1. Projectile Motion
SPH4U

2. Projectiles
A projectile is an object on which the only force acting is gravity. It is said to be “in free fall” even if part of its motion is upward.

3. Projectiles
A projectile may also be travelling horizontally while falling if its initial (launch) velocity had a horizontal component.

4. Parametric Equations
As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:

5. Parametric Equations
As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:
Vertical acceleration is –g.
Horizontal acceleration is zero.

6. Parametric Equations
As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:

7. Parametric Equations
As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:
Note that the variable common to both sets of equations is Dt.

8. The Parabola
Note that the path (or “trajectory”) of a projectile will therefore be parabolic.

9. Example 1: The Cliff
A ball is launched at 15.0 m/s [right] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?

10. Example 1: The Cliff
A ball is launched at 15.0 m/s [right] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands? First, find the time it takes the ball to fall 15.0 m:

11. Example 1: The Cliff
First, find the time it takes the ball to fall 15.0 m:

12. Example 1: The Cliff
First, find the time it takes the ball to fall 15.0 m:

13. Example 1: The Cliff
First, find the time it takes the ball to fall 15.0 m:

14. Example 1: The Cliff
First, find the time it takes the ball to fall 15.0 m:

15. Example 1: The Cliff
Now find the horizontal distance travelled in 1.75 s:

16. Example 1: The Cliff
Now find the horizontal distance travelled in 1.75 s:

17. The Initial Velocity
If the initial velocity is not parallel or perpendicular to the horizontal, it must be broken down into its horizontal and vertical components:

18. The Range
A ball is launched from ground level with an initial velocity of v1 at an angle q with the horizontal. How far does the ball travel horizontally before it lands?

19. The Range
A ball is launched from ground level with an initial velocity of v1 at an angle q with the horizontal. How far does the ball travel horizontally before it lands? Note that the answer to this question will not be a number but an algebraic expression.

20. The Range

21. The Range
We can discard this solution.

22. The Range
Substituting into the equation for horizontal distance:

23. The Range
Substituting into the equation for horizontal distance:

24. The Range Substituting into the equation for horizontal distance:
(by trig identities)

25. The Range
The advantage of deriving a general expression instead of calculating a number is that you can see the relationships between the variables. Ex.
So a projectile on the moon, with 1/6 the gravity, will travel 6 times further.

26. The Range
The advantage of deriving a general expression instead of calculating a number is that you can see the relationships between the variables. Ex.
So a projectile on the moon, with 1/6 the gravity, will travel 6 times further.
If the launch velocity is doubled, the projectile will travel 4 times further.

27. The Range
You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance?

28. The Range
You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance?
The maximum value of sin2q is 1.

29. The Range
You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance?
The maximum value of sin2q is 1.

30. The Range
However, the expression only applies to objects launched from ground level. It can not be applied in cases such as:

31. Example 2: Up Off the Cliff
A ball is launched at 15.0 m/s [37o above the horizontal] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?

32. Example 2: Up Off the Cliff
A ball is launched at 15.0 m/s [37o above the horizontal] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?

33. Example 2: Up Off the Cliff

34. Example 2: Up Off the Cliff

35. Example 2: Up Off the Cliff

36. Example 2: Up Off the Cliff

37. Example 2: Up Off the Cliff
Note that the ball stayed in the air longer and travelled further when launched slightly above the horizontal.

38. Example 2: At the Bottom
What was the ball’s impact velocity?

39. Example 2: At the Bottom What was the ball’s impact velocity?
Note that it is not zero upon impact.
It is only zero after impact, after the ground has exerted a normal force on it to stop it.

40. Example 2: At the Bottom What was the ball’s impact velocity?
Note that we still need to find the time first.

41. Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s

42. Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s

43. Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s

44. More Practice Textbook Questions p. 46 #3, 5 p. 50 #9, 10
Tomorrow: “Projectile Motion Lab”