Chapter 1.a

Vectors

Coordinate Systems

Used to describe the position of a point in space

Coordinate system consists of A fixed reference point called the origin Specific axes with scales and labels Instructions on how to label a point relative to the

origin and the axes

Cartesian Coordinate System

Also called rectangular coordinate system

x- and y- axes intersect at the origin

Points are labeled (x,y)

Polar Coordinate System

Origin and reference line are noted

Point is distance r from the origin in the direction of angle , ccw from reference line

Points are labeled (r,)

Polar to Cartesian Coordinates

Based on forming a right triangle from r and q

x = r cos q y = r sin q

Trigonometry Review

Given various radius vectors, find Length and angle x- and y-components Trigonometric functions:

sin, cos, tan

Cartesian to Polar Coordinates

r is the hypotenuse and q an angle

q must be ccw from positive x axis for these equations to be valid

2 2

tany

x

r x y

Example 3.1 The Cartesian coordinates of a

point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point.

Solution: From Equation 3.4,

and from Equation 3.3,

2 2 2 2( 3.50 m) ( 2.50 m) 4.30 mr x y

2.50 mtan 0.714

3.50 m216 (signs give quadrant)

y

x

Example 3.1, cont.

Change the point in the x-y plane

Note its Cartesian coordinates

Note its polar coordinates

Please insert active fig. 3.3 here

Vectors and Scalars

A scalar quantity is completely specified by a single value with an appropriate unit and has no direction.

A vector quantity is completely described by a number and appropriate units plus a direction.

Vector Example A particle travels from A to B

along the path shown by the dotted red line This is the distance

traveled and is a scalar The displacement is the

solid line from A to B The displacement is

independent of the path taken between the two points

Displacement is a vector

Vector Notation

Text uses bold with arrow to denote a vector: Also used for printing is simple bold print: A When dealing with just the magnitude of a

vector in print, an italic letter will be used: A or | |

The magnitude of the vector has physical units The magnitude of a vector is always a positive

number When handwritten, use an arrow:

A

A

A

Equality of Two Vectors

Two vectors are equal if they have the same magnitude and the same direction

if A = B and they point along parallel lines

All of the vectors shown are equal

A B

Adding Vectors

When adding vectors, their directions must be taken into account

Units must be the same Graphical Methods

Use scale drawings Algebraic Methods

More convenient

Adding Vectors Graphically

Choose a scale Draw the first vector, , with the appropriate length

and in the direction specified, with respect to a coordinate system

Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for

A

A

A

Adding Vectors Graphically, cont.

Continue drawing the vectors “tip-to-tail”

The resultant is drawn from the origin of to the end of the last vector

Measure the length of and its angle Use the scale factor to

convert length to actual magnitude

A

R

Adding Vectors Graphically, final

When you have many vectors, just keep repeating the process until all are included

The resultant is still drawn from the tail of the first vector to the tip of the last vector

Adding Vectors, Rules

When two vectors are added, the sum is independent of the order of the addition. This is the Commutative

Law of Addition A B B A

Adding Vectors, Rules cont.

When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped This is called the Associative Property of Addition

A B C A B C

Adding Vectors, Rules final

When adding vectors, all of the vectors must have the same units

All of the vectors must be of the same type of quantity For example, you cannot add a displacement to a

velocity

Negative of a Vector

The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero Represented as

The negative of the vector will have the same magnitude, but point in the opposite direction

A

0 A A

Subtracting Vectors

Special case of vector addition

If , then use Continue with standard

vector addition procedure

A B

A B

Subtracting Vectors, Method 2

Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector

As shown, the resultant

vector points from the tip of the second to the tip of the first

A B C

Multiplying or Dividing a Vector by a Scalar

The result of the multiplication or division of a vector by a scalar is a vector

The magnitude of the vector is multiplied or divided by the scalar

If the scalar is positive, the direction of the result is the same as of the original vector

If the scalar is negative, the direction of the result is opposite that of the original vector

Component Method of Adding Vectors

Graphical addition is not recommended when High accuracy is required If you have a three-dimensional problem

