Ch3_RubensteinD

=__Vectors__ =

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**__Homework 10/12/11 Lesson 1 A & B Vectors__**
** Vectors and Direction ** A vector quantity is a quantity that is fully described by both magnitude and direction. A scalar quantity is a quantity that is fully described by its magnitude. Ex: displacement, velocity, acceleration, and force. Vector quantities are often represented by scaled vector diagrams, also known as Free-Body Diagrams Characteristics of a Vector Diagram: **Conventions for Describing Directions of Vectors** Vectors can be directed due in all directions. However, there is a need for some form of a convention for identifying the direction of a vector that is __not__ due in one of the basic directions. There are a variety of conventions for describing the direction of any vector. **Representing the Magnitude of a Vector** The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. Vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. A scale is indicated and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention.
 * a scale is clearly listed
 * a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a //head// and a //tail//.
 * the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).
 * 1) The direction of a vector is often expressed as an angle of rotation of the vector about its tail from east, west, north, or south.
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its tail from due East.

<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">**Vector Addition** <span style="font-family: Arial,Helvetica,sans-serif;">Two vectors can be added together to determine the result (or resultant). <span style="font-family: Arial,Helvetica,sans-serif;">There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">** The Pythagorean Theorem **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">the Pythagorean theorem and trigonometric methods
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">the head-to-tail method using a scared vector diagram

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">** Using Trigonometry to Determine a Vector's Direction ** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">SOH-CAH-TOA <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- The ** sine function ** relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- The ** cosine function ** relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- The ** tangent function ** relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">** Use of Scaled Vector Diagrams to Determine a Resultant ** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Choose a scale and indicate it on a sheet of paper
 * 2) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Pick a starting location and draw the first vector //to scale// in the indicated direction. Label the magnitude and direction of the scale on the diagram
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> Starting from where the head of the first vector ends, draw the second vector //to scale// in the indicated direction. Label the magnitude and direction of this vector on the diagram.
 * 4) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Repeat steps 2 and 3 for all vectors that are to be added
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as **resultant (r)**
 * 6) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale
 * 7) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Measure the direction of the resultant using the counterclockwise convention

<span style="font-family: Arial,Helvetica,sans-serif; font-size: medium;">**__Homework 10/13/12 Lesson 1 C & D Vectors__**
<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">**Resultants** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The resultant is the vector sum of two or more vectors. It is //the result// of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R. As shown in the diagram, vector R can be determined by the use of a vector addition diagram. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Displacement vector R gives the same //result// as displacement vectors A + B + C. That is why it can be said that When displacement vectors are added, the result is a //resultant displacement//. But any two vectors can be added as long as they are the same vector quantity. In all such cases, the resultant vector is the result of adding the individual vectors. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">** Vector Components ** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">In this unit, we begin to see examples of vectors that are directed in //two dimensions// - upward and rightward, northward and westward, eastward and southward, etc. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to //transform// the vector into two parts with each part being directed along the coordinate axes. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%; line-height: 0px; overflow-x: hidden; overflow-y: hidden;"> <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Any vector directed in two dimensions can be thought of as having an influence in two different directions. 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.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">**__Homework 10/17/11 Lesson 1 E Vectors__**
<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">**Vector Resolution** <span style="font-family: Arial,Helvetica,sans-serif;">Any vector directed in two dimensions can be thought of as having two components. The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution that we will examine are
 * <span style="font-family: Arial,Helvetica,sans-serif;">the parallelogram method
 * <span style="font-family: Arial,Helvetica,sans-serif;">the trigonometric method

<span style="font-family: Arial,Helvetica,sans-serif;">**Parallelogram Method of Vector Resolution** <span style="font-family: Arial,Helvetica,sans-serif;">This method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. <span style="font-family: Arial,Helvetica,sans-serif;">1. Select a scale and accurately draw the vector to scale in the indicated direction. <span style="font-family: Arial,Helvetica,sans-serif;">2. Sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle <span style="font-family: Arial,Helvetica,sans-serif;">3. Draw the components of the vector. The components are the //sides// of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram <span style="font-family: Arial,Helvetica,sans-serif;">4. Label the components of the vectors with symbols to indicate which component represents which side <span style="font-family: Arial,Helvetica,sans-serif;">5. Measure the length of the sides of the parallelogram in //real// units. Label the magnitude on the diagram.

