1. Projectile Motion
Unit 6

2. Projectile Motion
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3. Horizontally launched
There are only two kinds of problems and we will approach each separately then mix for the review:
Horizontally launched
Departure angle of 0°
Launched at an angle
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4. Projectile Motion and the Velocity Vector
Any object that is moving through the air affected only by gravity is called a projectile.
The path a projectile follows is called its trajectory.
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5. Projectile Motion and the Velocity Vector
The trajectory of a thrown basketball follows a special type of arch-shaped curve called a parabola.
The distance a projectile travels horizontally is called its range.
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Range

6. Projectile Motion
It can be understood by analyzing the horizontal (x) and vertical (y) motions separately.
Figure Caption: Projectile motion of a small ball projected horizontally. The dashed black line represents the path of the object. The velocity vector at each point is in the direction of motion and thus is tangent to the path. The velocity vectors are green arrows, and velocity components are dashed. (A vertically falling object starting at the same point is shown at the left for comparison; vy is the same for the falling object and the projectile.)
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7. Projectile Motion
The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g.
This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.
Figure Caption: Multiple-exposure photograph showing positions of two balls at equal time intervals. One ball was dropped from rest at the same time the other was projected horizontally outward. The vertical position of each ball is seen to be the same at each instant.
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8. Solving Problems Involving Projectile Motion
Read the problem carefully, and choose the object(s) you are going to analyze.
Draw a diagram.
Choose an origin and a coordinate system.
Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g.
Examine the x and y motions separately.
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9. Solving Problems Involving Projectile Motion
6. List known and unknown quantities. Remember that vx never changes, and that vy = 0 at the highest point.
7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.
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10. The famous “T” chart distance acceleration Initial Velocity
quantity
Horizontal
X motion
Vertical
Y motion
distance
acceleration
0 m/s2
-10 m/s2
Initial Velocity
0 m/s
Final Velocity
time
Same
Same
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11. Horizontally Launched Projectiles
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12. Horizontal Speed
The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force.
Since there is no force, the horizontal acceleration is zero (ax = 0).
The ball will keep moving to the right at 5 m/sec.
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13. Horizontal Speed & distance
The horizontal distance a projectile moves can be calculated according to the formula:
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14. Vertical Speed
The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second.
After one second has passed, vy of the ball will be 9.8 m/sec.
After the 2nd second has passed, vy will be 19.6 m/sec and so on.
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15. Solving Problems Involving Projectile Motion
Example : Driving off a cliff.
A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high cliff. How fast must the motorcycle leave the cliff top to land on level ground below, 90.0 m from the base of the cliff where the cameras are? Ignore air resistance.
Figure 3-23.
Answer: The x velocity is constant; the y acceleration is constant. We know x0, y0, x, y, a, and vy0, but not vx0 or t. The problem asks for vx0, which is 28.2 m/s.
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16. Vector Review
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17. Vectors and Direction Key Question:
How do we accurately communicate length and distance?
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18. Vectors and Direction
A scalar is a quantity that can be completely described by one value: the magnitude.
You can think of magnitude as size or amount, including units.
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19. Vectors and Direction
A vector is a quantity that includes both magnitude and direction.
Vectors require more than one number.
The information “1 kilometer, 40 degrees east of north” is an example of a vector.
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20. Created by Kelly Rick

21. Created by Kelly Rick

22. Calculate a resultant vector
1) You are asked for the resultant vector.
2) You are given three displacement vectors.
3) Make a sketch of the ant’s path, and add the displacement vectors by components.
4) Solve:
x1 = (-2, 0) m x2 = (0,3) m x3 = (6,0) m
x1 + x2 + x3 = ( ) m and ( ) m = (4,3) m
The final displacement is 4 meters east and 3 meters north from where the ant started.
An ant walks 2 meters West, 3 meters North, and 6 meters East.
What is the displacement of the ant?
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23. Finding Vector Components Graphically
Draw a displacement vector as an arrow of appropriate length at the specified angle.
Mark the angle and use a ruler to draw the arrow.
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24. Created by Kelly Rick

