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					Kinematics in Two Dimensions                                                                                       01/09/2008 03:00 PM



                                                   Kinematics in Two Dimensions




                                                                             Kinematics in 2-dimensions. By the end of
                                                                             this you will


                                                                               1.    Remember your Trigonometry
                                                                               2.    Know how to handle vectors
                                                                               3.    be able to handle problems in 2-dimensions
                                                                               4.    understand projectile motion




  Swish! by alanfreed



 Two (or Three) Dimensional Motion

 For example: Projectiles, Planets, Pendulum, ....

 Need to have some new mathematical techniques to do this: however you may need to revise your
 basic trigonometry


                                                          Basic Trigonometry

 Basic Definitions of trig relationships

  A useful mnemomic

  SohCahToa
  θ =      a
           R
  Sin = opposite = d
        hypotenuse  R
  Cos = adjacent = R-δ
        hypotenuse  R
  Tan = opposite = d
        adjacent   R-δ

 Basic Trig.

 Useful relationships

        sin2θ + cos2θ = 1 (all values of θ)
        tan θ = sin θ / cos θ

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM



 These are not so useful

        cot θ = 1/tan θ
        sec θ = 1/cos θ
        cosec θ = 1/sin θ

  Special Values

         sin(450) = sin(π/4) = 1/√2
         cos(450) = cos(π/4) = 1/√2
         tan(45 0) = tan(π/4) = 1
         sin(300) = sin(π/6) = 1/2
         cos(600) = cos(π/3) = 1/2
         sin(600) = cos(300) = √3/2
         tan(30 0) = tan(π/3) = 1/√3
         tan(60 0) = tan(π/6) = √3
         sin(900) = sin(π/2) = 1
         cos(900) = cos(π/2) = 0
         tan(90 0) = tan(π/2) = ∞

  These are worth writing down, but they are very easy to work out.


  We often have a convenient simplification
  when the angles are small
  θ = a/R ~ sin(θ) = d/R ~ tan(θ) =                                d (R-δ)
  cos(θ) ~ 1

  Note that this only works if we measure angles in
  radians:

  2π radians = 360 0


                                                                      Vectors

 Need new mathematics to describe three dimensional objects:

 Scalars: quantities with only magnitude

 Vectors have direction as well

  Scalars                 Vectors
  Temperature Force

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM



  Speed                   Velocity
  Density                 Acceleration
  Time?                   Time?

 Need a new symbol: can use

        ~ which is obvious
        a
        a which gets confused with other symbols
        a, which is hard to read

 We will use both ~ and a,
                  a


 Need to be able to describe vectors in terms of scalar quantities: can do this in terms of
 components: the projection of the vector along each axis




  These are the components of the vector ~
                                         a

  Note that the components of a vector are scalars



 Also can do this in terms of length of the vector and angle(s).



  These two descriptions are related
  a x = a cos(θ)
  a y = a sin(θ)




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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM




  Adding vectors: put them nose to
  tail. Easy diagramatically




  If we want to add them
  algebraically, we just add the
  components:



                               ~ = ~ +~
                               c a b

         means

                          cx = a x + b x
                          cy = a y + b y


 To subtract vectors, Flip the vector round: the negative of a vector must add to the
 vector to give zero

 ~ - ~ = ~ + ( -a) = 0
 a a a          ~

 Note that this means that the negative of a vector just has all its components reversed

 The length of a vector is given by Pythagoras: 3-D analog of Pythagoras:




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Kinematics in Two Dimensions                                                                             01/09/2008 03:00 PM




                                                                                         Write this as
                                                                       q
                                                               j~j =
                                                                r         x 2 + y2 + z 2

 Note:

        The length of a vector is a scalar.
        Displacement is a vector.
        The length of the displacement vector is the distance


                                                                    Examples




  e.g. a boat sails 10 km North -East and 5
  km South: how must it sail to get to its
  start?




        How do we write this as a vector sum?




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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM




         N.E means "equal components along the x and y directions" so first
         step is



 e.g. Motion by a car.

 A car travels 5 km N, 10 km E, and then 15 km S. The components of vector ~ that describes this
                                                                           a
 are

   1 . a x = -10
        a y = 20

   2 . a x = 10
        a y = -10

   3 . a x = 10
        a y = 10

   4 . a x = -10
        a y = -10

        We write this as


                                                             Projectile Motion

 (can usually be treated as 2-D)




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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM




  Treat position, r,
  velocity v and
  acceleration as 2-D
  vectors. In
  general,motion in one
  direction can be
  treated independently
  of motion in a
  second.




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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM




  We can also see this
  quantitatively




  In general there are three independent vector
  quantities:

  The position r, the velocity v and the acceleration a.
  However we have to treat the components
  separately.




 It is easiest to treat the two motions as independent 1-D motions

                      vy = v0y + ay t
  vx = v0x + ax t                 1
              1       y = v0y t + 2 ay t2
  x = v0x t + 2 ax t2
                      a = À g(usually)
  ax = 0(usually) y
                      = À 9:8ms À2

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM
                               = À 9:8ms
 Can combine these to give (e.g) an equation for the range R:

        What is value of t when the height y = 0?

                                                                            2v0 sin ( Ò )
                                                                      t=
                                                                                 g

        (or t = 0: what does this mean?)
        Hence Range
                                                                              2
                                                                             v0 sin ( 2Ò )
                                                                     R=
                                                                                  g

 e.g. a ball is thrown at 15 ms-1, at 300 to the horizontal.

        How far does it go
        How long is it in the air?


                                                             Relative Velocity
 Example: a woman who can swim at 2 m/s is swimming
     How fast    in a river which flows at .7 m/s.
     does she
     swim
     upstream?
     How fast
     does she
     swim
     downstream?

 How about a round trip?

 A woman swims 100 m upstream at 2 m/s in a river with a current of .7 m/s, and then 100 m
 downstream to return to her starting point. Compared to swimming 200 m in still water, does her
 journey

   1 . Take longer?
   2 . Take a shorter time?
   3 . Take exactly the same time?


  As a somewhat
  more
  sophisticated
  example: a
  woman who can

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

  woman who can
  swim at 2 m/s is
  swimming in a
  river which flows
  at .7 m/s.

  At what angle
  should she swim to
  reach a point on the
  opposite bank
  immediately
  opposite the point
  from which she
  starts?

        Now we want to describe why things move , not just how.




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