Two Dimensional Motion and Vectors

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Two Dimensional Motion and Vectors

  1. 1. +Two-Dimensional Motion andVectors 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)

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