1. Special Cases of Force
Projectile Motion

2. Projectile Motion
A projectile is any object which once a force is used to throw it, hit it , propel it in some fashion, no other force acts on the object except for gravity.
Projection can be
horizontal, with no initial vertical velocity
vertical, with some initial vertical velocity

3. Projectile Motion: A special case of uniformly accelerated motion
This path is a parabola.
Projectile Motion: A special case of uniformly accelerated motion
If air resistance is negligible then only gravity
affects the path (trajectory) of a projectile.

4. Characteristics of Projectile Motion
The motion’s dimensions are vectors.
The projectile will move along it’s trajectory (path) in both the horizontal (x) direction and the vertical (y) direction at the same time.
The two directions are independent of each other.
Only time is the same between the horizontal and vertical motion. It is, after all, only one object.

5. Independent Motions
Horizontal
Vertical

6. Real Motion is the Combination of the Horizontal and Vertical Motions
The blue dot show the real motion
The path taken is the trajectory
This is horizontal projection

7. Initial Velocity – V Along the Trajectory
Angular projection - The initial velocity is the resultant of adding the two vector quantities together.
The projection includes an angle of projection 

8. Initial Velocity has an X and Y Component
The vertical component and the horizontal component are independent of each other.

9. Horizontal and Vertical Components of Velocity
Vertical velocity decreases at a constant rate
due to the influence of gravity.
It becomes zero.
Then increases in the negative direction
Negative velocity
gets larger
Vertical velocity = 0
Positive velocity
gets smaller

10. Calculating Components
You have learned to calculate components of a vector when we looked at inclined planes.
The components are calculated by using the trig functions.
The initial velocity acts as the hypotenuse of the right triangle. Vi

11. Continued Calculation
The vertical velocity and the horizontal velocity are the legs of the triangle.  Vy Vx Vi
Calculate the Vy using sin = o/h
Calculate Vx using cos = a/h

12. Horizontal Displacement of Projectiles
Horizontal projection or projection at an angle
graph of horizontal displacement v time
As time progresses, the object gets further and further from its starting point.
Time
Displacement

13. Vertical Displacement of Projectiles
Vertical Projection and Projection at an Angle
graph of vertical displacement (height) v time
The object rises into the air and then return to earth.
Time
Height

14. Vertical Displacement of Projectiles
Horizontal projection
Graph of vertical displacement (height) vs time
The object leaves a horizontal surface and fall to the ground.
Time
Height

15. Horizontal Velocity of Projectiles
Horizontal Projection and Projection at an Angle
Time
Velocity
The horizontal velocity of a projectile remains constant from the time it is projected until gravity brings it to the ground.
Remember: We are using an ideal situation where there is no air resistance

16. Vertical Velocity of Projectiles
Vertical Projection and Projection at an Angle
Time
Velocity
For objects projected directly upward or projected at some angle above the ground, the vertical velocity must begin positive, decrease to zero, and then increase in the negative direction. (Remember, gravity is negative)

17. Vertical Velocity of Projectiles
Horizontal projection
Time
Velocity
Horizontal projection begins with an initial vertical velocity of zero.
The vertical velocity then increases in a negative direction.

18. Horizontal Acceleration of Projectiles
Since we are idealizing the projection, we do not take into account any air resistance.
We can, therefore, say there is no horizontal acceleration.
Time
Acceleration

19. Vertical Acceleration of Projectiles
Vertical acceleration is the result of the pull of gravity, (-9.8 m/s2)
This is the same on the way up and on the way down.
Acceleration
-9.8
Time

20. Important Notes on Vertical and Angular Projection
The range (x) is the farthest the object will travel horizontally.
The maximum height (ymax ) is the farthest the object will travel vertically.
Y equals zero when it is at its lowest point.

21. Determining the Range
You can determine the displacement (range) of a projectile any any point along the trajectory.
X = horizontal distance (range)
vx = horizontal velocity
t = time

22. Determining the Height
You can determine the height (y) at any point in the trajectory!
y = vertical displacement
vy = initial vertical velocity
g = acceleration due to gravity
t = time

23. Other Important Notes on Vertical and Angular Projection
At the highest point of the trajectory, it is the exact midpoint of the time.
It takes the projectile half of the time to get to the top.
When the projectile gets to the top, it has to stop going up and start going down, so the velocity in the y-direction at the highest point is zero for a split second.
As the projectile falls, it is in free fall.

24. 3 Primary Factors Affecting Trajectory
Projection angle
aka release angle or take-off angle
Projection velocity
aka initial or take-off velocity
Projection height
aka above or below landing

25. Projection Angle
The optimal angle of
projection is dependent on
the goal of the activity.
For maximum height, the optimal angle is 90o.
For maximum distance, the optimal angle is 45o.

26. The effect of Projection angle on the Range of a projectile
10 degrees
30 degrees
40 degrees
45 degrees
60 degrees
75 degrees
The angle that maximizes Range is = 45 degrees

27. The effect of Projection velocity on the Range of a projectile
10 45 degrees Range ~ 10 m
20 45 degrees Range ~ 40 m
30 45 degrees Range ~ 90 m
100 90 80 70 60 50 40 30 20 10

28. Projectile Problems
Ignore air resistance.
ay = g = m/s2
Set up the two dimension separately
Origin x Origin y
Positive x Positive y
xi = constant yi =
vxi = vyi =
ax = ay = g

29. Projectile Problems – Two Dimensional Kinematics
Write general kinematic equations for each direction
Rewrite them for the problem at hand
Find the condition that couples the horizontal and vertical motions (usually time)

30. Equations of Constant Acceleration
This is the only equation to use for the horizontal part of the motion
x = vxt
d = ½ (vyf + vyi)t
d = vyit + ½ gt2
vvf2 = vvi2 + 2gd
These 3 equations are for the vertical part of the motion

31. The Monkey and the Banana
A zookeeper must throw a banana to a monkey hanging from the limb of a tree. The monkey has a habit of dropping from the tree the moment that the banana is thrown. If the monkey lets go of the tree the moment that the banana is thrown,will the banana hit the monkey?

32. When you take gravity into consideration you STILL aim at the monkey!
Monkey’s
Gravity free path is “floating” at height of limb
Banana’s
Gravity free path
Fall thru same height
It works! Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. Thus, the banana hits the monkey.

33. Homework Chapter 6 read to page 152
Problems: page 164, # 33,35,51,52,53,56
57,60.