Component method is an alternative method It uses projections of vectors along coordinate

axes

Components of a Vector, Introduction

A component is a projection of a vector along an axis Any vector can be

completely described by its components

It is useful to use rectangular components These are the projections

of the vector along the x- and y-axes

Vector Component Terminology

are the component vectors of They are vectors and follow all the rules for

vectors

Ax and Ay are scalars, and will be referred to as the components of

x yandA A

A

A

Components of a Vector

Assume you are given a vector

It can be expressed in terms of two other vectors, and

These three vectors form a right triangle

A

xA

yA

x y A A A

Components of a Vector, 2

The y-component is moved to the end of the x-component

This is due to the fact that any vector can be moved parallel to itself without being affected This completes the

triangle

Components of a Vector, 3

The x-component of a vector is the projection along the x-axis

The y-component of a vector is the projection along the y-axis

This assumes the angle θ is measured with respect to the x-axis If not, do not use these equations, use the sides of the

triangle directly

cosxA A

sinyA A

Components of a Vector, 4

The components are the legs of the right triangle whose hypotenuse is the length of A

May still have to find θ with respect to the positive x-axis

2 2 1and tan yx y

x

AA A A

A

Components of a Vector, final

The components can be positive or negative and will have the same units as the original vector

The signs of the components will depend on the angle

Unit Vectors

A unit vector is a dimensionless vector with a magnitude of exactly 1.

Unit vectors are used to specify a direction and have no other physical significance

Unit Vectors, cont.

The symbols

represent unit vectors They form a set of

mutually perpendicular vectors in a right-handed coordinate system

Remember,

kand,j,i

ˆ ˆ ˆ 1 i j k

Viewing a Vector and Its Projections

Rotate the axes for various views

Study the projection of a vector on various planes x, y x, z y, z

Unit Vectors in Vector Notation

Ax is the same as Ax

and Ay is the same as Ay etc.

The complete vector can be expressed as

i

j

ˆ ˆ

x yA AA i j

Adding Vectors Using Unit Vectors

Using Then

and so Rx = Ax + Bx and Ry = Ay + By

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

x y x y

x x y y

x y

A A B B

A B A B

R R

R i j i j

R i j

R i j

2 2 1tan yx y

x

RR R R

R

R A B

Adding Vectors with Unit Vectors

Note the relationships among the components of the resultant and the components of the original vectors

Rx = Ax + Bx

Ry = Ay + By

Three-Dimensional Extension

Using Then

and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Ax+Bz

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

x y z x y z

x x y y z z

x y z

A A A B B B

A B A B A B

R R R

R i j k i j k

R i j k

R i j k

2 2 2 1cos , .xx y z

RR R R R etc

R

R A B

Example 3.5 – 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.

Example 3.5 (A) Determine the components

of the hiker’s displacement for each day.

A

Solution: We conceptualize the problem by drawing a sketch as in the figure above. If we denote the displacement vectors on the first and second days by and respectively, and use the car as the origin of coordinates, we obtain the vectors shown in the figure. Drawing the resultant , we can now categorize this problem as an addition of two vectors.

B A

R

Example 3.5 We will analyze this

problem by using our new knowledge of vector components. Displacement has a magnitude of 25.0 km and is directed 45.0° below the positive x axis.

From Equations 3.8 and 3.9, its components are:

cos( 45.0 ) (25.0 km)(0.707) = 17.7 km

sin( 45.0 ) (25.0 km)( 0.707) 17.7 kmx

y

A A

A A

The negative value of Ay indicates that the hiker walks in the negative y direction on the first day. The signs of Ax and Ay also are evident from the figure above.

A

Example 3.5

The second displacement has a magnitude of 40.0 km and is 60.0° north of east.

Its components are:

cos60.0 (40.0 km)(0.500) = 20.0 km

sin 60.0 (40.0 km)(0.866) 34.6 kmx

y

B B

B B

B

Example 3.5

(B) Determine the components of the hiker’s resultant displacement for the trip. Find an expression for in terms of unit vectors.

