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RENE DESCARTES(1596-1650)

Motion In Two Dimensions

GALILEO GALILEI(1564-1642)

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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

Other examples: length, mass, power. Some are even negative (charge, energy, voltage, and temperature) but not directional.

Other examples: forces, fields (electric, magnetic, gravitational), and momentum.

In kinematics, time, distance and speed are scalars.

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

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Representing Vectors

The arrow’s length represents the

vector’s magnitude

An arrow is a simple way to represent a vector.

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 often θ. Two angle conventions are used:

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Vector Math

Vector EquivalenceTwo 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.

!a

!b

!a =!b

Vector Opposites

!a

!c

!a = −!c

equivalence allows vectors to be translated

opposites allows vectors to be subtracted

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Graphical Addition of VectorsVector Addition

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

!a

!b

!c = !a +

!b

!a !b

!c = !a +!b

click for applet click for applet

!a

!b

!c = !a +!b

simple vectoraddition

right trianglevector addition

non-right trianglevector addition

Vector SubtractionTo subtract a vector simply add the opposite vector.

!a −

!b

!c = !a −

!b

!a −!b

!c = !a −

!b

simple vectorsubtraction

non-right trianglevector subtraction

click for appletclick for web site

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click for applet

Head to Tail AdditionVectors 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 69.9 meters, 78.0˚, N

South Lawn Vector Walk

click for web site

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Resolving Vectors, Finding ResultantTo resolve a vector into component vectors, use trigonometry:

 sinθ =opphyp

=yr

⇒   y = r sinθ

cosθ =adjhyp

=xr  ⇒   x = r cosθ

!x

!y

!r

θ

If the vector components are known, the resultant can be found:

x2 + y2 = r2   ⇒   r = x2 + y2

tanθ =yx  ⇒   θ = tan−1 y

x⎛⎝⎜

⎞⎠⎟

Finding the resultant’s magnitude

Finding the resultant’s direction

The vector components are rectangular coordinates (x,y)The vector magnitude & direction are polar coordinates (r,θ)

Finding the horizontal component

Finding the vertical component

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Example of Vector Addition

!Ax

!Ay

!A

!A = 12, 30˚;

!B = 18, 60˚; find

!R =!A +!B

Ax = 12cos30˚= 10.4Ay = 12sin30˚= 6.0

!Bx

!By

!B Bx = 18cos60˚= 9.0

By = 18sin60˚= 15.6

Rx = 10.4 + 9.0 = 19.4Ry = 6.0 +15.6 = 21.6

R = 19.42 + 21.62 = 29.0

θ = tan−1 21.619.4

⎛⎝⎜

⎞⎠⎟

= 48.1̊

Try another:!A = 12, 30˚;

!B = 18,150˚; find

!R =!A +!B

R = (−5.2)2 +152 = 15.9; θ = 109˚!A = 12, 30˚;

!B = 18, 310˚; find

!R =!A +!B

R = 22.02 + (−7.79)2 = 23.3; θ = −19.5 or 340.5˚ !Rx

!Ry

!R

θ

Try it online:search for“PHET vectors”

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Projectile Motion – Horizontal Launchvx vx vx vx vx

vy

vy

vy

vy

vx

vy

vy

vx

vx

vy

v

v

Horizontal:constant velocity, ax = 0

Vertical:freefall acceleration,ay = g = –9.8 m/s2 velocity is tangent to

the path of motion

Δx = vxt

vyf = vyi + gt

Δy = vyit + 12 gt

2

Δy = 12 vyi + vyf( )t

vyf2 = vyi

2 + 2gΔy

v = vx2 + vy

2

θ = tan−1 vyvx

⎛⎝⎜

⎞⎠⎟

click for applet

click for applet

Projectile motion =constant (x) velocity+

freefall (y) acceleration

θ

resultant velocity:

v vy

vxvy

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Projectile Motion – Non Zero Launch Angle

click for applet click for applet

vx = vcosθvyi = vsinθ

velocity components:

vx

vx

vx

vxvx

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

vv

click for applet

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!vab

Relative VelocityAll 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

!vab +

!vbc =!vac

click for reference frame applet click for relative velocity applet

velocity of object a relative to

reference frame bvelocity of reference frame b relative to reference frame c

velocity of object a relative to

reference frame c

There is NO acceleration, motion is constant velocity.

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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

!vpg =

!vps +!vsg

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 groundvpa = velocity of plane relative to airvae = velocity of air relative to ground

!vpg =

!vpa +!vag

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

vpg = velocity of plane relative to groundvpa = velocity of plane relative to airvag = velocity of air relative to ground

Honors only: sometimes the vector sums are more complicated!Break vectors into x-components and y-components, solve with geometry and trig.

Pilots must fly with crosswind but not be sent off course.click for relative velocity applet

click for relative velocity applet

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