<span style="font-family: Arial,Helvetica,sans-serif;">**Trigonometric Method of Vector Resolution** <span style="font-family: Arial,Helvetica,sans-serif;">Trigonometric functions will be used to determine the components of a single vector. The method of employing trigonometric functions to determine the components of a vector are as follows:
 * 1) <span style="font-family: Arial,Helvetica,sans-serif;">Construct a //rough// sketch (no scale needed) of the vector in the indicated direction
 * 2) <span style="font-family: Arial,Helvetica,sans-serif;">Draw a rectangle about the vector such that the vector is the diagonal of the rectangle
 * 3) <span style="font-family: Arial,Helvetica,sans-serif;">Draw the components of the vector
 * 4) <span style="font-family: Arial,Helvetica,sans-serif;">Label the components of the vectors with symbols to indicate which component represents which side.
 * 5) <span style="font-family: Arial,Helvetica,sans-serif;">To determine the length of the side opposite the indicated angle, use the sine function. Substitute the magnitude of the vector for the length of the hypotenuse.
 * 6) <span style="font-family: Arial,Helvetica,sans-serif;">Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

<span style="font-family: Arial,Helvetica,sans-serif;">__** Homework 10/18/11 Lesson 1 G & H Vectors **__
<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">**Relative Velocity and Riverboat Problems** <span style="font-family: Arial,Helvetica,sans-serif;">A motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. The magnitude of the velocity of the moving object with respect to the observer on land will not be the same as the speedometer reading of the vehicle. Motion is relative to the observer. The observed speed of the boat must always be described relative to who the observer is. A tailwind is a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane is the vector sum of the velocity of the plane and the velocity of the wind. A headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Side Wind: To determine the resultant velocity, the plane velocity must be added to the wind velocity. The Pythagorean theorem can be used. The direction of the resulting velocity can be determined using a trigonometric function. <span style="font-family: Arial,Helvetica,sans-serif;">**Analysis of a Riverboat's Motion** <span style="font-family: Arial,Helvetica,sans-serif;">The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. The river current influences the motion of the boat and carries it downstream. Motorboat problems such as these are typically accompanied by three separate questions: <span style="font-family: Arial,Helvetica,sans-serif;">The second and third of these questions can be answered using the average speed equation.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif;">What is the resultant velocity of the boat?
 * 2) <span style="font-family: Arial,Helvetica,sans-serif;">If the Width of the river is x meters wide, then how much time does it take the boat to travel shore to shore?
 * 3) <span style="font-family: Arial,Helvetica,sans-serif;">What distance downstream does the boat reach the opposite shore?

<span style="font-family: Arial,Helvetica,sans-serif;">** Independence of Perpendicular Components of Motion ** <span style="font-family: Arial,Helvetica,sans-serif;">A force vector that is directed upward and rightward has an upward part and a rightward part. A component describes the affect of a single vector in a given direction. Any force vector that is exerted at an angle to the horizontal can be considered as having two components. The vector sum of these two components is always equal to the force at the given angle. The two components can be represented as legs of a right triangle formed by projecting the vector onto the x- and y-axis. The two perpendicular parts or components of a vector are independent of each other. A change in one component does not affect the other component. Changing a component will affect the motion in that specific direction. While the change in one of the components will alter the magnitude of the resulting force, it does not alter the magnitude of the other component. Any motion that is at an angle to the horizontal or the vertical can be thought of as having two perpendicular motions occurring simultaneously.

