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Preview. Section 1 Projectile Motion Section 2 Extra Questions. Pythagorean Theorem and Tangent Function. Resolving Vectors Into Components. Opposite of vector addition Vectors are resolved into x and y components.

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  1. Preview Section 1 Projectile Motion Section 2Extra Questions

  2. Pythagorean Theorem and Tangent Function

  3. Resolving Vectors Into Components

  4. Opposite of vector addition Vectors are resolved into x and y components For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that that the angle is 20.0˚. Answers: vx = 89 km/h vy = 32 km/h Resolving Vectors into Components

  5. 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?

  6. Projectile Motion • Projectiles: objects that are launched into the air • tennis balls, arrows, baseballs, wrestlers • Gravity affects the motion • Path is parabolic if air resistance is ignored • Path is shortened under the effects of air resistance

  7. Components of Projectile Motion • As the runner launches herself (vi), she is moving in the x and y directions.

  8. Analysis of Projectile Motion • Horizontal motion • No horizontal acceleration • Horizontal velocity (vx) is constant. • How would the horizontal distance traveled change during successive time intervals of 0.1 s each? • Horizontal motion of a projectile launched at an angle:

  9. Analysis of Projectile Motion • Vertical motion is simple free fall. • Acceleration (ag) is a constant -9.81 m/s2 . • Vertical velocity changes. • How would the vertical distance traveled change during successive time intervals of 0.1 seconds each? • Vertical motion of a projectile launched at an angle:

  10. Projectile Motion Click below to watch the Visual Concept. Visual Concept

  11. Projectile Motion - Special Case Initial velocity is horizontal only (vi,y = 0).

  12. Projectile Motion Summary • Projectile motion is free fall with an initial horizontal speed. • Vertical and horizontal motion are independent of each other. • Horizontally the velocity is constant. • Vertically the acceleration is constant (-9.81 m/s2 ). • Components are used to solve for vertical and horizontal quantities. • Time is the same for both vertical and horizontal motion. • Velocity at the peak is purely horizontal (vy= 0).

  13. 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

  14. Classroom Practice Problem(Projectile Launched at an Angle) • A golfer practices 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 102 m (170 m)

  15. Now 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?

  16. Preview • Multiple Choice • Short Response • Extended Response

  17. Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 9. What is the initial speed of the girl’s ball relative to the boy? A. 1.0 m/s C. 2.0 m/s B. 1.5 m/s D. 3.0 m/s

  18. Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 10. If air resistance is disregarded, which ball will hit the ground first? F. the boy’s ball H. neither G. the girl’s ball J. cannot be determined

  19. Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 10. If air resistance is disregarded, which ball will hit the ground first? F. the boy’s ball H. neither G. the girl’s ball J. cannot be determined

  20. Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 11. If air resistance is disregarded, which ball will have a greater speed (relative to the ground) when it hits the ground? A. the boy’s ball C. neither B. the girl’s ball D. cannot be determined

  21. Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 12. What is the speed of the girl’s ball when it hits the ground? F. 1.0 m/s H. 6.2 m/s G. 3.0 m/s J. 8.4 m/s

  22. Short Response, continued 15. A ball is thrown straight upward and returns to the thrower’s hand after 3.00 s in the air. A second ball is thrown at an angle of 30.0° with the horizontal. At what speed must the second ball be thrown to reach the same height as the one thrown vertically?

  23. Short Response, continued 15. A ball is thrown straight upward and returns to the thrower’s hand after 3.00 s in the air. A second ball is thrown at an angle of 30.0° with the horizontal. At what speed must the second ball be thrown to reach the same height as the one thrown vertically? Answer: 29.4 m/s

  24. Extended Response 16. A human cannonball is shot out of a cannon at 45.0° to the horizontal with an initial speed of 25.0 m/s. A net is positioned at a horizontal distance of 50.0 m from the cannon. At what height above the cannon should the net be placed in order to catch the human cannonball? Show your work.

  25. Extended Response 16. A human cannonball is shot out of a cannon at 45.0° to the horizontal with an initial speed of 25.0 m/s. A net is positioned at a horizontal distance of 50.0 m from the cannon. At what height above the cannon should the net be placed in order to catch the human cannonball? Show your work. Answer: 10.8 m

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