25. Finding the Magnitude of a Vector
When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem.
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26. Projectiles Launched at an Angle
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27. Projectile Motion
If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.
Vx=Constant velocity
ax=0
Figure Caption: Path of a projectile fired with initial velocity v0 at angle θ0 to the horizontal. Path is shown dashed in black, the velocity vectors are green arrows, and velocity components are dashed. The acceleration a = dv/dt is downward. That is, a = g = -gj where j is the unit vector in the positive y direction.
Horizontal: velocity is constant and acceleration is zero
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28. Projectile Motion
If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.
Vy=changing velocity
ay=-10
Figure Caption: Path of a projectile fired with initial velocity v0 at angle θ0 to the horizontal. Path is shown dashed in black, the velocity vectors are green arrows, and velocity components are dashed. The acceleration a = dv/dt is downward. That is, a = g = -gj where j is the unit vector in the positive y direction.
Vyi not 0m/s
Vertical: acceleration is -10m/s/s
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29. Components …… and vectors
Represents the velocity of an object in 2D space
Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single two-dimensional vector.
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30. Components …… and vectors
Now… we resolve the vector into its x and y components…why? Because the combination of the action in the x and y combined produce the 2D motion represented below
Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single two-dimensional vector.
Created by Kelly Rick

31. Components …… and vectors
Y-Component of motion
x-Component of motion
Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single two-dimensional vector.
Created by Kelly Rick

32. Components of the motion
Y-Component of motion
x-Component of motion
Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single two-dimensional vector.
Created by Kelly Rick

33. Practicing component form
Draw the x and y components for the following vector…
18m/s
37°
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34. Practicing component form
Draw the x and y components for the following vector…
18m/s
Y-component of 18m/s
37°
x-component of 18m/s
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35. Practicing component form
Draw the x and y components for the following vector…
Y-component of 18
Sine 37=Y-COMPONENT 18
Y- component= 18 (sin 37)
Y-component =10.8 18
SOH CAH TOA
Y-component of 18
37°
x-component of 18m/s
COS 37=X-COMPONENT 18
X- component= 18 (cos 37)
X-component =14.4
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36. Practicing component form…2
Draw the Θ and R components for the following vector… 8m
SOH CAH TOA Θ 6m
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37. Practicing component form…2
Draw the Θ and R components for the following vector… 8m
R=√x2+y2=10m
SOH CAH TOA Θ 6m
TanΘ=opp/adj
TanΘ = 8m/6m
Θ =Tan-1 (8/6)
Θ =53°
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38. Using Math to Find Resultants Using Math to Find Components
Note Taking Guide 1
Note Taking Guide 2
Using Math to Find Resultants
Using Math to Find Components
Vector Math Practice Sheet
Review
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39. 3-8 Solving Problems Involving Projectile Motion
Example 3-7: A kicked football.
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s, as shown. Calculate (a) the maximum height, (b) the time of travel before the football hits the ground, (c) how far away it hits the ground, (d) the velocity vector at the maximum height, and (e) the acceleration vector at maximum height. Assume the ball leaves the foot at ground level, and ignore air resistance and rotation of the ball.
Figure 3-24.
Answers: a. This can be calculated using vertical variables only, giving 7.35 m.
b. The equation for t is quadratic, but the only meaningful solution is 2.45 s (ignoring the t = 0 solution).
c. The x velocity is constant, so x = 39.2 m.
d. At the highest point, the velocity is the (constant) x velocity, 16.0 m/s.
e. The acceleration is constant, 9.80 m/s2 downward.
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40. Solving Problems Involving Projectile Motion
Conceptual Example : Where does the apple land?
A child sits upright in a wagon which is moving to the right at constant speed as shown. The child extends her hand and throws an apple straight upward (from her own point of view), while the wagon continues to travel forward at constant speed. If air resistance is neglected, will the apple land (a) behind the wagon, (b) in the wagon, or (c) in front of the wagon?
Figure 3-25.
Response: The child throws the apple straight up from her own reference frame with initial velocity vy0 (Fig. 3–25a). But when viewed by someone on the ground, the apple also has an initial horizontal component of velocity equal to the speed of the wagon, vx0. Thus, to a person on the ground, the apple will follow the path of a projectile as shown in Fig. 3–25b. The apple experiences no horizontal acceleration, so vx0 will stay constant and equal to the speed of the wagon. As the apple follows its arc, the wagon will be directly under the apple at all times because they have the same horizontal velocity. When the apple comes down, it will drop right into the outstretched hand of the child. The answer is (b).
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41. Created by Kelly Rick

42. Solving Problems Involving Projectile Motion
Conceptual Example : The wrong strategy.
A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move. (He hadn’t studied physics yet.) Ignore air resistance.
Figure 3-26.
Response: Both the water balloon and the boy in the tree start falling at the same instant, and in a time t they each fall the same vertical distance y = ½ gt2, much like Fig. 3–21. In the time it takes the water balloon to travel the horizontal distance d, the balloon will have the same y position as the falling boy. Splat. If the boy had stayed in the tree, he would have avoided the humiliation.
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43. Note Taking Guide II Projectile Problems Review
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