The document discusses the motion of projectiles fired at an angle. It explains that a projectile has both vertical and horizontal motion, with the vertical motion identical to if thrown straight up, undergoing acceleration from gravity. The horizontal motion is unaffected by gravity and maintains a constant velocity. The document provides an example of a ball thrown at a 60 degree angle, calculating its initial vertical velocity as 43 m/s and horizontal velocity as 25 m/s.

1. A Projectile Fired at an Angle A conceptual study

2. When an object is dropped, the only thing that is affecting its motion is gravity, g .

3. When an object is dropped, the only thing that is affecting its motion is gravity, g . Gravity is a force, F .

4. When an object is dropped, the only thing that is affecting its motion is gravity, g . Gravity is a force, F . Newton says that a force causes a mass, m , to accelerate:

5. So, when an object is dropped, it starts from rest (v i = 0 m/s), and it accelerates at ~+10 m/s each second.

6. So, when an object is dropped, it starts from rest (v i = 0 m/s), and it accelerates at ~+10 m/s each second. So, each second, the object is traveling 10 m/s faster than it was the second before.

7. Also, since the object is accelerating, it will travel a greater distance each second than it did the second before.

8. Also, since the object is accelerating, it will travel a greater distance each second than it did the second before. So, it’s obvious that, the longer an object falls, the faster it will go and the farther it will fall.

9. Let’s consider an object that’s thrown straight up. It goes up and then slows to a stop. At its highest point, its instantaneous velocity is 0 m/s. It then starts downward as if dropped from rest at that height. 0 m/s

10. During the upward part of the motion, the object slows from its initial upward velocity, v i , to 0 velocity. 0 m/s - g + g -v i +v i

11. During the upward part of the motion, the object slows from its initial upward velocity, v i , to 0 velocity. 0 m/s - g + g -v i +v i We know its accelerating because its velocity is changing.

12. During the upward part of the motion, the object slows from its initial upward velocity, v i , to 0 velocity. 0 m/s - g + g -v i +v i We know its accelerating because its velocity is changing. Since it’s decreasing in velocity, we say it’s accelerating negatively.

13. Each second, it is moving at 10 m/s slower than it was the second before. 0 m/s -40 m/s -30 m/s -20 m/s -10 m/s 40 m/s 30 m/s 20 m/s 10 m/s

14. Each second, it is moving at 10 m/s slower than it was the second before. 0 m/s -40 m/s -30 m/s -20 m/s -10 m/s 40 m/s 30 m/s 20 m/s 10 m/s Once it reaches the top and begins to fall back, it is moving at 10 m/s faster each second.

15. Each second, it is moving at 10 m/s slower than it was the second before. 0 m/s -40 m/s -30 m/s -20 m/s -10 m/s 40 m/s 30 m/s 20 m/s 10 m/s Once it reaches the top and begins to fall back, it is moving at 10 m/s faster each second. Whether moving up or down, the object accelerates at the same rate…10 m/s 2 .

17. Now, let’s consider objects that are thrown up and horizontally at the same time, a projectile fired at an angle.

18. Now, let’s consider objects that are thrown up and horizontally at the same time, a projectile fired at an angle. The objects’ vertical and horizontal motions are completely independent of each other.

19. The object will move vertically exactly the same as it would if thrown straight up. Vertical motion Horizontal motion

20. The object will move vertically exactly the same as it would if thrown straight up. It will move horizontally just like it would if it were rolling across a smooth, level, surface. Vertical motion Horizontal motion

21. It will accelerate vertically. It will move at a constant velocity horizontally. g v

22. Let’s say Kyle threw a ball at some velocity, v , at 60 ° above the horizontal. 60 ° v

23. Let’s say Kyle threw a ball at some velocity, v , at 60 ° above the horizontal. 60 ° v The thrown ball is moving upward and horizontally at the same time.

24. Let’s say Kyle threw a ball at some velocity, v , at 60 ° above the horizontal. 60 ° v The thrown ball is moving upward and horizontally at the same time. The ball has an initial vertical velocity, v y . v y

25. Let’s say Kyle threw a ball at some velocity, v , at 60 ° above the horizontal. 60 ° v The thrown ball is moving upward and horizontally at the same time. The ball has an initial vertical velocity, v y . v y It also has an initial horizontal velocity, v x . v x

26. v y = v sin  60 ° v v y v x

27. v y = v sin  60 ° v v y v x v x = v cos 

28. v y = v sin  60 ° v = 50 m/s v y v x v x = v cos  Let’s say Kyle through the ball at 50 m/s. Not bad!

29. v y = v sin  60 ° v = 50 m/s v y v x v x = v cos  Let’s say Kyle through the ball at 50 m/s. Not bad! So what’s the horizontal and vertical velocities of the throw?

30. 60 ° v = 50 m/s v y = 43 m/s v x

31. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s

32. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the horizontal velocity as time passes?

33. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the horizontal velocity, v x , as time passes? Nothing! There is no horizontal force, therefore, no horizontal acceleration.

34. v x v x v x v x v x v x v x v x v x

35. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the vertical velocity, v y , as time passes?

36. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the vertical velocity, v y , as time passes? It should decrease until it reaches the top of its flight, then increase as it falls.

37. v y v y v y v y v y v y v y v y v y = 0

38. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the vertical acceleration as time passes?

39. 60 ° v = 50 m/s v y = 43 m/s v x = 25 m/s What should happen to the vertical acceleration as time passes? Nothing. Acceleration is gravity, g , and remains at 10 m/s 2 .

40. g = 10 m/s g g g g g g g g