1

Chapter 3

Vectors and Two-Dimensional Motion

2

Vector vs. Scalar Review

All physical quantities encountered in this text will be either a scalar or a vectorA vector quantity has both magnitude (size) and directionA scalar is completely specified by only a magnitude (size)

3

Vector Notation

When handwritten, use an arrow:When printed, will be in bold print with an arrow: When dealing with just the magnitude of a vector in print, an italic letter will be used: A

Ar

Ar

4

Properties of Vectors

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

Movement of vectors in a diagramAny vector can be moved parallel to itself without being affected

5

All these vectors are identical. It is the same vectormoved (translated) to various positions, preserving it’s magnitude (length) and direction (angle)

6

More Properties of VectorsNegative Vectors

Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)

Resultant VectorThe resultant vector is the sum of a given set of vectors

( ); 0= − + − =A B A Ar r r r

= +R A Br r r

7

Adding VectorsWhen adding vectors, their directions must be taken into accountUnits must be the same Geometric Methods

faster, give quick estimateAlgebraic Methods

give numerical answer

8

Adding Vectors Geometrically (Triangle or Polygon Method)

Choose a scale Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate systemDraw 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 A

r

9

Graphically Adding Vectors, cont.

Continue drawing the vectors “tip-to-tail”The resultant is drawn from the origin of to the end of the last vectorMeasure the length of and its angle

Ar

Rr

10

Graphically Adding Vectors, cont.

When you have many vectors, just keep repeating the process until all are includedThe resultant is still drawn from the origin of the first vector to the end of the last vector

11

Notes about Vector Addition

Vectors obey the Commutative Law of Addition

The order in which the vectors are added doesn’t affect the result+ = +A B B A

r r r r

12

Vector SubtractionSpecial case of vector addition

Add the negative of the subtracted vector

Continue with standard vector addition procedure

( )− = + −A B A Br r r r

13

Quick quiz #2 page 55If vector B is added to vector A, the resultant vector has magnitudeA + B when A and B are:

(a) perpendicular to each other; (b) oriented in the same direction;(c) oriented in opposite directions; (d) none of these answers.

14

Multiplying or Dividing a Vector by a Scalar

The result of the multiplication or division is a vectorThe magnitude of the vector is multiplied or divided by the scalarIf the scalar is positive, the direction of the result is the same as of the original vectorIf the scalar is negative, the direction of the result is opposite that of the original vector

15

Problem 8 page 74

16

Components of a Vector

A component is a partIt is useful to use rectangular components

These are the projections of the vector along the x- and y-axes

17

All these vectors are identical, since themagnitude and direction of each of them is the same

X components of all these vectors are identical Y components of all these vectors are identical

18

Components of a Vector, cont.

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

Then,

cosA A θ=x

sinyA A θ=

x y= +A A Ar r r

19

More About Componentsof a Vector

The previous equations are valid only if θ is measured with respect to the x-axis, otherwise they change

(compare equations for components on two plots)

X or Y components of a vector can be positive or negative, depending on the direction of the vector relative to the direction of X or Y axis

20

More About Components, cont.

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

May still have to find θ with respect to the positive x-axisThe value will be correct only if the angle lies in the first or fourth quadrantIn the second or third quadrant, add 180°

2 2 1tan yx y

x

AA A A and

Aθ − ⎛ ⎞

= + = ⎜ ⎟⎝ ⎠

Ar

21

Adding Vectors Algebraically

Choose a coordinate system and sketch the vectorsFind the x- and y-components of all the vectorsAdd all the x-components

This gives Rx:

∑= xx vR

22

Adding Vectors Algebraically, cont.

Add all the y-componentsThis gives Ry:

Use the Pythagorean Theorem to find the magnitude of the resultant:Use the inverse tangent function to find the direction of R:

∑= yy vR

2y

2x RRR +=

xRtan=θ y1 R−

23

Problem 16, page 74

5°

21°

Dallas

Atlanta

Chicago

730 miles

560 miles

X (East)

Y (North) y’

?