<span style="font-family: Arial,Helvetica,sans-serif;">10/19/11 Lab: Vector Activity
<span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">This was the information given to us by the other group. We had to start by the railing near the stairs, close to the window, but we did not know where the "legs" were going to take us.
 * Date: 10/19/11**
 * Partners: Maddy Weinfeld, Jenna Malley, John Chiavelli**

<span style="font-family: Arial,Helvetica,sans-serif;">We then walked the course given to us by the other group and discovered that the ending location was in the northeast corner near the windows. Then, we measured the displacement between our starting location and the final location and found that it was 22.99 meters. Then, we analyzed the information and tested our results in different ways:

Results: 23.04 m @ 2 degrees
 * <span style="font-family: Arial,Helvetica,sans-serif;">Graphical Method: **

Results: 23.16 m @ 1.7 degrees
 * Analytical Method:**

Measured vs. Graphical: Percent Error = **0.22%**
 * Percent Error Calculations (measured distance compared to both graphical results and analytical results)**

Measured vs. Analytical: Percent Error = **0.73%** =__ Projectile Motion __=

__**Homework 10/19/11 Lesson 2 A & B Projectile Motion**__
**What is a Projectile?** 1. What is a projectile? 2. What are the different types of projectiles? 3. What is the difference between the free-body diagrams of projectiles moving in all different directions? 4. What does inertia have to do with projectile motion? 5. What would happen if gravity did not have an effect on projectile motion? The main idea of this passage is that the __only__ force that acts upon a projectile is gravity. 1. A projectile is an object upon which the only force acting is gravity. Projectiles are the most common example of an object that is moving in two dimensions. A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by gravity. 2. There are a variety of examples of different types of projectiles. Three of these are an object dropped from rest, an object thrown straight up vertically into the air, and an object that is thrown upward at an angle to the horizontal. 3. Regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards,t he free-body diagram of a projectile always looks the same. 4. Inertia has to do with projectile motion because objects in projectile motion continue by means of its own inertia. Meaning, although gravity is the only force that acts upon an object in projectile motion, inertia is what gives it its horizontal aspect. 5. If gravity did not have an effect on projectile motion than the object would just infinitely continue in its initial path.
 * Questions: **
 * Main Idea: **
 * Answers to Questions: **

**Characteristics of a Projectile's Trajectory** 1. What are the two components of a projectile's motion? 2. Are the two components dependent or independent of each other? 3. Why does the presence of gravity not affect the horizontal component of a projectile? 4. What effect does gravity have on a non-horizontally launched projectile? 5. What would the result be if gravity did not have an effect on a non-horizontally launched projectile? The main idea of this passage is that projectiles travel with a parabolic trajectory due to the fact that the downward force of gravity accelerates them downward from their otherwise straight-line, gravity-free trajectory. Also, the force of gravity does not affect the horizontal component of motion. 1. The two components of a projectile's motion are the horizontal and vertical components. There are two components because many projectiles not only undergo a vertical motion, but also undergo a horizontal motion. 2. The two components are completely independent of each other, therefore, they must be discussed separately. 3. The force of gravity does not affect the horizontal component of a projectile because it acts downward and is unable to alter the horizontal motion. There must be a horizontal force to cause a horizontal acceleration. 4. Gravity causes an object in non-horizontal projectile to fall below its inertial path, forming a downward acceleration. Gravity is what causes an object to have a parabolic trajectory. 5. It would have no affect on the horizontal motion, and the object would just travel in the same direction it initially was traveling in However, the vertical component would be affected and the object would travel along a straight-line, inertial path.
 * Questions: **
 * Main Idea: **
 * Answers to Questions: **

**__Homework 10/20/11 Lesson 2 C Projectile Motion__**
1. How do the numerical values of the x and y components change with time? 2. What purpose does a vector diagram have? 3. How do the numerical values of velocity differ in a horizontally launched projectile compared to a non-horizontally launched projectile? 4. How do the numerical values of velocity and displacement differ? 5. What equation is used to describe the horizontal and vertical displacement of a projectile? The main idea of this passage is that the horizontal and vertical components of the velocity vector change with time during the course of an object's trajectory. 1. The horizontal component, or the x component, stays constant over the course of time. However, the vertical component, or the y component, decreases at a rate of 9.8 m/s/s. The vertical velocity changes by 9.8 m/s each second and the horizontal velocity never changes. 2. A vector diagram is used to represent how the x and the y components of the velocity change with time. The lengths of the vector arrows are representative of the magnitudes of that quantity. 3. There is no difference. The horizontal component still remains constant and the vertical component still decreases at a rate of 9.8 m/s/s. 4. For velocity, the horizontal component stays constant while the vertical component changes at a rate of 9.8 m/s/s. For displacement, the horizontal component changes at a constant rate as time goes on while the vertical component changes according to the equation stated below. 5. Vertical displacement: **y = (.5)(g)(t)^2** Horizontal displacement: **x = vi (of x) t**
 * Horizontal and Vertical Components of Velocity / of Displacement **
 * Questions: **
 * Main Idea: **
 * Answers to Questions: **