Solution: The resultant displacement for the trip has components given by Equation 3.15:

Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km

Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km

In unit-vector form, we can write the total displacement as

R

ˆ ˆR = (37.7 + 16.9 ) kmi j

R A B

R

R

Example 3.5

Using Equations 3.16 and 3.17, we find that the resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east.

R

Let us finalize. The units of are km, which is reasonable for a displacement. Looking at the graphical representation in the figure above, we estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of in our final result. Also, both components of are positive, putting the final position in the first quadrant of the coordinate system, which is also consistent with the figure.

R

R

R

Chapter 1.b

Motion in Several Dimensions

Motion in Two Dimensions

Using + or – signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion Will look at vector nature of quantities in more detail

Still interested in displacement, velocity, and acceleration

Will serve as the basis of multiple types of motion in future chapters

Position and Displacement

The position of an object is described by its position vector,

The displacement of the object is defined as the change in its position

r

f ir r r

General Motion Ideas

In two- or three-dimensional kinematics, everything is the same as as in one-dimensional motion except that we must now use full vector notation Positive and negative signs are no longer

sufficient to determine the direction

Average Velocity

The average velocity is the ratio of the displacement to the time interval for the displacement

The direction of the average velocity is the direction of the displacement vector

The average velocity between points is independent of the path taken This is because it is dependent on the displacement, also

independent of the path

avg t

rv

Instantaneous Velocity

The instantaneous velocity is the limit of the average velocity as Δt approaches zero

As the time interval becomes smaller, the direction of the displacement approaches that of the line tangent to the curve

0lim

t

d

t dt

r rv

Instantaneous Velocity, cont

The direction of the instantaneous velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion

The magnitude of the instantaneous velocity vector is the speed The speed is a scalar quantity

Average Acceleration

The average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.

f i

avgf it t t

v v va

Average Acceleration, cont

As a particle moves, the direction of the change in velocity is found by vector subtraction

The average acceleration is a vector quantity directed along

f iv v v

v

Instantaneous Acceleration

The instantaneous acceleration is the limiting value of the ratio as Δt approaches zero

The instantaneous equals the derivative of the velocity vector with respect to time

0lim

t

d

t dt

v va

tv

Producing An Acceleration

Various changes in a particle’s motion may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change

Even if the magnitude remains constant Both may change simultaneously

Kinematic Equations for Two-Dimensional Motion

When the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motion

These equations will be similar to those of one-dimensional kinematics

Motion in two dimensions can be modeled as two independent motions in each of the two perpendicular directions associated with the x and y axes Any influence in the y direction does not affect the motion

in the x direction

Kinematic Equations, 2

Position vector for a particle moving in the xy plane

The velocity vector can be found from the position vector

Since acceleration is constant, we can also find an expression for the velocity as a function of time:

ˆ ˆx yr i j

ˆ ˆx y

dv v

dt

rv i j

f i tv v a

Kinematic Equations, 3

The position vector can also be expressed as a function of time: This indicates that the position vector is the sum

of three other vectors: The initial position vector The displacement resulting from the initial velocity The displacement resulting from the acceleration

21

2f i it tr r v a

Projectile Motion

An object may move in both the x and y directions simultaneously

The form of two-dimensional motion we will deal with is called projectile motion

Assumptions of Projectile Motion

The free-fall acceleration is constant over the range of motion It is directed downward This is the same as assuming a flat Earth over the

range of the motion It is reasonable as long as the range is small

compared to the radius of the Earth The effect of air friction is negligible With these assumptions, an object in

projectile motion will follow a parabolic path This path is called the trajectory

Projectile Motion Diagram

Analyzing Projectile Motion Consider the motion as the superposition of the

motions in the x- and y-directions The actual position at any time is given by:

The initial velocity can be expressed in terms of its components vxi = vi cos q and vyi = vi sin q

The x-direction has constant velocity ax = 0

The y-direction is free fall ay = -g

212f i it t r r v g

Effects of Changing Initial Conditions

The velocity vector components depend on the value of the initial velocity Change the angle and

note the effect Change the magnitude

and note the effect

Analysis Model

The analysis model is the superposition of two motions Motion of a particle under constant velocity in the

horizontal direction Motion of a particle under constant acceleration in

the vertical direction Specifically, free fall

Projectile Motion Vectors

The final position is the

vector sum of the initial position, the position resulting from the initial velocity and the position resulting from the acceleration

