A. the result.
S S
C C
S

S
C)

C
!

!
(B S

S
B)

S S
!

B!C
!
S

S S
(A S

A!B
A

S S
B B
S S
A A
he bookkeeping
nents separately.
Vectors Properties and Operations
Doing addition:
Almost always the right answer is to break each vector into
components and sum each component independently.

(3.14) y

ant vector are

S
By R S
(3.15) Ry
B

s, we add all the Ay S

A
ctor and use the x
omponents with Ax Bx

Rx
btained from its
Figure 3.16 This geometric
Vectors Properties and Operations

Doing addition:
Almost always the right answer is to break each vector into
components and sum each component independently.
Vector Addition Example
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. What is the magnitude
#»
and direction of the hiker’s resultant displacement R for the trip?

0
Based on S&J Example 3.5, pg 69.
 2
y 1 R z2
Vector Addition Example
A hiker begins a trip by first walking 25.0 km southeast from her
2
1 1 31 cm 2 2 1 1 7.0 cm 2 2 5 40 cm
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. What is the magnitude
#»
and direction of the hiker’s resultant displacement R for the trip?
N
om her car. She stops
e walks 40.0 km in a W E
forest ranger’s tower. y (km) S

for each day. 20

Tower
S
10 R S
B

a sketch as in Figure 0 x (km)

S Car
45.0! 20 30 40
econd days by A and
" 10 S
es, we obtain the vec- A 60.0!
he resultant vector as " 20 Tent

egorize this
0
problem
Based Figure3.5,
on S&J Example 3.17pg 69.
(Example 3.5) The
Vector Addition Example
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. What is the magnitude
#»
and direction of the hiker’s resultant displacement R for the trip?

Ax = A cos(−45.0◦ ) = 17.7 km
Ay = A sin(−45.0◦ ) = −17.7 km

Bx = B cos(60.0◦ ) = 20.0 km
Ay = A sin(60.0◦ ) = 34.6 km

0
Based on S&J Example 3.5, pg 69.
Vector Addition Example
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. What is the magnitude
#»
and direction of the hiker’s resultant displacement R for the trip?

Ax = A cos(−45.0◦ ) = 17.7 km
Ay = A sin(−45.0◦ ) = −17.7 km

Bx = B cos(60.0◦ ) = 20.0 km
Ay = A sin(60.0◦ ) = 34.6 km

#»
R = (Ax + Bx )î + (Ay + By )ĵ
= (17.7 + 20)î + (−17.7 + 34.6)ĵ km
= 37.7î + 17.0ĵ km
= 41.3 km at 24.2◦ north of east
0
Based on S&J Example 3.5, pg 69.
 The negative of the vector A is defined
S
as
S
the vector that when
S
added Sto A gives
zero for the vector sum. That is, A 1 1 2 A 2 5 0. The vectors A and 2 A have the
Vectors Properties and Operations
same magnitude but point in opposite directions.

Subtracting Vectors
Negation
If #»
uThe −We#»
=operation
vector. vdefine
then #» has
of vector subtraction
theuoperation
makes
theSA same use of theSdefinition#»
2 B as magnitude
S
vector 2 B added
of the negative
astovvector
butSApoints
:
of a
in
S S S S
the opposite direction. A 2 B 5 A 1 12 B 2 (3.7)
The geometric construction for subtracting two vectors in this way is illustrated in
Figure 3.10a.
Subtraction
#» S #» S #» #»
Another way of looking atSvectorSsubtraction is to notice that the difference
A −A B
2 B A + (−
=between twoB)
vectors A and B is what you have to add to the second vector

S S S
We would draw Vector C # A " B is
S
B here if we were the vector we must
S S S S
adding it to A. B add to B to obtain A.
S
A
S S S
S C#A"B
S B
S S "B
A"B
S S
Adding "B to A FigureS3.10
S
is equivalent to A vector B fro
S S
subtracting B tor 2 B is eq
S S
from A. vector B an
site directio
a b looking at ve
Summary

• vectors
• vector addition and subtraction

Assignment due Thursday, Jan 16.

(Uncollected) Homework Serway & Jewett,

• Ch 3, onward from page 71. Objective Qs: 1 & 3; Conc Qs: 5
Probs: 31, 55, 65