rene descartes (1736-1806)
DESCRIPTION
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 PresentationTRANSCRIPT
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
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.
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:
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
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
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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
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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
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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
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
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
⎛⎝⎜
⎞⎠⎟
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Projectile motion =constant motion +
freefall motion
θ
resultant velocity:
v vy
Projectile Motion – Non Zero Launch Angle
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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
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
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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
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
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.
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