Ch3_ChiavelliJ

**Chapter 3 Wiki Page**

Lesson 1 Vector Notes: Method 1
toc a. **Vector**- quantity described by both magnitude and direction

**Scalar**- fully described by magnitude

Vector Diagram or Free-Body Diagrams *Pictured above is a scaled vector diagram portraying a displacement vector

Key Points:- a scale is present- vector arrow (head and tail) drawn in specific direction- magnitude and direction clearly labeled

Convention for Describing Vector Direction 1. As an angle of rotation of the vector about its “tail” from east, west, north, or south EX: 40 degrees North of West 2. As a counterclockwise angle of rotation of the vector about its “tail” from due East EX: 30 degrees = vector rotated 30 degrees in a counterclockwise direction relative to due East The length of the arrow (usually scaled) displays magnitude of a vector

Topic Sentence: Scaled vector diagrams are can describe direction in two somewhat similar ways and portray magnitude by the length of the scaled arrow.

b. Vector Addition

The sum of two vectors is known as the resultant --- net force experienced by an object determined by the vector sum of all individual forces acting on the object With regard to free body diagrams: The Pythagorean Theorem

For adding only 2 vectors that make a right angle to each other: the theorem

 Using Trigonometry to Determine a Vector’s Direction

SOH-CAH-TOA key to finding direction of resultant vector However, the measure of an angle produced by trigonometry may not always be the direction of the vector (must take angle of rotation into account)

When two vectors do not make right angles to each other: Scaled Vector Diagrams to Determine Resultant Or Head-to-Tail Method- used to determine vector sum or resultant

Must draw a vector to scale at a starting position, where first vector ends and the next one begins Repeats and when finished, all vectors added head-to-tail Resultant drawn from tail of first vector to head of last vector (start to finish) Scale is used to convert and direction calculated by protractor from counterclockwise angle rotation from due East

How to Apply: 1. Choose scale 2. Choose starting location and draw vector including direction and magnitude 3. Using head-to-tail technique, draw second vector 4. Repeat for all vectors except resultant 5. Draw resultant from starting point to head of last vector 6. Find direction and magnitude using best method possible

>>>>>>>>>

Topic Sentence: There are several ways to determine the magnitude and direction of vectors depending on the individual nature of the scenario.

Lesson 1 Vector Notes- Homework Part c and d c.  Vector Basics and Operations

Resultant- is the vector sum of 2 or more vectors



A + B + C = R

When displacement vectors are added, the result is a resultant displacement

Formula applies to all vector addition scenarios (displacement, force, velocity, etc.)



Topic Sentence: The vector sum of all individual vectors is the resultant and can be applied to a variety of vector situations.

d. Transforming Vectors into 2 Parts



A vector directed in 2 dimensions has an influence in 2 different directions (known as components)

Combined influence of components= influence of single 2D vector

Example of Concept in Action:





Topic Sentence: A vector in 2D has two components, in which a single vector illustrates the influence of that vector in a specific direction.

e. **Vector Resolution**- the process of determining the magnitude of a vector

**Parallelogram Method:** 1. Choose scale and draw vector with magnitude and direction 2. Sketch a parallelogram around the vector: at tail sketch vertical/horizontal lines AND at head sketch vertical/horizontal lines WHICH WILL MEET TO FORM RECTANGLE (a special parallelogram) 3. Draw/label components of the vector and also show direction through arrowheads 4. Measure length of sides and determine magnitude in real* units



**Trigonometric Method:** 1. Draw rough sketch of vector in right direction, labeling magnitude and angle formed with horizontal 2. Draw a rectangle about the vector so that the vector is the diagonal of the rectangle: sketch horizontal/vertical lines at head and tail of vector 3. Draw/label components of the vector and also show direction through arrowheads 4. Use the sine function to determine length of side opposite indicated angle, and use magnitude of vector for length of hypotenuse 5. Repeat using cosine function to find additional length of adjacent



Topic Sentence: Vector resolution is the tool employed to determine the magnitude of a vector, often accomplished by using the parallelogram or trigonometric methods.

g. **Relative Velocity** Motion is relative to the observer

Speed/velocity will be different from the perspective of a "stationary observer" than that of the actual plane or boat or object in motion

Therefore, the observed speed must be described relative to the observer

Tailwind= wind approaching from behind

Headwind= wind approaching from in front

Side wind= wind approaching from the side

In a side wind scenario, vector addition can be used to determine the resultant

Plane/wind : boat/water current

**Average Speed = distance / time**

Topic Sentence: Understanding relative velocity and implementing the tactics of vector addition make it possible to solve word problems incorporating speed, time, and distance. (riverboat/plane)

h.

