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Two-Dimensional Motion and
Vectors Chapter 3 pg. 81-105
+

            What do you think?

 How    are measurements such as mass and
    volume different from measurements such ...
+
    Introduction to Vectors


     Scalar- a quantity that has magnitude but
     no direction
      Examples:   volum...
+
    Vector Properties

     Vectors   are generally drawn as arrows.
      Length represents the magnitude
      Arro...
+
    Finding the Resultant Graphically

                     Method
                        Draw each vector in the pro...
+
    Vector Addition

                  Vectorscan be moved parallel
                  to themselves without changing
  ...
+
    Vector Addition

                       Vectorscan be added
                       in any order.
                  ...
Sample Resultant Calculation
                    A toycar moves with a
                    velocity of .80 m/s across a
 ...
+




3.2 Vector Operations
+

                What do you think?

   What is one disadvantage of adding vectors by the graphical
    method?

   Is...
+
    Vector Operations

     Use a traditional x-y coordinate system as shown below
     on the right.
     The Pythago...
+ Pythagorean Theorem and Tangent Function
+
    Pythagorean Theorem and Tangent
    Function
     We can use the inverse of the tangent
     function to find the a...
+
    Example


     An archaeologist climbs the great pyramid
     in Giza. The pyramid height is 136 m and
     width i...
+
    Example


     Given:

     Δy=   136m
      width    is 2.30 X 102m for whole pyramid
     So, Δx    = 115m
   ...
+
    Example


 Calculate:               θ=   tan-1 (opp/adj)

d2   =Δx2 + Δy2           θ=   tan-1 (136/115)
       ...
+
    Example


     While following the directions on a
     treasure map a pirate walks 45m north
     then turns and w...
+
    Resolving Vectors Into Components
+
    Resolving Vectors into Components


     Component:  the horizontal x and vertical yparts that
     add up to give ...
+
    Example

     Given:   v= 95 km/h          θ= 35.0°
     Unknown     vx=??vy= ??
     Rearrange    the equations
...
+
    Example


    vy=(sin    θ)(v)      vx=    (cosθ)(v)
     vy=   (sin35°)(95)    vx   = (cos 35°)(95)
     vy=  ...
+
    Example


     Howfast must a truck travel to stay
     beneath an airplane that is moving 105
     km/h at an angl...
+




3.3 Projectile Motion
+

                    What do you think?

     Suppose   two coins fall off of a table simultaneously. One
     coin fal...
+
    Projectile Motion

     Projectiles:   objects that are launched into the air
        tennis balls, arrows, baseba...
+
    Projectile Motion


     Pathis parabolic if air resistance is
     ignored
     Path is shortened under the effec...
Components of Projectile Motion

                As the runner launches
                herself (vi), she is
            ...
+
    Projectile Motion


     Projectile
               motion is free fall with an initial
     horizontal speed.
    ...
+
    Projectile Motion


     Components  are used to solve for vertical
     and horizontal quantities.
     Timeis th...
+
    Example


     The Royal Gorge Bridge in Colorado rises
     321 m above the Arkansas river. Suppose
     you kick ...
+
    Example


     Given:   d = 321m    a = 10m/s2
    vi= 5m/s         t = ??             vf = ??
     REMEMBER      ...
+
    Classroom Practice Problem
    (Horizontal Launch)
     People   in movies often jump from buildings into
     pool...
+
    Projectiles Launched at an Angle


