1. What is Projectile Motion?

2. Instructional Objectives:
Students will be able to:
Define Projectile Motion
Distinguish between the different types of projectile motion
Apply the concept to a toy car and measure its velocity

3. Vocabulary
Projectile – any object projected through the air, whose motion is due to the force of gravity acting on the object and the object’s inertia.
Inertia – the tendency of an object at rest to remain at rest or an object in motion to remain in motion with uniform speed in a straight line.

4. Vocabulary
Trajectory - the name given to the path followed by a projectile
Range – the horizontal displacement of a projectile
Launch angle – the angle, with respect to the horizontal, at which an object is projected into the air.
trajectory θ
range

5. Projectile Motion
When an object is thrown or launched into the air, it undergoes projectile motion. The pathway of any projectile is a parabola (excluding air resistance).

6. Types of Projectile Motion
Horizontal
Motion of a ball rolling freely along a level surface
Horizontal velocity is ALWAYS constant
Vertical
Motion of a freely falling object
Force due to gravity
Vertical component of velocity changes with time
Parabolic
Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

7. Equations Used to Analyze Projectile Motion
For horizontal motion: Use the constant speed equation, v = Δx/t
For vertical motion: Use the Big 4 kinematics equations for uniformly accelerated motion
Assume no air resistance
Assume “g” is constant throughout the object’s motion

8. Vertically Launched Projectiles
An object dropped or thrown vertically downward or projected vertically upward is considered a projectile.
The motion of such an
object is analyzed using
the equations for free-fall.

9. Horizontally Launched Projectiles
HORIZONTAL AND VERTICAL MOTIONS ARE ALWAYS
INDEPENDENT OF EACH OTHER!
Horizontal motion is due to inertia.
Vertical motion is due to gravity.

10. Analyzing Horizontal and Vertical Motions
Horizontal Motion (x) Vertical Motion (y)
vix ≠ 0 vfx ≠ 0
viy = 0
vfy ≠ 0
vix = vfx
g = m/s2
a = 0
To find ∆y or t,
∆x = range
use equations for free-fall
vconstant = ∆x/t
vf = vi + gt
vconstant = vix
vf2 = vi2 + 2g∆y
∆x = vixt
∆y = vit + ½gt2
∆y = ½ (vi + vf) t

11. The distance a horizontally projected object falls vertically in every second is equal to the distance a freely-falling object falls vertically in every second.
0 m m m m m
Initial horizontal velocity = 5.0 m/s
Initial vertical velocity = 0 m/s
Total time of “flight” = 4.0 s
Horizontal distance traveled /second = 5.0 m
Horizontal distance traveled in 4.0 s = 20. m
4.9 m
19.6 m
44.1 m
78.4 m

12. Projectiles Launched at an Angle
viy=0
vfy=0 Δy
viy
vix
Range, Δx

13. vi is the initial velocity.
In the previous diagram:
θ is the launch angle.
vi is the initial velocity.
vix is the x-component of the initial velocity.
viy is the y-component of the initial velocity.
vfy is the y-component of the final velocity and
equals zero at the top of the trajectory.
vfx is the x-component of the final velocity
and is equal to vix throughout the object’s
motion. (No acceleration horizontally!)
∆y is the vertical displacement of the object.

14. Analyzing Horizontal and Vertical Motions
vix = vi cos θ viy = vi sin θ
vfx = vix vfy = 0
a = a = g
Δx = 2 vixt Δy = viyt + ½ gt2
Δx = 2 vitcos θ Δy = vitsin θ + ½ gt2
vfy2 = viy2 + 2gΔy
0 = (visin θ)2 + 2gΔy
Δy = ½ vi2sin2θ 2g
tup = tdown
ttotal = tup + tdown =2t

15. What’s Important:
The trajectory has a symmetrical shape.
The height of the trajectory is the
highest point reached by the object.
At the top of the trajectory, the
object’s vertical velocity (vfy) = 0.
Time up = time down
Total time in the air = 2t
The range = 2 vixt = 2 vitcos θ
θ up = θ down

17. Projectile Motion
After breaking the initial velocity into x and y components separately, the pathway of the object can be tracked by using a series of equations:

18. Examples of Projectile Motion
Launching a Cannon ball

19. Projectile Motion Equations
Variables:
Y = distance in meters vertically
Vy = y component of velocity
Viy = y component of initial velocity
Vix = x component of initial velocity
g = acceleration of gravity
t = time during flight

20. Equations X- Component Y- Component Vectors X or Horizontal:
Velocity: Vx remains constant; therefore
vi(x) = vf(x) (Newton’s 1st law of motion)
X- Component
Y- Component
Vectors
Note: g= 9.8 m/s2

21. Factors Affecting Projectile Motion
What two factors would affect projectile motion?
Angle
Initial velocity
Initial Velocity
Angle

22. Projectile Motion
In projectile problems, vertical and horizontal motions are completely independent of one another.
For example: An object will accelerate downward at the exact same rate regardless of whether it is dropped, thrown forward slowly, or thrown forward very quickly.
Likewise: An object will continue to move forward the same distance every second whether it is rolling along a flat surface or moving vertically.

23. Projectile Motion
Time is the only variable which will be the same for the two motions. The time it takes for an object to hit the ground is the same amount of time it has to move forward.

24. Projectile Motion Equations
One method for solving for horizontally projected objects:
1) Write down all variables, and identify them as horizontal or vertical.
2) Solve one dimension (x or y, depending on what is given in the problem) for time.
3) Use that time in the other dimension to find unknown quantity

25. Class Exercise
An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal
Calculate the horizontal and vertical components of the initial velocity
After 2.0 seconds, how far has the object traveled in the horizontal direction?
How high is the object at this point?

26. Solution
Part a
Part b
Part c

27. Projectile Launched at an Upward Angle from an Elevated Surface

28. θ
Treat problem as
2 separate motions. θ

29. Remember: For ANY problem:
You cannot interchange x and y components for initial and final velocities!
You cannot interchange ∆x and ∆y values!
You must solve for horizontal and vertical quantities separately!
The only link between horizontal and vertical motion is TIME !