A component describes the influence of a single vector in a specific direction

Perpendicular components of vectors are independent of one another to say that the changing of one DOES NOT affect the other




 * d=v * t **

Topic Sentence: The perpendicular components of vectors are independent of one another, and any change in either will not have an effect on the latter

= Lesson 2 Vector Notes Method 3 = a. **Questions Developed in Preview** What are the specifications for projectile motion? What is the law of inertia? How does gravity affect projectile motion? Besides gravity, do any other forces act upon an object in projectile motion? How does horizontal motion affect projectile motion?

The main idea of the passage is to cover the basics and fundamentals of projectile motion.

**Projectile:** an object upon which the only force acting is gravity (as long as air resistance is negligible)

Once projected or dropped, continues in motion by its own inertia AND is influenced only by gravity



(dropped from rest, thrown vertically upward, thrown upward at angle to horizontal)

A single force is acting upon the object gravity, IF NOT THE CASE..... then not projectile motion

A free-body diagram would consistently represent projectile motion (single force) as the following:  No force is required to keep an object in motion, BUT required to establish acceleration


 * Newton's Law of Inertia **: <span style="font-family: 'Times New Roman',Times,serif;">An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force

<span style="font-family: 'Times New Roman',Times,serif;">Gravity is the downward force upon a projectile, influencing its vertical motion (causing vertical acceleration) and is responsible for its parabolic trajectory

<span style="font-family: 'Times New Roman',Times,serif;">A projectile has a constant horizontal velocity, due to the tendency of objects in motion to remain in motion at a constant velocity

<span style="font-family: 'Times New Roman',Times,serif;">A projectile is acted upon by a single force being gravity. An object that is projected or dropped continues in motion by its own inertia and is influenced solely by gravity. <span style="font-family: 'Times New Roman',Times,serif;">An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. <span style="font-family: 'Times New Roman',Times,serif;">Gravity is the only force that acts upon a projectile. <span style="font-family: 'Times New Roman',Times,serif;">No, and air resistance is negligible. <span style="font-family: 'Times New Roman',Times,serif;">Although only briefly discussed, horizontal motion of a projectile is due to the frequency of objects in motion to stay in motion with a constant horizontal velocity.
 * <span style="font-family: 'Times New Roman',Times,serif;">What are the specifications for projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">What is the law of inertia? **
 * <span style="font-family: 'Times New Roman',Times,serif;">How does gravity affect projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">Besides gravity, do any other forces act upon an object in projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">How does horizontal motion affect projectile motion? **

<span style="font-family: 'Times New Roman',Times,serif;">b. **Questions Developed in Preview** <span style="font-family: 'Times New Roman',Times,serif;">How does the acceleration of gravity affect the trajectory of projectile motion? <span style="font-family: 'Times New Roman',Times,serif;">How is projectile motion different from free fall? <span style="font-family: 'Times New Roman',Times,serif;">What are the proper terms to describe projectile trajectory? <span style="font-family: 'Times New Roman',Times,serif;">What is the difference between horizontal and non-horizontal projectile launch? <span style="font-family: 'Times New Roman',Times,serif;">Why does the trajectory of projectile motion maintain a parabolic shape?

<span style="font-family: 'Times New Roman',Times,serif;">The main idea of the passage is to explore the trajectory of projectile motion, and further discover the details of the concept.

<span style="font-family: 'Times New Roman',Times,serif;">Projectile motion consists of VERTICAL and HORIZONTAL motion

__//<span style="font-family: 'Times New Roman',Times,serif;">Horizontally Launched Projectiles: //__ <span style="font-family: 'Times New Roman',Times,serif;">Free-falling objects will accelerate according to the acceleration of gravity

<span style="font-family: 'Times New Roman',Times,serif;">Gravity accelerates an object downward, but does not have an effect on its horizontal motion

<span style="font-family: 'Times New Roman',Times,serif;">Cannonball example: projectile travels with constant horizontal velocity & downward vertical acceleration

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<span style="font-family: 'Times New Roman',Times,serif;">//__Non Horizontally Launched Projectiles__// <span style="font-family: 'Times New Roman',Times,serif;">A scenario in which a projectile is launched upward and at an angle to the horizontal

<span style="font-family: 'Times New Roman',Times,serif;">Travels with parabolic trajectory

<span style="font-family: 'Times New Roman',Times,serif;">Presence of gravity does not influence horizontal motion of projectile



