1. PROJECTILE MOTION
2-D MOTION

2. Projectile Motion Curve path or trajectory
x and y motion at the same time
Use the 4 kinematics equations
x direction: x, v, t
y direction: y, vf, vo, a, t
2 types
Launched horizontally
Launched at an angle

3. Launched Horizontally
x-direction
NO a
vº = vf
y-direction
Δ uniformly
a = g
vº = 0 m/s
ty = tx

4. Launched Horizontally

5. Launched Horizontally

6. Example 1
A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff 78.4 m high.
How long does it take the stone to reach the bottom of the cliff?
How far from the base of the cliff does the stone strike the ground?
What are the horizontal and vertical components of the velocity of the stone just before it hits the ground?

7. Example 1
Draw Diagram
Make a table x y vx vº t vf a

8. Example 1 x y 78.4 m vx 5.0 m/s vº 0 m/s t vf a 9.8 m/s2
Solve for time
y = vot + ½gt2
= ½gt2
2y (78.4 m)
t = =
√ g √ m/s/s
= 4.0 s

9. Example 1 x y 78.4 m vx 5.0 m/s vº 0 m/s t 4.0 s vf a 9.8 m/s2
Solve for x
x = vxt
= (5.0 m/s) (4.0 s)
= 20. m

10. Example 1 x y 20. m 78.4 m vx 5.0 m/s vº 0 m/s t 4.0 s vf a 9.8 m/s2
Solve for vx and vy final
vx is constant = 5.0 m/s
vy is changing
vy = vº + gt
= gt
= (9.8m/s/s) (4.0 s)
= 39 m/s

11. Example 2
A steel ball rolls with constant velocity across a tabletop m high. It rolls off and hits the ground m horizontally from the edge of the table. How fast was the ball rolling?

12. Example 2 Draw Diagram Make a table x y 0.352 m 0.950 m vx vº 0 m/s tvf a
9.8 m/s2

13. Example 2 x y 0.352 m 0.950 m vx vº 0 m/s t vf a 9.8 m/s2
Solve for time
y = vot + ½gt2
= ½gt2
2y (0.950 m)
t = =
√ g √ m/s/s
= s

14. Example 2 x y 0.352 m 0.950 m vx vº 0 m/s t vf a 9.8 m/s2 0.440 s
Solve for vx x
vx = t
0.352 m
vx =
0.440 s
= m/s

15. Example 3
An auto, moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 43.9 m below the road and 87.7 m beyond the edge of the road.
How long did the auto take to fall?
How fast was it going when it left the road?
What was its acceleration 10 m below the edge of the road?

16. Example 3 Draw Diagram Make a table x y 87.7 m 43.9 m vx vº 0 m/s t vf

17. Example 3 x y 87.7 m 43.9 m vx vº 0 m/s t vf a 9.8 m/s2 Solve for time
y = vot + ½gt2
= ½gt2
2y (43.9 m)
t = =
√ g √ m/s/s
= 2.99 s

18. Example 3 x y 87.7 m 43.9 m vx vº 0 m/s t 2.99 s vf a 9.8 m/s2
Solve for vx x
vx = t
87.7 m
vx =
2.99 s
= 29.3 m/s

19. Example 3 x y 87.7 m 43.9 m vx 29.3 m/s vº 0 m/s t 2.99 s vf a
Acceleration is only due to gravity.

20. Launched at an Angle x-direction y-direction ty = tx Solve x and y
NO a
vº constant
y-direction
Δ uniformly
a = g
tup = tdown
ty = tx
Solve x and y

21. Example 4
A player kicks a football from ground level with a velocity of magnitude of 27.0 m/s at an angle of 30.0º above the horizontal.
Find the hang time.
Find the distance the ball travels before it hits the ground.
Find its maximum height.

22. Example 4
Draw Diagram
Make a table x y vx vº t vf a
9.8 m/s2

23. Launched at Angle Solve for initial vx and vy vx = v cosθ
= (27 m/s) cos 30º
= 23 m/s
vy = v sinθ
= (27 m/s) sin 30º
= 14 m/s x y vx
23 m/s vº
14 m/s t vf a
9.8 m/s2

24. Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time
vy = vo + gt
- 14 m/s = 14 m/s + (-9.8 m/s2) t
-28 m/s = -9.8 m/s2 t
t = 2.8 s

25. Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time
y = vot + ½gt2
= ½gt2
2y (43.9 m)
t = =
√ g √ m/s/s
= 2.99 s

26. Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time
y = vot + ½gt2
= ½gt2
2y (43.9 m)
t = =
√ g √ m/s/s
= 2.99 s

27. Example 5
A rude tourist throws a peach pit horizontally with a 7.0 m/s velocity out of an elevator cage.
If the elevator is not moving, how long will the pit take to reach the ground, 17.0 m below?
How far from the elevator will the pit land?
He throws the next pit when the elevator is at the same height but moving upward at a constant 8.5 m/s velocity. How long will it take this pit to land?
How far away will this pit land?

28. Example 5 Draw Diagram Make a table x y 87.7 m 43.9 m vx vº 0 m/s t vf

29. Graphing Projectile Motion
Height
Horizontal Distance

30. Graphing Projectile Motion
Height
Time

31. Graphing Projectile Motion
Vertical
Speed
Time

32. Graphing Projectile Motion
Horizontal
Velocity
Time

33. Graphing Projectile Motion
Horizontal
Distance
Time