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RENE DESCARTES (1736-1806)

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Motion In Two Dimensions. RENE DESCARTES (1736-1806). Vectors in Physics. All physical quantities are either scalars or vectors. Scalars. A scalar quantity has only magnitude. Common examples are length, area, volume, time, mass, energy, and temperature. - PowerPoint PPT Presentation
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RENE DESCARTES (1736-1806) Motion In Two Dimensions GALILEO GALILEI (1564-1642)
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Page 1: RENE DESCARTES (1736-1806)

RENE DESCARTES(1736-1806)

Motion In Two Dimensions

GALILEO GALILEI(1564-1642)

Page 2: RENE DESCARTES (1736-1806)

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.

Page 3: RENE DESCARTES (1736-1806)

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:

Page 4: RENE DESCARTES (1736-1806)

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.

va vb

va =vb

Vector Opposites

va vc va =−vc

equivalence allows vectors to be translated

opposites allows vectors to be subtracted

Page 5: RENE DESCARTES (1736-1806)

Graphical Addition of VectorsVector Addition

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

va vb

vc =va+vb

va vb

vc =va+vb

click for appletclick for applet

va vb vc =va+

vb

simple vectoraddition

right trianglevector addition

non-right trianglevector addition

Vector SubtractionTo subtract a vector simply add the opposite vector.

va −vb

vc =va−vb va −

vb

vc =va−vb

simple vectorsubtraction

non-right trianglevector subtraction

click for appletclick for web site

Page 6: RENE DESCARTES (1736-1806)

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˚

South Lawn Vector Walk

click for web site

Page 7: RENE DESCARTES (1736-1806)

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

 sinθ =opphyp

=yr

⇒  y=rsinθ

cosθ =adjhyp

=xr  ⇒  x =rcosθ

vx

vy vr

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

Page 8: RENE DESCARTES (1736-1806)

Example of Vector Addition

vAx

vAy

vA

vA=24, 30˚;

vB=36, 60˚; find

vR =

vA+

vB

Ax =24cos30˚=20.78Ay=24sin30˚=12.0

vBx

vBy

vB Bx =36cos60˚=18.0

By=36sin60˚=31.18

Rx =20.78 +18.0 =38.78Ry=12.0 + 31.18 =43.18

R= 38.782 + 43.182 =58.04

θ =tan−1 43.1838.78

⎛⎝⎜

⎞⎠⎟

= 48.07˚

Honors:

vA=24, 30˚;

vB=36,150˚; find

vR =

vA+

vB

R= (−10.39)2 + 302 =31.75; θ =109.1̊

vA=24, 30˚;

vB=36, 310˚; find

vR =

vA+

vB

R= 43.922 +(−15.57)2 =46.61; θ =340.5˚ vRx

vRy

vR

θ

Page 9: RENE DESCARTES (1736-1806)

Projectile Motion – Horizontal Launchvx 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 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 motion +

freefall motion

θ

resultant velocity:

v vy

Page 10: RENE DESCARTES (1736-1806)

Projectile Motion – Non Zero Launch Angle

click for applet click for applet

vx =vcosθvyi =vsinθ

velocity components:

vx

vx

vx

vx

vx

vx

vx

vx

vx

θ

vy

vy

vy

vy

vy vy

vy

vy

vx

θ

vyiv

vertical velocity, vy is zero here! v

v

v

v

v

v

vv

Page 11: RENE DESCARTES (1736-1806)

vvab

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

vvab + vvbc =

vvac

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

Page 12: RENE DESCARTES (1736-1806)

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

vvpg =vvps +

vvsg

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

vvpe =vvpa +

vvae

Page 13: RENE DESCARTES (1736-1806)

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.

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.

click for relative velocity applet


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