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Kinematics in Two Dimensions (PowerPoint)

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					   Chapter 3

Projectile Motion




   Alta High Conceptual Physics
              Vectors and Scalars
   Vector vs. Scalars
       Vectors have both magnitude and direction
            Displacement, Velocity, Acceleration, Force,
             and Momentum
       Scalars have only magnitude
            Mass, Time, and Temperature




                        Alta High Conceptual Physics
                 Vector Addition
   Two Methods
       Graphical Method
           Requires a ruler and a protractor
           Process
                Convert each vector into a line that fits a scale of
                 your choosing
                Draw Vectors head to tail, measuring the angle
                 exactly
                Draw a resultant
                Measure the resultant and the resulting angle
                Convert the measurement back into a vector answer

                          Alta High Conceptual Physics
                 Graphical Addition
      A man drives 125 km west and then turns
      45º North of West and travels an additional
      100 km. What is his displacement?
      Step one: Set Scale
         1 cm = 25 Km


             100 km = 4 cm


45°

                          Alta High 5 cm
                       125 km = Conceptual Physics
                 Graphical Addition
      Step Two: Move Vectors Head to Tail
      Step Three: Draw Resultant from Tail of First
                  Vector to the head of the last.
100 km = 4 cm




    45°


                      125 km = 5 cm



                          Alta High Conceptual Physics
                   Graphical Addition
      Step Four:    Measure the Measure the Resultant
                    and the resulting angle.
      Step Five:    Using the Scale, convert the
100 km = 4 cm       measurement into an answer.

                    8.3 cm
                                       8.3 cm * 25 Km/cm = 208 Km
    45°


                       125 km = 5 cm         26.2°

          Final Answer: His displacement is 208
                              of West
          Km @ 19.9° North High Conceptual Physics
                         Alta
       Components of Vectors
   Often we need to change a single
    vector into two vectors in order to make
    addition easier. These two vectors will
    be a right angles to each other and
    are called component vectors or just
    components for short.
   The breaking down of a vector this way
    is called resolution.
                 Alta High Conceptual Physics
       Vector Resolution
Step One: Draw Original Vector


           45º




                    100 Km 45°
                    North of West


                 Alta High Conceptual Physics
       Vector Resolution
Step Two: Draw parallelogram to
Resolve Vector into components




               100 Km 45°
               North of West


            Alta High Conceptual Physics
           Vector Resolution
 Step Three: Label and measure
             components
                                      North component = 70.7 km
                  45°


West component = 70.7 km



                        100 Km 45°
                        North of West


                   Alta High Conceptual Physics
         Mathematical Addition
   Mathematical Addition of Vectors
    Requires a basic knowledge of
    Geometry – You must know:
       Sin θ = Opposite/Hypotenuse
       Cos θ = Adjacent/Hypotenuse
       Tan θ = Sin θ/ Cos θ =Opposite/Adjacent
       ArcSin, ArcCos, ArcTan


                   Alta High Conceptual Physics
         Mathematical Addition
   Mathematical addition requires that you
    be able to
       Draw a rough sketch of the original vectors
       Draw a parallelogram
       Draw the Resultant
       Use Triangle geometry to find the
        magnitude and direction of the resultant


                    Alta High Conceptual Physics
               Example
What is the result of adding two vectors if
 the first vector is 100 km to the west
 and the second vector is 50 km to the
 north?




               Alta High Conceptual Physics
          Mathematical Addition
 Step One: Sketch the vectors
 Step Two: Put vectors on the same axis and draw the
 parallelogram




100 Km West
                  +        50 Km North

              Step One                                  Step Two
                         Alta High Conceptual Physics
         Mathematical Addition
  Step Three: Draw the Resultant
  Step 4: Use Right triangle geometry to find the
          length of the resultant

                           Step Four:
               50 Km
                           Since this is a right triangle you can
                           find the resultant using Pythagorean
100 Km                     theorem
                             Pythagorean Theorem: C2 = A2 + B2
               111.8 Km              C = SQRT(A2 + B2) =
                              SQRT{(100)2+(50)2} =111.8 Km

Step Three

                       Alta High Conceptual Physics
    Graphical vs. Mathematical
             Methods
   Graphical Method
       Less Complicated Math
       Good Approximation
       Not as exact as you would like
   Mathematical Method
       No need for ruler or protractor
       Less time dedicated to sketch
       More Accurate result
       Mathematics is more difficult
                      Alta High Conceptual Physics
           Projectile Motion
   A projectile is any object thrown or
    launched
   Uses the same kinematic equations you
    have already learned
   Requires that you use two sets of
    equations; one in the horizontal and
    one in the vertical
   These two sets are related only by time

                Alta High Conceptual Physics
             Projectile Motion
   A projectile which is dropped or thrown
    horizontally has an initial velocity in the Y
    direction (V0y) = 0
   Acceleration in the horizontal direction (ax) =
    0; hence Velocity in the horizontal direction
    (Vx) is constant
    This means that only one kinematic equation
    applies in the horizontal direction:
                     Xx = Vxt

                   Alta High Conceptual Physics
            Projectile Motion
   While it is moving forward, it is also falling
    just as if you had dropped it.
   So in the Y direction:
                     x = ½ at2
   Where a is the acceleration due to gravity.
   The one thing that both motions have in
    common is time. When the projectile hits the
    ground it stops moving in both directions!

                   Alta High Conceptual Physics
          Projectile Motion
A ball is thrown horizontally off a building
    which is 200 m high with a velocity of
    10 m/s
(a) How long does it take to reach the
    ground?
(b) How far from the building will it land?

(c) What is its velocity just before it hits
    the ground (remember magnitude and
    direction)
               Alta High Conceptual Physics
       Fast Moving Projectiles:
              Satellites
   A satellite is a fast moving projectile
   Since the projectile moves in a curved path,
    and the earth is a curved surface, if the
    projectile moved fast enough its motion
    would follow the curve of the earth
   Such a projectile would fall around the
    earth rather than to the earth
   Such a projectile becomes a satellite and
    orbits the earth.

                   Alta High Conceptual Physics
            Problem Types
   Adding Vectors
   Resolving vectors into components
   Projectile Motion




                Alta High Conceptual Physics

				
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