2. Projectile Motion Projectile motion: a combination of horizontal motion with constant horizontal velocity and vertical motion with a constant downward acceleration due to gravity. Projectile motion refers to the motion of an object that is thrown, or projected, into the air at an angle. We restrict ourselves to objects thrown near the Earth’s surface as the distance traveled and the maximum height above the Earth are small compared to the Earth’s radius so that gravity can be considered to be constant.

3. Horizontal Component of Velocity

4. Horizontal and Vertical

5. Projectile Motion The motion of a projectile is determined only by the object’s initial velocity and gravity. The vertical motion of a projected object is independent of its horizontal motion. The vertical motion of a projectile is nothing more than free fall. The one common variable between the horizontal and vertical motions is time.

6. Path of a Projectile A projectile moves horizontally with constant velocity while being accelerated vertically. A right angle exists between the direction of the horizontal and vertical motion; the resultant motion in these two dimensions is a curved path. The path of a projectile is called its trajectory. The trajectory of a projectile in free fall is a parabola.

8. Path of a Projectile

9. v o = initial velocity or resultant velocity v x = horizontal velocity v yi = initial vertical velocity v yf = final vertical velocity R= maximum horizontal distance (range) x = horizontal distance  y = change in vertical position y i = initial vertical position y f = final vertical position  = angle of projection (launch angle) H = maximum height g = gravity = 9.8 m/s 2

10. Path of a Projectile

11. The horizontal distance traveled by a projectile is determined by the horizontal velocity and the time the projectile remains in the air. The time the projectile remains in the air is dependent upon gravity. Immediately after release of the projectile, the force of gravity begins to accelerate the projectile vertically towards the Earth’s center of gravity.

12. Path of a Projectile The velocity vector v o changes with time in both magnitude and direction. This change is the result of acceleration in the negative y direction (due to gravity). The horizontal component ( x component) of the velocity v o remains constant over time because there is no acceleration along the horizontal direction The vertical component ( v y ) of the velocity v o is zero at the peak of the trajectory. However, there is a horizontal component of velocity, v x, at the peak of the trajectory.

13. Path of a Projectile

15. In the prior diagram, r is the position vector of the projectile. The position vector has x and y components and is the hypotenuse of the right triangle formed when the x and y components are plotted. The velocity vector v o  t would be the displacement of the projectile if gravity were not acting on the projectile. The vector 0.5  g  t 2 is the vertical displacement of the projectile due to the downward acceleration of gravity. Together, this determines the vertical position for the projectile: Δy = (v y ·t) – (0.5·g·t 2 )

16. Path of a Projectile

17. Acceleration of a Projectile g g g g g x y Acceleration points down at 9.8 m/s 2 for the entire trajectory of all projectiles.

18. Velocity of a projectile vovo vfvf v v v x y Velocity is tangent to the path for the entire trajectory.

19. Position graphs for 2-D projectiles x y t y t x

20. Velocity graphs for 2-D projectiles t Vy t Vx

21. Acceleration graphs for 2-D projectiles t ay t ax

22. Fig. 04.24

23. Problem Solving: Projectile Motion Analyze the horizontal motion and the vertical motion separately. If you are given the velocity of projection, v o, you may want to resolve it into its x and y components. Think for a minute before jumping into the equations. Remember that v x remains constant throughout the trajectory, and that v y = 0 m/s at the highest point of any trajectory that returns downward.

24. Horizontal velocity component: v x is constant because there is no acceleration in the horizontal direction if air resistance is ignored.

25. Vertical velocity component: At the time of launch: After the launch: If v y positive, direction of vertical motion is up; if v y negative, direction of vertical motion is down; if v y = 0, projectile is at highest point.

26. Horizontal position component: If you launch the projectile horizontally: then v o = v x v yi = 0 m/s  = 0 o

27. Vertical position component:

28. Relationship Between Vertical and Horizontal Position: this equation is only valid for launch angles in the range 0  <  < 90 

29. Range (total horizontal displacement)

31. Maximum Height

32. When Do The Range & Maximum Height Equations Work? Works when  y = 0. Does not work when  y  0.

33. Determining v o from v x and v y If the vertical and horizontal components of the velocity are known, then the magnitude and direction of the resultant velocity can be determined. Magnitude:

34. Determining v o from v x and v y Direction: from the horizontal Direction: from the vertical

35. Range and Angle of Projection

36. The range is a maximum at 45  because sin (2·45) = 1. For any angle  other than 45 , a point having coordinates (x,0) can be reached by using either one of two complimentary angles for , such as 15  and 75  or 30  and 60 .

37. Range and Angle of Projection The maximum height and time of flight differ for the two trajectories having the same coordinates (x, 0). A launch angle of 90° (straight up) will result in the maximum height any projectile can reach.

39. For Objects Shot Horizontally: v x constant  y negative;  y = - height

40. Zero Launch angle A zero launch angle implies a perfectly horizontal launch. vovo

41. For Objects Shot Horizontally: When hits at bottom: V yf should be negative v o = resultant velocity

42. For Objects Shot Horizontally:  with horizontal:  with vertical:

43. For Situations in Which  y = 0 m

44. For Situations In Which  y Positive

45. At any point in the flight:

46. For Situations In Which  y Negative

47. At launch: After launch: When it hits ground: