Vectors and Scalars

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Vectors and Scalars

• A scalar is a quantity that is completely
  specified by a positive or negative number with
  an appropriate unit and has no direction.
• A vector is a physical quantity that must be
  described by a magnitude (number) and appropriate
  units plus a direction.

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Some Notes About Scalars

• Some examples
• Temperature
• Volume
• Mass
• Time intervals
• Rules of ordinary arithmetic are used to
  manipulate scalar quantities

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Vector Example

• A particle travels from A to B along the path
  shown by the dotted red line
• This is the distance traveled and is a scalar
• The displacement is the solid line from A to B
• The displacement is independent of the path taken
  between the two points
• Displacement is a vector

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Other Examples of Vectors

• Many other quantities are also vectors
• Some of these include
• Velocity
• Acceleration
• Force
• Momentum

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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 or
  
• The magnitude of the vector has physical units
• The magnitude of a vector is always a positive
  number

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Equality of Two Vectors

• Two vectors are equal if they have the same
  magnitude and the same direction
• if A B and they point along parallel
  lines
• All of the vectors shown are equal

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Adding Vectors

• When adding vectors, their directions must be
  taken into account
• Units must be the same
• Graphical Methods
• Use scale drawings
• Algebraic Methods
• More convenient

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Adding Vectors Graphically

• Continue drawing the vectors tip-to-tail
• The resultant is drawn from the origin of to
  the end of the last vector
• Measure the length of and its angle
• Use the scale factor to convert length to actual
  magnitude

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Adding Vectors Graphically, final

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

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Adding Vectors, Rules

• When two vectors are added, the sum is
  independent of the order of the addition.
• This is the commutative law of addition

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Adding Vectors, Rules cont.

• When adding three or more vectors, their sum is
  independent of the way in which the individual
  vectors are grouped
• This is called the Associative Property of
  Addition

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Adding Vectors, Rules final

• When adding vectors, all of the vectors must have
  the same units
• All of the vectors must be of the same type of
  quantity
• For example, you cannot add a displacement to a
  velocity

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Negative of a Vector

• The negative of a vector is defined as the vector
  that, when added to the original vector, gives a
  resultant of zero
• Represented as
• The negative of the vector will have the same
  magnitude, but point in the opposite direction

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Subtracting Vectors

• Special case of vector addition
• Continue with standard vector addition procedure

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Multiplying or Dividing a Vector by a Scalar

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

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Multiplying Vectors

• Two vectors can be multiplied in two different
  ways
• One is the scalar product
• Also called the dot product
• The other is the vector product
• Also called the cross product
• These products will be discussed as they arise in
  the text

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Components of a Vector

• A component is a part
• It is useful to use rectangular components
• These are the projections of the vector along the
  x- and y-axes

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Vector Component Terminology

• are the component vectors of
• They are vectors and follow all the rules for
  vectors
• Ax and Ay are scalars, and will be referred to as
  the components of A
• The combination of the component vectors is a
  valid substitution for the actual vector

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Components of a Vector, 2

• 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
• When using this form of the equations, q must be
  measured ccw from the positive x-axis

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Components of a Vector, 3

• The components are the legs of the right triangle
  whose hypotenuse is
• May still have to find ? with respect to the
  positive x-axis
• Use the signs of Ax and Ay

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Components of a Vector, final

• The components can be positive or negative and
  will have the same units as the original vector
• The signs of the components will depend on the
  angle

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Unit Vectors

• A unit vector is a dimensionless vector with a
  magnitude of exactly 1.
• Unit vectors are used to specify a direction and
  have no other physical significance

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Unit Vectors, cont.

• The symbols
• represent unit vectors in the x, y and z
  directions
• They form a set of mutually perpendicular vectors
  

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Unit Vectors in Vector Notation

• is the same as Ax and is the same as
  Ay etc.
• The complete vector can be expressed as

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Adding Vectors Using Unit Vectors

• Using
• Then
• Then Rx Ax Bx and Ry Ay By

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Adding Vectors with Unit Vectors Diagram
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Adding Vectors Using Unit Vectors Three
Directions

• Using
• Rx Ax Bx , Ry Ay By and Rz Az Bz
• 
  etc.

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Exercises of chapter 1

• 1,4, 6, 8, 13, 17, 26, 35, 37, 43, 49, 52,
• 57, 62, 64, 66, 68.

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Chapter 2

• Motion in One Dimension

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Kinematics

• To describe motion while ignoring the agents that
  caused the motion
• For now, we will consider motion in one dimension
  - along a straight line -, and use the particle
  model.
• A particle is a point-like object, has mass but
  infinitesimal size

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Position

• Defined in terms of a frame of reference
• The reference frame must has an origin.
• The objects position is its location with
  respect to the origin of the frame of reference.

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Position-Time Graph

• The position-time graph shows the motion of the
  particle (car)
• The smooth curve is a guess as to what happened
  between the data points

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Displacement

• Defined as the change in position during some
  time interval Dt
• Represented as ?x
• Xi is the initial position and Xf is the final
  position.
• Different than distance the length of a path
  followed by a particle

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Average Velocity

• The average velocity is rate at which the
  displacement occurs
• The dimensions are length / time L/T
• The slope of the line connecting the initial and
  final points in the position time graph

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Average Velocity, cont

• Gives no details about the motion
• Gives the result of the motion
• It can be positive or negative
• It depends on the sign of the displacement
• It can be interpreted graphically

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Displacement from a Velocity - Time Graph

• The displacement of a particle during the time
  interval ti to tf is equal to the area under the
  curve between the initial and final points on a
  velocity-time curve

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Instantaneous Velocity

• The limit of the average velocity as the time
  interval becomes infinitesimally short, or as the
  time interval approaches zero
• The instantaneous velocity indicates what is
  happening at every point of time

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Instantaneous Velocity, graph

• The instantaneous velocity is the slope of the
  line tangent to the x vs. t curve
• This would be the green line
• The blue lines show that as ?t gets smaller, they
  approach the green line

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Instantaneous Velocity, equations

• The general equation for instantaneous velocity
  is
• When velocity is used in the text, it will mean
  the instantaneous velocity

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Instantaneous Velocity, Signs

• At point A, vx is positive
• The slope is positive
• At point B, vx is zero
• The slope is zero
• At point C, vx is negative
• The slope is negative

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Instantaneous Speed

• The instantaneous speed is the magnitude of the
  instantaneous velocity vector
• Speed can never be negative

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Fig 2.5
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Constant Velocity

• If the velocity of a particle is constant
• Its instantaneous velocity at any instant is the
  same as the average velocity over a given time
  period
• vx v x avg Dx / Dt
• Also, xf xi vx t
• These equations can be applied to particles or
  objects that can be modeled as a particle moving
  under constant velocity

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Fig 2.6
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Average Acceleration

• Acceleration is the rate of change of the
  velocity
• It is a measure of how rapidly the velocity is
  changing
• Dimensions are L/T2
• SI units are m/s2