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.
We restrict ourselves to objects thrown near the Earth’s surface so that gravity can be considered to be constant.

4. 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.

5. Horizontal velocity vector (remains constant)
Vertical velocity vector (steadily increasing)

6. Gravity only affects vertical motion.
Projectile motion problems are best solved by
treating horizontal and vertical motion separately.
*IMPORTANT*
Gravity only affects vertical motion.
There are two general types
of projectile motion situations.
1. object launched horizontally
2. object launched at an angle

7. 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. Rx = vxt h = 0.5gt2 Object Launched Horizontally horizontal vertical
vx = initial horizontal velocity
h = initial height
above ground
t = total time in the air
Rx = horizontal range
IMPORTANT FACTS
There is no horizontal acceleration.
There is no initial vertical velocity.
The horizontal velocity is constant.
Time is the same for both vertical and horizontal.
horizontal
vertical
Rx = vxt
h = 0.5gt2

9. Motion in 2D…
Throw a ball on a moving truck demo
Monkey shooting

12. For Objects Shot Horizontally:
vx constant
Dy negative
Dy = -height

13. For Objects Shot Horizontally:
When hits at bottom:
Vyf should be negative
vo = resultant velocity

14. For Objects Shot Horizontally:
q with horizontal:
q with vertical:

15. Motion in two dimensions
Projectile Motion
Motion in two dimensions
© 2006 Certiport.com

16. When a projectile is launched at an angle the object moves both up and down and side to side at the same time

17. The vertical component (Vy) controls how high the object rises
The vertical component (Vy) controls how high the object rises. This is the objects maximum height. It occurs half way through the projectiles path **The path of a projectile is called its trajectory

18. This means, when a projectile is launched at an angle, it’s velocity has a vertical and horizontal component

19. Since the maximum height (dy) occurs half way through the trajectory, vertical equations use time for only half the trip total time=4s
ty=2s

20. The horizontal component (Vh) controls how far over the object travels
The horizontal component (Vh) controls how far over the object travels. This is called the object’s range. It occurs at the end of the entire trajectory.

21. Since the object’s range (dx), occurs over the entire trip, horizontal equations must use the time for the total trip.
tx=4s

22. Changing the angle at which the projectile is launched effects the vertical and horizontal distances the object travels

23. Fill in the x & y velocity vectors

25. Path of a Projectile

26. Path of a Projectile vo = initial velocity or resultant velocity
vx = horizontal velocity
vyi = initial vertical velocity
vyf = final vertical velocity
R= maximum horizontal distance (range)
x = horizontal distance
Dy = change in vertical position
yi = initial vertical position
yf = final vertical position
q = angle of projection (launch angle)
H = maximum height
ag = gravity = 9.8 m/s2

27. Path of a Projectile
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.

28. Path of a Projectile
The velocity vector vo 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 vo remains constant over time because there is no acceleration along the horizontal direction
The vertical component (vy) of the velocity vo is zero at the peak of the trajectory. However, there is a horizontal component of velocity, vx, at the peak of the trajectory.

29. Path of a Projectile

30. Horizontal velocity component:
vx is constant because there is no acceleration in the horizontal direction if air resistance is ignored.

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

32. Horizontal position component:
If you launch the projectile horizontally:
then vo = vx
vyi = 0 m/s
 = 0o

33. Vertical position component:

34. Determining vo from vx and vy
If the vertical and horizontal components of the velocity are known, then the magnitude and direction of the resultant velocity can be determined.
Magnitude:

35. Determining vo from vx and vy
Direction: from the horizontal
Direction: from the vertical

36. Range and Angle of Projection

37. Range and Angle of Projection
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 50 .

38. 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 Situations In Which Dy Positive
At any point in the flight:

40. What angle gives the projectile maximum range?

41. Excel Spreadsheet

42. And the winner is ….
A 45 degree angle gives the projectile maximum range
In nature, frogs naturally jump at approximately a degree angle to help them cover maximum distance on flat ground