24

Motion in Two Dimensions

Using + or – signs generally is not sufficient to describe motion in more than one dimension

Vectors can be used to describe motion in two dimensions

Still interested in displacement, velocity, and acceleration

25

Displacement

The position of an object is described by its position vector, The displacementof the object is defined as the change in its position

rr

f iΔ = −r r rr r r

26

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

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

The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and is in the direction of motion

av tΔ

≡Δr

vrr

27

AccelerationThe average acceleration is defined as the rate at which the velocity changes

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

The vector of acceleration is in the direction of ΔV : may be in any direction with respect to the direction of motion

av tΔ

=Δv

arr

vv

v

v

V

V

y

x

28

Ways an Object Might Accelerate

The magnitude of the velocity (the speed) can changeThe direction of the velocity can change

Even when the magnitude remains constant

Both the magnitude and the direction can change

29

Projectile Motion

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

Consider combination motion in vertical and horizontal directions. When the only acceleration involved is due to the gravity, it is called projectile motion

30

Assumptions of Projectile Motion

Ignore air frictionIgnore the rotation of the earthWith these assumptions, an object in projectile motion will follow a parabolic path

31Example of projectile motion…

32

Rules of Projectile MotionThe x- and y-directions of motion are completely independent of each otherIn x-direction the motion is with constant velocity

ax = 0In y-direction the motion is with constant acceleration due to gravity g

ay = -gThe initial velocity can be broken down into its x- and y-components

cos sinOx O O Oy O Ov v v vθ θ= =

33

Projectile Motion: velocity change (and velocity projections)

cos sinOx O O Oy O Ov v v vθ θ= =

34

Projectile motion: velocity (red) and acceleration (purple)

g=-9.80m/s²

v

v

g=-9.80m/s²g=-9.80m/s²

v

35

Equation (X projection)

x-direction ax = 0

x = vxotThis is the only equation in the x-direction since there is uniform (constant) velocity in that direction

constantvcosvv xooxo ==θ=

36

Equations (Y projection)y-direction

free fall problema = -g

take the positive direction as upwarduniformly accelerated motion, so the motion equations from Chapter 2 all hold (just replace X by Y and use velocity projections to Y axis)

ooyo sinvv θ=

37

Equations for motion in one dimension (left), and projectile motion (right)

( )

= +

Δ = = +

Δ = +

= + Δ

2

2 2

1212

2

o

o

o

o

v v at

x vt v v t

x v t at

v v a x

( )

= +

Δ = = +

Δ = +

= + Δ

2

2 2

1212

2

o

o

o

o

v v at

x vt v v t

x v t at

v v a x

- gy y

y

y

y

g

gΔ

-

-

a=-g, g=9.80m/s²

y y

general case (chapter 2)

x = voxt

y

y

y

38

Velocity of the Projectile

The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point

2 2 1tan yx y

x

vv v v and

vθ −= + =

39

Projectile Motion at Various Initial Angles

The maximum range occurs at a projection angle of 45o

Complementary values of the initial angle result in the same range

The heights will be different

40

Some Variations of Projectile Motion (example 3.5 page 64)

An object is fired horizontally (in this example)

The initial velocity in this case is all in the x-direction

vo = vx and vy = 0

41

Non-Symmetrical Projectile Motion (example 3.7 page 66)

Use “standard” equations for x- and y- motion and solve the system of equations (like it is done in the book)

May break (although it is not necessary) the y-direction into parts and then use equations for each part separately

Option1: up and downOption2: symmetrical back to initial height and then the rest of the height

42

Example of accelerations in two directions(example 3.8 page 67)

a=20 m/s² in X direction

g=9.80 m/s²in Y-direction

43

Problem 58 page 78

44

Relative VelocityRelative velocity is about relating the measurements of two different observers

It is important to specify the frame of reference, since the motion may look different in different frames of referenceIt may be useful to use a moving frame of reference instead of a stationary one

45

Example: two cars

Assume the following notation:E is an observer, stationary with respect to the earthA and B are two moving cars

46

Relative Position

is the position of car A as measured by E

is the position of car B as measured by E

is the position of car A as measured by car B

AErr

ABrrBErr

AB AE EB= −r r rr r r

47

Relative PositionThe position of car A relative to car B is given by the vector subtraction equation

48

Relative Velocity Equations

The rate of change of the displacements gives the relationship for the velocities

AB AE EB= −v v vr r r

49

Example 3.10 page 70

50

Example 3.11 and Exercise 3.11 page 71