     We will make a triangle and use our sin,
     cos, tan equations to find ou...
+
     Classroom Practice Problem
     (Projectile Launched at an Angle)
     A golferpractices driving balls off a cliff...
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Two Dimensional Motion and Vectors Slide 1 Two Dimensional Motion and Vectors Slide 2 Two Dimensional Motion and Vectors Slide 3 Two Dimensional Motion and Vectors Slide 4 Two Dimensional Motion and Vectors Slide 5 Two Dimensional Motion and Vectors Slide 6 Two Dimensional Motion and Vectors Slide 7 Two Dimensional Motion and Vectors Slide 8 Two Dimensional Motion and Vectors Slide 9 Two Dimensional Motion and Vectors Slide 10 Two Dimensional Motion and Vectors Slide 11 Two Dimensional Motion and Vectors Slide 12 Two Dimensional Motion and Vectors Slide 13 Two Dimensional Motion and Vectors Slide 14 Two Dimensional Motion and Vectors Slide 15 Two Dimensional Motion and Vectors Slide 16 Two Dimensional Motion and Vectors Slide 17 Two Dimensional Motion and Vectors Slide 18 Two Dimensional Motion and Vectors Slide 19 Two Dimensional Motion and Vectors Slide 20 Two Dimensional Motion and Vectors Slide 21 Two Dimensional Motion and Vectors Slide 22 Two Dimensional Motion and Vectors Slide 23 Two Dimensional Motion and Vectors Slide 24 Two Dimensional Motion and Vectors Slide 25 Two Dimensional Motion and Vectors Slide 26 Two Dimensional Motion and Vectors Slide 27 Two Dimensional Motion and Vectors Slide 28 Two Dimensional Motion and Vectors Slide 29 Two Dimensional Motion and Vectors Slide 30 Two Dimensional Motion and Vectors Slide 31 Two Dimensional Motion and Vectors Slide 32 Two Dimensional Motion and Vectors Slide 33 Two Dimensional Motion and Vectors Slide 34
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Two Dimensional Motion and Vectors