212f i it t r r v g

Projectile Motion – Implications

The y-component of the velocity is zero at the maximum height of the trajectory

The acceleration stays the same throughout the trajectory

Range and Maximum Height of a Projectile

When analyzing projectile motion, two characteristics are of special interest

The range, R, is the horizontal distance of the projectile

The maximum height the projectile reaches is h

Height of a Projectile, equation

The maximum height of the projectile can be found in terms of the initial velocity vector:

This equation is valid only for symmetric motion

2 2sin

2i iv

hg

Range of a Projectile, equation

The range of a projectile can be expressed in terms of the initial velocity vector:

This is valid only for symmetric trajectory

2 sin2i ivR

g

More About the Range of a Projectile

Range of a Projectile, final

The maximum range occurs at qi = 45o

Complementary angles will produce the same range The maximum height will be different for the two

angles The times of the flight will be different for the two

angles

Uniform Circular Motion

Uniform circular motion occurs when an object moves in a circular path with a constant speed

The associated analysis motion is a particle in uniform circular motion

An acceleration exists since the direction of the motion is changing This change in velocity is related to an acceleration

The velocity vector is always tangent to the path of the object

Changing Velocity in Uniform Circular Motion

The change in the velocity vector is due to the change in direction

The vector diagram shows

f i v v v

Centripetal Acceleration

The acceleration is always perpendicular to the path of the motion

The acceleration always points toward the center of the circle of motion

This acceleration is called the centripetal acceleration

Centripetal Acceleration, cont

The magnitude of the centripetal acceleration vector is given by

The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

2

C

va

r

Period

The period, T, is the time required for one complete revolution

The speed of the particle would be the circumference of the circle of motion divided by the period

Therefore, the period is defined as 2 r

Tv

Tangential Acceleration

The magnitude of the velocity could also be changing In this case, there would be a tangential acceleration The motion would be under the influence of both

tangential and centripetal accelerations Note the changing acceleration vectors

Total Acceleration

The tangential acceleration causes the change in the speed of the particle

The radial acceleration comes from a change in the direction of the velocity vector

Total Acceleration, equations

The tangential acceleration:

The radial acceleration:

The total acceleration:

Magnitude

Direction Same as velocity vector if v is increasing, opposite if v is

decreasing

t

dva

dt

2

r C

va a

r

2 2r ta a a

Relative Velocity

Two observers moving relative to each other generally do not agree on the outcome of an experiment

However, the observations seen by each are related to one another

A frame of reference can described by a Cartesian coordinate system for which an observer is at rest with respect to the origin

Different Measurements, example

Observer A measures point P at +5 m from the origin

Observer B measures point P at +10 m from the origin

The difference is due to the different frames of reference being used

Different Measurements, another example The man is walking on the

moving beltway The woman on the beltway

sees the man walking at his normal walking speed

The stationary woman sees the man walking at a much higher speed The combination of the

speed of the beltway and the walking

The difference is due to the relative velocity of their frames of reference

Relative Velocity, generalized

Reference frame SA is stationary

Reference frame SB is moving to the right relative to SA at This also means that SA

moves at – relative to SB

Define time t = 0 as that time when the origins coincide

ABv

BAv

Notation

The first subscript represents what is being observed

The second subscript represents who is doing the observing

Example The velocity of A as measured by observer B

ABv

Relative Velocity, equations

The positions as seen from the two reference frames are related through the velocity

The derivative of the position equation will give the velocity equation

is the velocity of the particle P measured by observer A is the velocity of the particle P measured by observer B

These are called the Galilean transformation equations

PAu

PA PB BAt r r v

PA PB BA u u v

PBu

Acceleration in Different Frames of Reference

The derivative of the velocity equation will give the acceleration equation

The acceleration of the particle measured by an observer in one frame of reference is the same as that measured by any other observer moving at a constant velocity relative to the first frame.