1. AP Physics I
Kinematics in 2D

2. 3.5 Projectile Motion

3. Projectile Motion
So Far we have talked about Vertical Projectile Motion (We called this Free Fall)
All velocity is in the Up and Down Direction (y direction)
NO Horizontal Velocity!!!!
We Used the Kinematics Equations
P-T, V-T, and A-T Graphs have only BEEN IN THE Y- DIRECTION!!!!!
Now We have to add the Horizontal Piece!!! We get 3 new P-T, V-T and A-T graphs and an additional graph….. Height vs Range (Y and X graph)
YOU HAVE 7 Graphs To Understand!!!!!!!!!!!!!

4. Skateboard ball Demo (class video) Case 1: No Horizontal Motion
Projectile Motion
Skateboard ball Demo (class video)
Case 1: No Horizontal Motion
Case 2: Horizontal Motion

5. Creating projectile motion
Throw the ball straight up while moving on rollerblades. As long as you do not change your speed or direction while the ball is in flight, it will land back in your hands.
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6. Projectile motion
Projectile is an object launched into the air with some horizontal velocity ( since under the influence if gravity it will have vertical velocity as well)
Only acceleration will be g at -9.8m/s2
The path of the projectile is called a trajectory and is the shape of a parabola (quadratic).
No horizontal acceleration…Horizontal velocity is constant throughout flight
We ignore air resistance in this class 99.9% of the time.

7. Projectile motion
Here come all the P-T, V-T and A-T Graphs for both the vertical (you have already seen these when we did free fall) and horizontal directions.
Complete in Notability
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8. Qualitative analysis of projectile motion in the y-axis
A ball thrown straight up in the air by a person moving horizontally on a skateboard will land back in the person's hand.
Earth exerts a gravitational force on the ball, so its upward speed decreases until it stops at the highest point, and then its downward speed increases until it returns to your hands.
With respect to you on the skateboard, the ball simply moves up and down.

9. Qualitative analysis of projectile motion in the x-axis
A ball thrown straight up in the air by a person moving horizontally on a skateboard will land back in the person's hand.
The ball also moves horizontally.
No object exerts a horizontal force on the ball. Thus, according to Newton's first law, the ball's horizontal velocity does not change once it is released and is the same as the person's horizontal component of velocity.

10. Qualitative analysis of projectile motion in the x- and y-axes
A ball thrown straight up in the air by a person moving horizontally on skateboard will land back in the person's hand.
The ball continues moving horizontally as if it were not thrown upward.
The ball moves up and down as if it does not move horizontally.
Horizontal and vertical motions of the ball are independent of each other.

11. Analyzing Projectile Motion Using a X and Y graph (Vertical vs horizontal position)
The angle of the initial velocity above the horizontal (i.e., above the x-axis) is the launch angle.
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12. Projectile motion
Projectiles are objects launched at an angle relative to a horizontal surface.
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13. Example 1:
A heavy red ball is released from rest 2.0 m above a flat, horizontal surface. At exactly the same instant, a yellow ball with the same mass is fired horizontally at 3.0 m/s.
Which ball hits the ground first?
The red ball hits first.
The yellow ball hits first.
They hit at the same time.
Answer: C

14. Example 2:
A heavy red ball is released from rest 2.0 m above a flat, horizontal surface. At exactly the same instant, a yellow ball with the same mass is fired horizontally at 3.0 m/s. Which ball hits the ground first?
The red ball hits first.
The yellow ball hits first.
They hit at the same time.
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15. Conceptual Exercise 3.7: Throwing a ball
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16. Conceptual Exercise : Throwing a ball
You throw a tennis ball as a projectile. Arrows represent the ball's instantaneous velocity and acceleration and the force or forces exerted on the ball by other objects when at the three positions shown in the diagram.
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17. Projectile motion 4 types of problems
1. Throw/move something that is not on the ground (cliff) with no initial vertical velocity (no vy)…only horizontal velocity (vx)
2. Launch a object at the same height that it returns. (Kick a football).
3. Launch something at an initial height but stops at a height different than the initial launch height.
4: Free Fall (already discussed) (Drop/throw something with no horizontal velocity)

18. Projectile Motion: 1st Type
Vertical 
fall
Projectile 
Motion vx vy v
a = g
A projectile is an object 
moving in two dimensions under the influence of Earth's gravity. Its path is a parabola.

