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					Two & Three
 Week 4 -- Lesson 2
 By The End Of This Lecture, You Should Be Able To
• - Determine 2-D and 3-D displacements, velocities,
    and accelerations
• - Determine the magnitudes of displacement, velocity,
    and acceleration vectors
• - Combine relative velocities to express the velocity
    of a particle relative to a coordinate system that is
    itself moving with respect to anther coordinate
• Up to this point we have focused on motion that
  occurred along a single axis. The axis might have been
  horizontal or vertical, but the motion remained along a
  single straight line.
  Most motions do not occur along a single line.
  Examples of objects that have motions that would not be
  along a straight line include a baseball that has been
  struck by a bat, a car as it rounds a curve, a missile that
  has been fired, the space shuttle after it has been
  launched, the earth as it orbits the sun and so forth.
        Displacement In Two
• We begin by trying to describe the
  displacement of an object that moves in two
  dimensions. In the next slide, an object
  moves from P1 to P2 and then from P2 to P3.
  We want to know the displacement of the
  object from P1 at the end of the trip.
Adding Vectors Graphically
Adding Vectors Graphically
Adding Vectors Graphically
Adding Vectors Graphically
             Displacement In Two
• Remember that when determining the displacement of an
  object, you are interested in only two locations: the point
  where the object starts, and the point where the object
  ends up. With this in mind, we can determine the
  displacement of the object from its starting position by
  noting that the path the object actually traced out from P1
  to P2 and then to P3 has the same endpoints (and is
  therefore equivalent for our purposes) as if the object had
  gone directly from P1 to P3.
 Displacement In Two Dimensions
• In other words,

                  
                C  A B
• Notice that each of the terms in the expression
  has an arrow over them. This is to indicate that
  the quantity is a vector quantity. Vector algebra
  is different than ordinary algebra. You must be
  careful when adding and subtracting vector
Displacement In Two Dimensions
• If A has a magnitude of 3 units and B has a
  magnitude of 4 units, C does not necessarily
  have a magnitude of 7 units. In fact, C may
  have any magnitude between 1 unit ( |A-B| ) and
  7 units (A+B) depending on how A and B are
  directed. So, if vector algebra works differently
  than ordinary algebra, how do you add or
  subtract vectors? The first step is to break the
  vectors into parts that lie along common axes.
        Displacement In Two
• This process is called breaking a vector into
  components (or parts of a vector that are
  oriented at right angles to one another). The
  next slide shows a three dimensional vector
  broken into components along the x, y, and
  z-axes. The vector A can be written as a
  sum of its three components:
Components Of A Vector

A  A x ˆ  A y ˆ  Az k
        i       j      ˆ
        Displacement In Two
• By breaking the vectors A and B into
  components, each vector has components
  that lie along the same axes. Components
  that lie along a common axis add (if they
  are directed the same way) and subtract (if
  they are oppositely directed) the same way
  ordinary numbers do.
Adding Vectors By Components
Adding Vectors By Components
Adding Vectors By Components
Adding Vectors By Components

       C x  Ax  Bx
Adding Vectors By Components

       C y  Ay  B y
Adding Vectors By Components

      C  C C
Adding Vectors By Components


             opposite 
           1                 1  C y 
             adjacent   tan  C 
      tan                     
                                x
• You need to be very careful when breaking a
  vector into components and when combining
  components to form a vector. In chapter 3, the
  text has some expressions that look like the
         Ax  A cos      Ay  A sin 

                          Ay 
                   tan  
                         A 
                          x
• And it would appear these expressions are
  completely general. Then, you read in the
  paragraph following and find out that they are
  only valid provided the angle is expressed
  relative to the positive x-axis. The angles we
  deal with are not always expressed this way. To
  be sure you are applying relationships correctly,
  you should draw the triangles and go back to the
  definitions of the sin, cos, and tan functions you
  learned in trig. (review chapter 1 pg 17-18)
     Vector Analysis of Motion
• It will be essential to your physics study to
  practice the resolution of physical quantities
  into components.
• Try Chapter 3 problems Pg 72-73 1-15
            Relative Motion
• One of the more straight-forward examples
  dealing with vectors is that of relative
  motions that occur at constant velocities.
  Suppose a person is walking at 1.0 m/s from
  left to right on a flatbed car that is moving
  at 10 m/s from right to left relative to the
  ground. A person on the ground would see:


vPC = velocity of person relative to car
vCG = velocity of car relative to ground
vPG = velocity of person relative to ground

                         
             v PG  v PC  vCG
Relative Motion Example Continued
Notice that the expression shown at the bottom of
 the last slide is an addition of vectors. The
 vectors always add, regardless of the directions
 of the individual vectors involved. The
 directionality of each vector is taken into account
 by the vector operation. When the vector
 expression is reduced to a scalar equation (this
 can only be done using components), then you
 add the magnitudes if the vectors point in the
 same direction and subtract if the vectors point
 opposite directions:
Relative Motion Example Continued
       vPG  vPC  vCG
              m     m
      vPG  1  10
              s      s
         vPG  9
                 mˆ
        v PG  9 i
• Or, the person appears to be heading at 9
  m/s to the left when viewed by an observer
  on the ground.
• Two dimensional velocities work the same
  way, except that you must break the vectors
  into components before combining them.
  Give it a try with the problem on the next
        Try It On Your Own
• A plane needs to fly the 675 mile distance
  from New Orleans to Chicago in 1 hour, 45
  minutes in order to keep a flight schedule.
  There is a 20 mph wind blowing from the
  Northwest. What constant velocity must the
  plane have in order to make the trip in the
  prescribed time?
• Two and three dimensional vector quantities are
  defined analogous to their one dimensional
• When dealing with vector quantities, break all
  vectors into components before trying to work
  with them. Then, if needed, put the parts back
  together to form a vector.
• Be careful to always draw triangles and apply the
  definitions of sin, cos, and tan when working with
  vectors to avoid traps and pitfalls.
• Chapter 3 Problems
• 1-38 odds pg 45
• Read Lab Activity on Projectiles

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