  1. 1. + Two-Dimensional Motion and Vectors Chapter 3 pg. 81-105
  2. 2. + What do you think?  How are measurements such as mass and volume different from measurements such as velocity and acceleration?  How can you add two velocities that are in different directions?
  3. 3. + Introduction to Vectors  Scalar- a quantity that has magnitude but no direction  Examples: volume, mass, temperature, speed  Vector - a quantity that has both magnitude and direction  Examples:acceleration, velocity, displacement, force
  4. 4. + Vector Properties  Vectors are generally drawn as arrows.  Length represents the magnitude  Arrow shows the direction  Resultant - the sum of two or more vectors  Make sure when adding vectors that  Youuse the same unit  Describing similar quantities
  5. 5. + Finding the Resultant Graphically  Method  Draw each vector in the proper direction.  Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length.  Draw the resultant from the tip of the first vector to the tail of the last vector.  Measure the resultant.  The resultant for the addition of a + b is shown to the left as c.
  6. 6. + Vector Addition  Vectorscan be moved parallel to themselves without changing the resultant.  the red arrow represents the resultant of the two vectors
  7. 7. + Vector Addition  Vectorscan be added in any order.  The resultant (d) is the same in each case  Subtraction is simply the addition of the opposite vector.
  8. 8. Sample Resultant Calculation  A toycar moves with a velocity of .80 m/s across a moving walkway that travels at 1.5 m/s. Find the resultant speed of the car.
  9. 9. + 3.2 Vector Operations
  10. 10. + What do you think?  What is one disadvantage of adding vectors by the graphical method?  Is there an easier way to add vectors?
  11. 11. + Vector Operations  Use a traditional x-y coordinate system as shown below on the right.  The Pythagorean theorem and tangent function can be used to add vectors.  More accurate and less time-consuming than the graphical method
  12. 12. + Pythagorean Theorem and Tangent Function
  13. 13. + Pythagorean Theorem and Tangent Function  We can use the inverse of the tangent function to find the angle.  θ= tan-1 (opp/adj)  Another way to look at our triangle d  d2 =Δx2 + Δy2 Δy θ Δx
  14. 14. + Example  An archaeologist climbs the great pyramid in Giza. The pyramid height is 136 m and width is 2.30 X 102m. What is the magnitude and direction of displacement of the archaeologist after she climbs from the bottom to the top?
  15. 15. + Example  Given:  Δy= 136m  width is 2.30 X 102m for whole pyramid  So, Δx = 115m  Unknown: d = ?? θ= ??
  16. 16. + Example  Calculate:  θ= tan-1 (opp/adj) d2 =Δx2 + Δy2  θ= tan-1 (136/115)  θ= 49.78° d = √Δx2 + Δy2 d = √ (115)2 +(136)2 d = 178m
  17. 17. + Example  While following the directions on a treasure map a pirate walks 45m north then turns and walks 7.5m east. What single straight line displacement could the pirate have taken to reach the treasure?
  18. 18. + Resolving Vectors Into Components
  19. 19. + Resolving Vectors into Components  Component: the horizontal x and vertical yparts that add up to give the actual displacement  Forthe vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that the angle is 35.0˚. 35°
  20. 20. + Example  Given: v= 95 km/h θ= 35.0°  Unknown vx=??vy= ??  Rearrange the equations  sin θ= opp/ hyp  opp=(sin θ) (hyp)  cosθ= adj/ hyp  adj= (cosθ)(hyp)
  21. 21. + Example vy=(sin θ)(v) vx= (cosθ)(v)  vy= (sin35°)(95)  vx = (cos 35°)(95)  vy= 54.49 km/h  vx = 77.82 km/h
  22. 22. + Example  Howfast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of 25° to the ground?
  23. 23. + 3.3 Projectile Motion
  24. 24. + What do you think?  Suppose two coins fall off of a table simultaneously. One coin falls straight downward. The other coin slides off the table horizontally and lands several meters from the base of the table.  Which coin will strike the floor first?  Explain your reasoning.  Would your answer change if the second coin was moving so fast that it landed 50 m from the base of the table? Why or why not?
  25. 25. + Projectile Motion  Projectiles: objects that are launched into the air  tennis balls, arrows, baseballs, javelin  Gravity affects the motion  Projectile motion:  The curved path that an object follows when thrown, launched or otherwise projected near the surface of the earth
  26. 26. + Projectile Motion  Pathis parabolic if air resistance is ignored  Path is shortened under the effects of air resistance
  27. 27. Components of Projectile Motion  As the runner launches herself (vi), she is moving in the x and y directions.
  28. 28. + Projectile Motion  Projectile motion is free fall with an initial horizontal speed.  Vertical and horizontal motion are independent of each other. the acceleration is constant (-10 m/s2 )  Vertically  We use the 4 acceleration equations  Horizontally the velocity is constant  We use the constant velocity equations
  29. 29. + Projectile Motion  Components are used to solve for vertical and horizontal quantities.  Timeis the same for both vertical and horizontal motion.  Velocity at the peak is purely horizontal (vy= 0).
  30. 30. + Example  The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas river. Suppose you kick a rock horizontally off the bridge at 5 m/s. How long would it take to hit the ground and what would it’s final velocity be?
  31. 31. + Example  Given: d = 321m a = 10m/s2 vi= 5m/s t = ?? vf = ??  REMEMBER we need to figure out :  Up and down aka free fall (use our 4 acceleration equations)  Horizontal (use our constant velocity equation)
  32. 32. + Classroom Practice Problem (Horizontal Launch)  People in movies often jump from buildings into pools. If a person jumps horizontally by running straight off a rooftop from a height of 30.0 m to a pool that is 5.0 m from the building, with what initial speed must the person jump?  Answer: 2.0 m/s
  33. 33. + Projectiles Launched at an Angle  We will make a triangle and use our sin, cos, tan equations to find our answers  Vy = V sin θ  Vx = V cosθ  tan = θ(y/x)
  34. 34. + Classroom Practice Problem (Projectile Launched at an Angle)  A golferpractices driving balls off a cliff and into the water below. The edge of the cliff is 15 m above the water. If the golf ball is launched at 51 m/s at an angle of 15°, how far does the ball travel horizontally before hitting the water?  Answer: 1.7 x 102m (170 m)
  • wagsc

    Oct. 25, 2014
  • dennielle_20

    Aug. 10, 2014

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