19. Projectile Motion: 1st Type
Projectile motion can be 
understood by analyzing the vertical and horizontal motion separately.
The speed in the x-direction is constant.
The speed in the y-direction is changing.
Vertical 
fall
Projectile 
Motion vx vy v
a = g

20. Example 1
A mountain lion leaps horizontally from a 7.5 m high rock with a 
speed of 4.5 m/s. How far from the base of the rock will he land?

21. Example 1
A mountain lion leaps horizontally from a 7.5 m high rock with a speed of 4.5 m/s. How far from the base of the rock will he land?
Step 1: determine the time it will take for the lion to reach the ground.

22. Example 1
A mountain lion leaps horizontally from a 7.5 m high rock with a 
speed of 4.5 m/s. How far from the base of the rock will he land?
Step 2: Determine how far from the base he will land.

23. Example 2
A cannon ball is shot from a cannon at a height of 15 m with a velocity of 20 m/s. How far away will the cannon ball land?
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Answer

24. Justify your answer qualitatively, with no equations or calculations.
A marble rolls off a table from a height of 0.8 m with a velocity of 3 m/s. Then another marble rolls off the same table with a velocity of 4 m/s. Which values are the same for both marbles?
Justify your answer qualitatively, with no equations or calculations. A
The final speeds of the marbles. B
The time each takes to reach the ground. C
The distance from the base of the table where each lands. [
Answer
B. Both marbles have only a velocity in the horizontal direction; none in the vertical direction. The time it takes to reach the ground depends only on the initial velocity in the vertical direction, the height of the table, and the acceleration due to gravity. All three of these things are the same for both.

25. Projectile Motion: 2nd Type
(Takes off at the Same Height as it hits the ground)
If an object is launched at an angle with the horizontal, 
the analysis is similar except that the initial velocity has a 
vertical component. vx vy v
vx = v

26. Projectile Motion (One More Time with Launch Angle)
Text: p. 82
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27. Projectile Motion
Velocity leaving is the same as the velocity when it hits the ground…just negative
Remember:
Constant velocity in horizontal direction (x)
Free-fall motion in the vertical direction.(y)
At the top
Vy=0
Vx=same as it always was

28. Projectile motion can be described by two kinematics equations:vx vy v
vx = v
Projectile motion can be described by two kinematics equations:
horizontal component:  vertical component:

29. vx vy v
vx = v
Flying time:
Horizontal range:
Maximum Height:

30. Ignoring air resistance, the horizontal component of a projectile's velocity:
is zero. B
remains constant. C
continuously increases. D
continuously decreases.
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Answer B

31. A ball is thrown with a velocity of 20 m/s at an angle of 60° above the horizontal. What is the horizontal component of its instantaneous velocity at the exact top of its trajectory? A
10 m/s B
17 m/s C
20 m/s D
zero
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Answer A

32. Ignoring air resistance, the magnitude of the horizontal component of a projectile's acceleration:
is zero. B
remains a non-zero constant. C
continuously increases. D
continuously decreases.
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Answer A

33. (Mulitple Answer) An Olympic athlete throws a javelin at six different angles above the horizontal, each with the same speed: 20°, 30°, 40°, 60°, 70° and 80°. Which two throws cause the javelin to land the same distance away? Be prepared to justify your answer. A
30° and 80° B
20° and 70°
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Answer
B and D; since the range equation is V02 sin(2θ)/g, complementary angles will have the same answer C
30° and 70° D
30° and 60°

34. It is less than its initial speed.
When a football in a field goal attempt reaches its maximum height, how does its speed compare to its initial speed? (Justify your answer.) A
It is zero. B
It is less than its initial speed. C
It is equal to its initial speed. D
It is greater than its initial speed.
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Answer
B; the initial velocity is V0=√(v0x2+v0y2)
the velocity at the maximum height is just V0x

