Objects in projectile motion follow a parabolic trajectory under the influence of gravity. The horizontal and vertical motions are independent and can be treated as separate one-dimensional motions. For a projectile launched at an angle, the horizontal velocity (vx) and initial vertical velocity (vyi) can be calculated using trigonometric functions of the launch angle and initial speed. The maximum horizontal distance of travel, or range, can then be determined by calculating the time of flight from the vertical motion equations and using that time in the horizontal motion equation.

1. Projectile Motion

2. Objects in projectile motion follow a
parabolic path called a trajectory.

3. If released at the same instant, a bullet shot from a
gun will hit the ground at the same time as a bullet
dropped from the same height.

4. The horizontal and vertical motions of a
projectile are independent, meaning they do
not affect each other.
Horizontal motion
Vertical motion

5. The 2-D motion of a projectile can be separated
into two 1-D motions: horizontal and vertical.

6. HPM: Horizontal Projectile Motion

7. The horizontal motion of a projectile is always
constant, if we neglect air resistance.

8. For projectiles shot at 0°, all of the initial velocity
is in the x direction. Thus, Vyi = 0 m/s.

9. For projectiles shot at 0°, vertical displacement
and velocity will always be negative.

10. To hit the target, when should you release
the package?

11. Rules for Projectile Motion
• Treat horizontal and vertical as two
separate sides of the problems
• TIME is the key, and the only variable that
can be used for both horizontal and
vertical
• Horizontal Motion is always constant
• vx is constant
• ax = 0 m/s2
• Objects follow a parabolic shape

12. Horizontal Projectile Motion
• All of the initial velocity is in the x direction,
Vyi = 0 m/s
• Vertical displacement and velocity will
always be negative

13. Example problem

14. APM: Angled Projectile Motion

15. For projectiles shot at an angle, initial
velocity is both vertical and horizontal.

16. Horizontal velocity and initial vertical velocity can
be found using trig functions.
Vx = Vi Cos θ
Vyi
Vx
Vi
θ
Vyi = Vi Sin θ

17. At the end of the problem, you can recombine
horizontal and vertical velocity to get the total 2-D
velocity.
Vx
Vyf
Vf
Vx
Vyf
Vf
Vf
2 = Vx
2 + Vyf
2
θ
θ = tan-1(Vyf / Vx)

18. For APM, vy at any height is the same while going
up and coming down except for direction.

20. Velocities of projectile motion
Note: Vy = 0 at the highest point.

21. With air resistance, the actual path
is shorter.

22. Angled Projectile Motion
• Initial velocity is both vertical and
horizontal
• Use trig functions to find vyi and vx
• Vx = Vi Cos θ
• Vyi = Vi Sin θ
• Remember clues
• vy at the top is 0 m/s
• vy at any height is the same while going up
and coming down except for direction

23. Example Problem
Happy Gilmore hits his shot at 55.0 m/s with an
angle of 50.0° to the ground. How far did the
ball travel before it lands?
•vi = 55.0 m/s
•θ = 50.0°
•ay = -9.81 m/s2
•∆x = ?

24. vi = 55.0 m/s ay = -9.81 m/s2 ∆x = ?
θ = 50.0°
• Find vx and vyi
vx = vi Cos θ
= 55.0 m/s Cos (50.0°)
= 35.4 m/s
vyi = vi Sin θ
= 55.0 m/s Sin (50.0°)
= 42.1 m/s

25. vi = 55.0 m/s ay = -9.81 m/s2 ∆x = ?
θ = 50.0° vyi = 42.1 m/s vx = 35.4 m/s
• What do we need to find ∆x?
• Time! Find time from the vertical side
• ∆y = vyi ∆t + ½ ay ∆t2
0 m = (42.1 m/s) ∆t + ½ (-9.81 m/s2) ∆t2
- (42.1 m/s) ∆t = ½ (-9.81 m/s2) ∆t2
(42.1 m/s) = ½ (9.81 m/s2) ∆t
∆t = 8.58 s

26. vi = 55.0 m/s ay = -9.81 m/s2 ∆x = ?
θ = 50.0° vyi = 42.1 m/s vx = 35.4 m/s
∆t = 8.58 s
• Now we can find ∆x
• ∆x = vx ∆t
= (35.4 m/s)(8.58 s)
= 304 m

27. Where Should We Aim the Cannon?
At or Above the monkey?

28. Above the Monkey

29. At the Monkey

30. Explanation of monkey experiment

31. Ranges of projectiles versus angle.

32. Another interesting application

33. If speed is great enough…

34. That’s how the space shuttle and
satellites orbit the earth.