RENE DESCARTES(1736-1806)

Motion In Two Dimensions

GALILEO GALILEI(1564-1642)

Vectors in Physics

A scalar quantity has only magnitude.

All physical quantities are either scalars or vectors

A vector quantity has both magnitude and direction.

Scalars

Vectors

Common examples are length, area, volume, time, mass, energy, voltage, and temperature.

Common examples are acceleration, force, electric field, momentum.

In kinematics, distance and speed are scalars.

In kinematics, position, displacement, and velocity are vectors.

Representing Vectors

The arrow’s length represents the

vector’s magnitude

A simple way to represent a vector is by using an arrow.

The arrow’s orientation represents the vector’s direction

“StandardAngle”

“BearingAngle”

θ0˚

θ

90˚

180˚

270˚

E, 90˚

N, 0˚

W, 270˚

S, 180˚

In physics, a vector’s angle (direction ) is called “theta” and the symbol is θ. Two angle conventions are used:

Vector Math

Vector Equivalence

Two vectors are equal if they have the same length and the same direction.

Two vectors are opposite if they have the same length and the opposite direction.

va

vb

va =

vb

Vector Opposites

va

vc

va =−vc

equivalence allows vectors to be translated

opposites allows vectors to be subtracted

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Head to Tail Addition

Vectors add according to the “Head to Tail” rule.

The tail of a vector is placed at the head of the previous vector.

The resultant vector is from the tail of the first vector to the head of the last vector. (Note that the resultant itself is not head to tail.)

For the Vector Field Trip, the resultant vector is 68.6 meters, 79.0˚

South Lawn Vector Walk

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Graphical Addition of Vectors

Vector AdditionVectors add according to the “Head to Tail” rule. The resultant vector isn’t always found with simple arithmetic!

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simple vectoraddition

right trianglevector addition

non-right trianglevector addition

Vector SubtractionTo subtract a vector simply add the opposite vector.

simple vectorsubtraction

right trianglevector subtraction

Resolving VectorsIt is useful to resolve or “break down” vectors into component vectors.

(Same as finding rectangular coordinates from polar coordinates in math.)

vx

vy

vr

θ

sinθ=opphyp

=yr

cosθ=adjhyp

=xr

tanθ=oppadj

=yx

Finding a Resultant

Often a vector’s components are known, and the resultant of these components must be found.

(Same as finding polar coordinates from rectangular coordinates in math.)

vx

vy

vr

θ

x2 +y2 =r2

tanθ=yx

θ =tan−1 yx

⎛ ⎝

⎞ ⎠

Finding the magnitude of the resultant:

Finding the direction of the resultant:

r = x2 +y2

Resolving Vectors

Example: A baseball is thrown at 30 m/s at an angle of 35˚ from the ground. Find the horizontal and vertical components of the baseball’s velocity.

vx

vy

30 m/ s

35̊

sin35̊=y

30

y=30sin35̊=17.2 m/ s

cos35̊=x

30

x=30cos35̊=24.6 m/ s

Horizontal velocity:

Vertical velocity:

Finding a Resultant

Example: A model rocket is moving forward at 10 m/s and downward at 4 m/s after it has reached its peak. What overall velocity (magnitude and direction) does the rocket have at this moment?

10 m/ s

4 m/ s

vr

θx2 +y2 =r2

102 +42 =r2

tanθ=410

=0.40

θ =tan−1 0.40( ) =21.8̊

Find the rocket’s speed (magnitude)

Find the rocket’s angle (direction)

r = 116=10.8 m/ s

θ =21.8˚ below horizontal (or -21.8˚)

Projectile Motion

vx vx vx vx vx

vy

vy

vy

vy

vx

vy

vy

vx

vx

vy

v

v

Horizontal:constant motion, ax = 0

Vertical:freefall motion,ay = g = •9.8 m/s2 velocity is tangent

to the path of motion

Δx = vxt

vyf =vyi + gt

Δy=vyit+ 12 gt2

Δy= 12 vyi + vyf( )t

vyf2 =vyi

2 + 2gΔy

v = vx2 + vy

2

θ =tan−1 vy

vx

⎛

⎝⎜⎞

⎠⎟

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Projectile motion =constant motion +

freefall motion

θ

resultant velocity:

v vy

Projectile Motion at an Angle

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vx =vcosθvyi =vsinθ

velocity components:

vx

vx

vx

vx

vx

vx

vx

vx

vx

θ

vy

vy

vy

vy

vyvy

vy

vy

vx

θ

vyiv

vertical velocity, vy is zero here! v

v

v

v

v

v

v

v

Relative Velocity

All velocity is measured from a reference frame (or point of view).

Velocity with respect to a reference frame is called relative velocity.

A relative velocity has two subscripts, one for the object, the other for the reference frame.

Relative velocity problems relate the motion of an object in two different reference frames.

refers tothe object

refers to thereference frame

click for reference frame applet click for relative

velocity applet

velocity of object a relative to

reference frame b

velocity of reference frame b relative to reference frame c

velocity of object a relative to

reference frame c

Relative VelocityAt the airport, if you walk on a moving sidewalk, your velocity is increased by the motion of you and the moving sidewalk.

vpg = velocity of person relative to groundvps = velocity of person relative to sidewalkvsg = velocity of sidewalk relative to ground

When flying against a headwind, the plane’s “ground speed” accounts for the velocity of the plane and the velocity of the air.

vpe = velocity of plane relative to earthvpa = velocity of plane relative to airvae = velocity of air relative to earth

Relative Velocity

When flying with a crosswind, the plane’s “ground speed” is the resultant of the velocity of the plane and the velocity of the air.

vpe = velocity of plane relative to earthvpa = velocity of plane relative to airvae = velocity of air relative to earth

Sometimes the vector sums are more complicated!

Pilots must fly with crosswind but not be sent off course.