35. You are throwing a ball for the second time
You are throwing a ball for the second time. If the ball leaves your hand with twice the velocity it had on your first throw, its horizontal range R (compared to your first serve) would be A
1.4 R B
R/2 C 2R D 4R D

36. It is impossible to tell from the information given.
A stone is thrown horizontally from the top of a tower 
at the same instant a ball is dropped vertically. 
Which object is traveling faster when it hits the level 
ground below? (Justify your answer.) A
It is impossible to tell from the information given. B
the stone C
the ball D
Neither, since both are traveling at the same speed.
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Answer
B; both the stone and the ball have will have the same vertical component of the velocity when they hit the ground but the stone will also have a horizontal component

37. A plane flying horizontally at a speed of 50
A plane flying horizontally at a speed of 50.0 m/s and at an elevation of 160 m drops a package. Two seconds later it drops a second package. How far apart will the two packages land on the ground? A
100 m B
170 m C
180 m D
210 m
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Answer A

38. Determine the total time in the air.
A projectile is fired with an initial speed of 30 m/s at an angle of 30° above the horizontal.
Determine the total time in the air.
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Answer

39. Determine the maximum height reached by the projectile.
A projectile is fired with an initial speed of 30 m/s at an angle of 30° above the horizontal.
Determine the maximum height reached by the projectile.
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Answer

40. Determine the maximum horizontal distance covered by the projectile.
A projectile is fired with an initial speed of 30 m/s at an angle of 30° above the horizontal.
Determine the maximum horizontal distance covered by the projectile.
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Answer

41. A projectile is fired with an initial speed of 30 m/s at an angle of 30° above the horizontal.
Determine the velocity of the projectile 2s after firing.
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Answer vx vy v

42. Projectile Motion: 3rd Typevy vx v
vx = v
If an object is launched at an angle with the horizontal and from an initial height, the analysis is similar except when finding the total time in the air you will need to use the quadratic formula. vx vy v vx vy v vx vy v vx vy v

43. Determine the total time in the air.
A projectile is fired from the edge of a cliff 200 m high with an initial speed of 30 m/s at an angle of 45° above the horizontal.
Determine the total time in the air.
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Answer
Answer: t total= 28sec

44. Determine the maximum horizontal range.
A projectile is fired from the edge of a cliff 200 m high with an initial speed of 30 m/s at an angle of 45° above the horizontal.
Determine the maximum horizontal range.
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Answer

45. Determine the magnitude of the velocity just before impact.
A projectile is fired from the edge of a cliff 200 m high with an initial speed of 30 m/s at an angle of 45° above the horizontal.
Determine the magnitude of the velocity just before impact.
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Answer

46. A projectile is fired from the edge of a cliff 200 m high with an initial speed of 30 m/s at an angle of 45° above the horizontal.
Determine the angle the velocity makes with the horizontal just before impact.
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Answer

47. Example 3.8: Best angle for farthest flight
You want to throw a rock the farthest possible horizontal distance. You keep the initial speed of the rock constant and find that the horizontal distance it travels depends on the angle at which it leaves your hand.
What is the angle at which you should throw the rock so that it travels the longest horizontal distance, assuming you throw it with the same initial speed?
I'm leaving this for the instructor to solve in class

48. Example 3.8: Best angle for farthest flight
I'm leaving this for the instructor to solve in class
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49. The Range of a Projectile
The range of a projectile is the horizontal distance traveled.
For smaller objects air resistance is critical, and the maximum range comes at an angle less than 45°.

50. Example 3.9: Shot from a cannon
Stephanie Smith Havens is to be shot from an 8-m-long cannon at 100 km/h. The barrel of the cannon is oriented 45° above the horizontal. She hopes to be launched so that she lands on a net that is 40 m from the end of the cannon barrel and at the same elevation (our assumption).
Estimate the speed with which she needs to leave the cannon to make it to the net.
This is left for the instructor to solve as an example

51. Example 3.9: Shot from a cannon
This is left for the instructor to solve as an example
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