<span style="font-family: 'Times New Roman',Times,serif;">Parabolic trajectory is a result of downward force of gravity and acceleration, and the downward displacement in position as opposed to the lack of gravity

<span style="font-family: 'Times New Roman',Times,serif;">Constant horizontal velocity remains same (inertia from start), as no other horizontal forces act upon the projectile

<span style="font-family: 'Times New Roman',Times,serif;">The downward force of gravity and acceleration (9.8 m/s<span style="font-family: 'Times New Roman',Times,serif; vertical-align: super;">2 ) are responsible for the trajectory of a projectile in motion. <span style="font-family: 'Times New Roman',Times,serif;">They are generally similar concepts, yet projectile motion has 2 components, while free fall by itself only concerns one. <span style="font-family: 'Times New Roman',Times,serif;">The proper term to describe the shape of projectile motion is //parabolic//. <span style="font-family: 'Times New Roman',Times,serif;">Non-horizontal launch is when an object is launched upward and at an angle to the horizontal. <span style="font-family: 'Times New Roman',Times,serif;">This is a result of the downward force of gravity and acceleration.
 * <span style="font-family: 'Times New Roman',Times,serif;">How does the acceleration of gravity affect the trajectory of projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">How is projectile motion different from free fall? **
 * <span style="font-family: 'Times New Roman',Times,serif;">What are the proper terms to describe projectile trajectory? **
 * <span style="font-family: 'Times New Roman',Times,serif;">What is the difference between horizontal and non-horizontal projectile launch? **
 * <span style="font-family: 'Times New Roman',Times,serif;">Why does the trajectory of projectile motion maintain a parabolic shape? **

<span style="font-family: 'Times New Roman',Times,serif;">c. Part 1
 * <span style="font-family: 'Times New Roman',Times,serif;">Expressing Vectors Through Numbers **

<span style="font-family: 'Times New Roman',Times,serif;">1. What are the key concepts concerned with projectile motion?

<span style="font-family: 'Times New Roman',Times,serif;">2. What are the characteristics of horizontal launch motion?

<span style="font-family: 'Times New Roman',Times,serif;">3. What are the characteristics of non-horizontal launch motion?

<span style="font-family: 'Times New Roman',Times,serif;">4. How is horizontal motion affected by projectile motion?

<span style="font-family: 'Times New Roman',Times,serif;">5. How does the vertical component of the velocity vector change throughout the trajectory?

<span style="font-family: 'Times New Roman',Times,serif;">The main idea of the passage is to communicate the numerical factors of projectile trajectory and tie in the conceptual aspects.

<span style="font-family: 'Times New Roman',Times,serif;">Projectile Fast Facts: <span style="font-family: 'Times New Roman',Times,serif;">-only force acting upon projectile is gravity <span style="font-family: 'Times New Roman',Times,serif;">- no horizontal forces acting upon objects, no horizontal acceleration <span style="font-family: 'Times New Roman',Times,serif;">- horizontal velocity is constant <span style="font-family: 'Times New Roman',Times,serif;">- vertical acceleration due to gravity= -9.8m/s/s <span style="font-family: 'Times New Roman',Times,serif;">- vertical velocity changes by acceleration of gravity each second <span style="font-family: 'Times New Roman',Times,serif;">- horizontal motion independent of vertical motion
 * <span style="font-family: 'Times New Roman',Times,serif;">- **<span style="font-family: 'Times New Roman',Times,serif;">have parabolic trajectory as a result of gravity

<span style="font-family: 'Times New Roman',Times,serif;">Horizontal Launch

<span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif;">Horizontal velocity remains constant <span style="font-family: 'Times New Roman',Times,serif;">Vertical velocity changes by -9.8m/s/s every second

<span style="font-family: 'Times New Roman',Times,serif;">Non-Horizontal Launch

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<span style="font-family: 'Times New Roman',Times,serif;">Horizontal velocity remains constant <span style="font-family: 'Times New Roman',Times,serif;">Vertical velocity changes by -9.8m/s/s every second <span style="font-family: 'Times New Roman',Times,serif;">Notice initial velocity

<span style="font-family: 'Times New Roman',Times,serif;">**__Horizontal velocity constant AND vertical velocity changes by 9.8m/s/s every second__**

