1. Chapter 3 Projectile motion Two Kinematics in Dimensions

2. Projectile Motion • Projectile motion is motion in two directions • Motion in the x-direction is independent of the y-direction • Motion in the y-direction is independent of the x-direction

3. What is a Projectile? • A projectile is any object that is placed into free flight and is being affected by gravity. • The path of a projectile is called the trajectory. Trajectory

4. More on Trajectories • The trajectory of a projectile forms the shape of a parabola. Parabola

5. Half Parabola Different Trajectories • Depending on initial location and angle, projectiles can form different paths, all of which possess some parabolic shape. This path is called a “half” parabola.

6. Trajectory The Half Parabola • An object is launched horizontally and allowed to fall. • There is no vertical velocity at the start, v1y is zero. • The time of flight is equal to the time it would take to drop the object from rest.

7. Partial Parabola Different Trajectories • Depending on initial location and angle, projectiles can form different paths, all of which still possess some parabolic shape. This path is called a “partial” parabola.

8. Different Trajectories • Depending on initial location and angle, projectiles can form different paths, all of which still possess some parabolic shape. This path is called a “full” parabola. Full Parabola

9. Vy Vx The Full Parabola • The key to the full parabola is symmetry. • Try to identify some points of symmetry. Throw

10. Half Parabola Full Parabola Partial Parabola Quick Comparison of Paths • Half Parabola • Launched Horizontally from some height • Full Parabola • Launched at an angle from ground level • Symmetrical trajectory • Partial Parabola • Launched at an angle from some height

11. Half Parabola Timing • The time of flight of a half parabolic path is equal to that of simply dropping the object from the same height. Horizontal velocity (vx) has no affect on flight time because it is not affected by gravity.

12. Jill drops the yellow ball off of a cliff. What happens to the ball? Does it have constant velocity?

13. Now Jill drops the yellow ball and throws the red ball horizontally. Which ball will hit the ground first?

14. Half Parabola Summary • Objects must be dropped from some height d1y. • The vertical reference point is usually the ground or floor. • The time of projectile’s flight is identical for that of simply dropping the object the same distance (straight down). • Horizontal velocity remains constant in all projectile problems (v1x = v2x).

15. A stone is thrown horizontally from the top of a 78.4m high cliff at 5m/s. a) How long does it take to reach the bottom?

16. A stone is thrown horizontally from the top of a 78.4m high cliff at 5m/s. a) How long does it take to reach the bottom?

17. A stone is thrown horizontally from the top of a 78.4m high cliff at 5m/s. b) How far from the base does it land?

18. A stone is thrown horizontally from the top of a 78.4m high cliff at 5m/s. c) What are the final vy and vx ?

19. Sample Partial • A cannon nestled in the side of a cliff (d1y = 65m) fires a cannon ball at 26 . How long until the ball splashes into the sea? Fire

20. Zap! Zap! 0 Sample Problem • A toy car is raced off a table (1.1m high) onto the floor below. • How long did it take for the car to crash on the floor?

21. Projectile Motion Type Not all object are launch horizontally Objects can be launched at an angle

22. Recall the trajectory of the golf ball when hit with a 3 iron. • What would the trajectory of a 9 iron look like? • The loft of the club changed the launch angle.

23. Object 1 was launched at 60o • Object 2 was launched at 30o

24. Object 1 was launched from a 25m high cliff at 0o • Object 2 was launched at 60o

25. Projectile motion generator

26. Physix Instruments Analytical Vector Addition • You will need to understand the basic trig functions. SOH CAH hyp opp TOA adj

27. v1y v1 v1x Initial Velocity Breakdown • When an object is launched at some angle, it’s initial velocity (v1) can be broken down into two components. • Horizontal Component (Vx) • Vertical Component (Vy) • What shape is formed? • Consider also the launch angle (q). Please Note: horizontal and vertical components are independent of one another. The only commonality is time. Right Triangle q

28. Important! v1 v1y v1x Initial Velocity Breakdown (Cont.) • Consider the breakdown from the previous slide again. • There are trigonometric relationships between the sides and angles of a right triangle. q

29. 13 y 22.62° x Practicing Trig Functions • Consider the triangle below. • Solve for the unknown values. Searching for x Searching for y

30. v1 v1y v1x Dart-X Sample Velocity Breakdown • A dart gun is fired at an angle of 30° with a muzzle velocity of 40m/s. • Calculate the components of the velocity? Horizontal Component (x) Vertical Component (y) q Make sure your calculator is in Degree mode!

31. Object launched at an angleMulti media studio • Parabolic motion of projectiles • Non-Horizontally launched projectiles • Maximum Range • Monkey and the Zoo Keeper

32. V1=40 v1y =25o v1x • Given an initial velocity of 40m/s and an angle of 25 • find v1x & v1y Searching for y Searching for x

33. v1 v1y v1x Sample Full Parabola Problem • A golf ball is struck at an angle of q = 36° with the horizontal at a velocity of 45m/s. • What are the components of the velocity (v1x and v1y)? Horizontal Component Strike Vertical Component q

34. Practice Problems Homework • WS 7b 1-2 • WS 7c 1-3

35. X & Y are Independent

36. Problem Solving Strategies • Solve for the horizontal component Vxi • Use trig functions • Solve for the vertical component Vyi • Use trig functions • Solve each direction (x & y) separately • Symmetry can be used when the launching & landing places are the same height.

37. A football player kicks a ball at 27m/s at an angle of 30°. • Find the hang time • find the horizontal distance the ball travels. • The maximum height of the ball.

38. Problem Solving Strategies V=27m/s Step 1:Solve for the horizontal and vertical components (V1x & V1y) V1y= ?m/s V1x= ?m/s Searching for y Searching for x

39. Problem Solving Strategies Symmetry can be used when the launching & landing places are the same height. Vy 15.0m/s 12.5m/s 10.0m/s 7.50m/s 5.00m/s 2.50m/s 0.00m/s

40. A football player kicks a ball at 27m/s at an angle of 30°. • Find the hang time • find the horizontal distance the ball travels. • The maximum height of the ball.

41. A football player kicks a ball at 27m/s at an angle of 30°. • Find the hang time Symmetry

42. A football player kicks a ball at 27m/s at an angle of 30°. b) Find the horizontal distance

43. A football player kicks a ball at 27m/s at an angle of 30°. c) Find the maximum height What is true about the vertical velocity at the maximum height? Vy Vy=0m/s 15.0m/s 12.5m/s 10.0m/s 7.50m/s 5.00m/s 2.50m/s 0.00m/s

44. A football player kicks a ball at 27m/s at an angle of 30°. Find the max height

45. An arrow is shot at 44m/s at an angle of 60° • Find the maximum height of the arrow. • Find the horizontal distance the arrows travels. • Find the hang time

46. Problem Solving Strategies V=44m/s Step 1:Solve for the horizontal and vertical components (V1x & V1y) V1y= ?m/s V1x= ?m/s Searching for y Searching for x

47. An arrow is shot at 44m/s at an angle of 60° • Find the hang time • find the horizontal distance the arrows travels. c) The maximum height of the arrow.

48. A football player kicks a ball at 44m/s at an angle of 60°. • Find the hang time Symmetry

49. A football player kicks a ball at 44m/s at an angle of 60°. b) Find the horizontal distance

50. Recall, vy=0 at dy max A football player kicks a ball at 44m/s at an angle of 60°. Find the max height