<span style="font-family: 'Times New Roman',Times,serif;">Symmetrical nature of projectile motion portrayed below: <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif;">Horizontal velocity remains constant, while vertical velocity changes by 9.8m/s/s every second. <span style="font-family: 'Times New Roman',Times,serif;">This scenario occurs when a projectile is launched at a horizontal, or simply dropped from rest. <span style="font-family: 'Times New Roman',Times,serif;">This scenario occurs when a projectile is launched at an angle to the horizontal. <span style="font-family: 'Times New Roman',Times,serif;">Horizontal velocity remains constant. <span style="font-family: 'Times New Roman',Times,serif;">The vertical velocity of the velocity vector changes by 9.8m/s/s every second.
 * <span style="font-family: 'Times New Roman',Times,serif;">1. What are the key concepts concerned with projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">2. What are the characteristics of horizontal launch motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">3. What are the characteristics of non-horizontal launch motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">4. How is horizontal motion affected by projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">5. How does the vertical component of the velocity vector change throughout the trajectory? **

<span style="font-family: 'Times New Roman',Times,serif;">c. Part 2

<span style="font-family: 'Times New Roman',Times,serif;">1. What is the vertical displacement of a projectile based on? <span style="font-family: 'Times New Roman',Times,serif;">2. What are the key equations that pertain to horizontal projectile motion? <span style="font-family: 'Times New Roman',Times,serif;">3. In what way will an initial vertical component (non-horizontal launch) affect the resulting displacement? <span style="font-family: 'Times New Roman',Times,serif;">4. What are the key equations that pertain to non-horizontal projectile motion? <span style="font-family: 'Times New Roman',Times,serif;">5. What is unique about the trajectory of a projectile?

<span style="font-family: 'Times New Roman',Times,serif;">The main idea of the passage is to evaluate the displacement of projectile trajectory, and integrating key equations.

Δd= //vi//t + ½ at 2

<span style="font-family: 'Times New Roman',Times,serif;">Use this equation for the vertical displacement of a //horizontally// launched projectile

Δd= //vi//t + ½ at 2

<span style="font-family: 'Times New Roman',Times,serif;">Use this equation for the vertical displacement of a //non-horizontally// launched projectile

<span style="font-family: 'Times New Roman',Times,serif;">The symmetrical nature of projectile trajectory is key to understanding the factors of the concept

<span style="font-family: 'Times New Roman',Times,serif;">The displacement is dependent upon the initial velocity, time, and acceleration **.** Δd= //vi//t + ½ at 2 <span style="font-family: 'Times New Roman',Times,serif;">It will only change the general trajectory of the projectile. The displacement can be easily found by using the correct formula. Δd= //vi//t + ½ at 2 <span style="font-family: 'Times New Roman',Times,serif;">The trajectory of a projectile is symmetrical.
 * <span style="font-family: 'Times New Roman',Times,serif;">1. What is the vertical displacement of a projectile based on? **
 * <span style="font-family: 'Times New Roman',Times,serif;">2. What are the key equations that pertain to horizontal projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">3. In what way will an initial vertical component (non-horizontal launch) affect the resulting displacement? **
 * <span style="font-family: 'Times New Roman',Times,serif;">4. What are the key equations that pertain to non-horizontal projectile motion? **
 * <span style="font-family: 'Times New Roman',Times,serif;">5. What is unique about the trajectory of a projectile? **

= Vector Addition Activity =


 * Data: **


 * Graphical Method: **


 * Analytical Method: **


 * Percent Error Calculations: **

=Activity: Ball In Cup Part 1=

<span style="font-family: 'Times New Roman',Times,serif;">**Video of Trial** <span style="font-family: 'Times New Roman',Times,serif;">media type="file" key="working ball in a cup.m4v" width="300" height="300"
 * <span style="font-family: 'Times New Roman',Times,serif;">Data and Calculations: **

<span style="font-family: 'Times New Roman',Times,serif;">**Percent Error Calculations:** <span style="font-family: 'Times New Roman',Times,serif;">**Discussion of Results** <span style="font-family: 'Times New Roman',Times,serif;">The percent error of our calculations was 0.481%. This figure is indicative of good results, and therefore precise calculations in terms of determining the correct location of the cup for "catching" the projectile ball. We were successful in determining the initial velocity and hang time of the projectile off a table at first and then used that information to predict its range from the counter top. With careful measurements and alignment, we were able to project the ball into the cup for three consecutive trials. Based on our ability to launch the ball into the cup with minimal percent error, our laboratory was a success.

=<span style="font-family: 'Times New Roman',Times,serif;">Lab: Shooting The Grade = <span style="font-family: 'Times New Roman',Times,serif;">Lab Partners: Dani Rubenstein, Jenna Malley, and Maddie Weinfeld

<span style="font-family: 'Times New Roman',Times,serif;">**__Purpose/Rationale:__** To launch a ball at a specified angle (30 degrees) and speed, so that the ball passes through 5 rings in midair consecutively, and lands in a cup on the floor.

<span style="font-family: 'Times New Roman',Times,serif;">The basis of our theoretical projections are rooted in the principles of projectile motion. The data shown above was collected using the carbon paper to calculate the average range of the projectile ball when shot out of the device at a thirty-degree angle. Then the average range was used to calculate hang time and each component of the initial velocity. We then calculated the specific location of the projectile at six intervals (five rings and one final cup). These locations each had an x and y component.

<span style="font-family: 'Times New Roman',Times,serif;">**__Materials and Methods:__** <span style="font-family: 'Times New Roman',Times,serif;">The key materials in this laboratory are primarily a projectile launcher (set at a 30 degree angle), small yellow ball, five masking tape hoops, and a plastic cup. Using calculations from the average range and hang time, the initial velocity components were used to determine the exact position of the projectile at 6 time intervals. We then measured meticulously both horizontally and vertically in order to hang the hoops with twine at 5 specific locations and place the cup in its designated position. By continually testing the projectile, followed by adjustment, and three confirmation runs, we proceeded in aligning the parabolic trajectory.

<span style="font-family: 'Times New Roman',Times,serif;">









media type="file" key="6 test run for procedure.mov"

<span style="font-family: 'Times New Roman',Times,serif;">**__Observations and Data From Initial Velocity__** <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">The data shown above was collected using the carbon paper to calculate the average range of the projectile ball when shot out of the device at a thirty-degree angle. Then the average range was used to calculate hang time and each component of the initial velocity.

<span style="font-family: 'Times New Roman',Times,serif;">**__Observations and Data From Performance__**

<span style="font-family: 'Times New Roman',Times,serif;">**Video Trial:** <span style="font-family: 'Times New Roman',Times,serif; line-height: 0px; overflow: hidden;">media type="file" key="Movie on 2011-11-02 at 09.40

<span style="font-family: 'Times New Roman',Times,serif;">**Table of Results:** <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">*Note: we only were able to succeed in getting the ball through the 5th ring on the first day and did not achieve the ball in cup factor

<span style="font-family: 'Times New Roman',Times,serif;">**__Physics Calculations__** <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif;">**__Error Analysis:__** <span style="font-family: 'Times New Roman',Times,serif;">





<span style="font-family: 'Times New Roman',Times,serif;">**__Percent Error Table of Results:__** <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">**__Conclusion:__** <span style="font-family: 'Times New Roman',Times,serif;">We succeeded in getting the ball through five hoops one time, but were successful in getting the ball through four hoops consecutively. As our video confirms, we technically only achieved four hoops. The total cumulative error was 1.33%, which is very good considering the various factors contributing to error. The majority of error occurred in terms of measuring the correct position of the cup, as evident by an average error of 1.81%, reflecting the smaller likelihood of getting the ball in the cup. The most likely source of possible error would be the measuring of its position, having to do so in 3-D space. Having used the weight to perfect a vertical position was evidently not good enough, nor was the levelness of our horizontal measurements. It is clear that the farther away from the projectile launcher, the more likely it is to generate the correct measurements. Another source of error could have been any sporadic shooting by the projectile launcher, and therefore not being exactly the same every time. Additionally, the stature of the cup being at a virtually right angle with the ground, and not tilted towards the projectile launcher could have made it more difficult to achieve the cup factor. In changing the laboratory, I would continue to use the weight for vertical measurements, and even employ the use of a meter stick for the horizontal measurements. Also, I would give the stature of the cup more consideration when performing the initial calculations. Lastly, I would consider the notion that the latter rings tend to be harder to achieve, and therefore calculate their locations more precisely. In terms of real-life application, a military personnel might have to shoot a target that would have a parabolic trajectory, and would have to use the same basic principles of projectile motion to achieve their mission.

=<span style="font-family: 'Times New Roman',Times,serif;">Gourd-a-rama Project = <span style="font-family: 'Times New Roman',Times,serif;">

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<span style="font-family: 'Times New Roman',Times,serif;">If I were to construct a new gourd vehicle, I would consider several factors. First, I would institute larger wheels, rather than the small wheels put in place. Secondly, I would have made a stronger and more supportive body to hold the gourd. Lastly, I would have used thicker and sturdier twine to hold the gourd in place. These improvements are based on my personal observations of my vehicle during two runs on the ramp.