**10/24/11 Activity: Ball in Cup**
- Measure the initial velocity of a ball - Apply concepts from two-dimensional kinematics to predict the impact point of a ball in projectile motion - Take into account trial-to-trial variations in the velocity measurement when calculating the impact point
 * Partners: Maddy Weinfeld, John Chiavelli, Jenna Malley**
 * Objectives:**


 * Pre-Lab Questions**


 * 1. If you were to drop a ball, releasing it from rest, what information would be needed to predict how much time it would take for the ball to hit the floor? What assumptions must you make?** They only additional information that would be needed would be the distance of the starting height of the object. Because we can assume that the ball is falling at -9.81 m/s/s and has an initial velocity of 0 m/s, we are able to solve for the time it would take for the ball to hit the floor. Also, you can assume that no outside forces besides gravity are impacted the fall of the ball.


 * 2. If the ball in Question 1 is traveling at a known horizontal velocity when it starts to fall, explain how you would calculate how far it will travel before it hits the ground.** To answer this question, we would need to find the distance of the horizontal component of the ball's motion. Unlike the vertical component, the horizontal component changes at constant speed so we could use the equal v = d/t. Because we solved for it in the previous question, we would already have the time so we now would have enough information to calculate how far the ball would travel before it hit the ground.


 * 3. A single Photogate can be used to accurately measure the time interval for an object to break the beam of a Photogate. If you wanted to know the velocity of the object, what additional information would you need?** If we wanted to know the velocity of the object we would need to also know the angle at which it was being launched and the horizontal distance at which it lands. Without these two factors, we would not be able to solve for the velocity of the object.

- Measure the height of where the ball is being launched from the ground - Set up a chart of the situation, so we can collect the vi, a, t, and d of both the x and y components - Use photo copy paper to mark the exact spot of where the ball is landing after being launched - Launch the ball at least five times - Measure the distances of all the markings on the paper from the starting point of the ball and calculate an average - Use the equation d = vit + 1/2at^2 to solve for the time with the information from the y component - Use the time and plug in same equation but using the information from the x component - Calculate and record vi
 * 4. Write your procedure and what data you need to collect. How will you analyze your results in terms of precision and/or in terms of accuracy?**

We will analyze our results in terms of precision by completing numerous different trials of our procedure. By launching the ball at least five times, we guarantee that our results have consistency, evidently meaning that they are precise. In addition, we will also analyze our results in terms of accuracy. To prove that our results are accurate, we can take our calculations and use them in different situations and see if we get the results that we predicted.


 * Calculations for Part A (Find the initial velocity of the ball)**

However, these calculations were WRONG! At first, we did not take the height of the cup into account, which was causing the calculation we received to be too far back from where the ball was actually landing. Then, we measured the cup and learned that it was .095 meters. We then used this number while finding our result, and got a more accurate answer, which is shown below: When we calculated the result, we found that the cup would have to be 2.08 m away from the starting point of the ball. However, once we actually tried it out, we found that the cup had to be 2.09 m away, which was almost the exact result we got from our calculations.
 * Calculations for Part B (Change the height and calculated where to place the cup in order for the ball to land in)**

This is a video of when the ball landed in the cup 2.09 m away from the starting point. media type="file" key="working ball in a cup.m4v" width="300" height="300"
 * Video**

We found our percent error to be .0451%, which shows that our results were very precise.
 * Percent Error**

This experiment tested to see if we could calculate the theoretical result of a projectile and apply it to a real life situation. In part one, we easily were able to calculate and figure out that the initial velocity of the ball was 4.86 m/s. Then, we were able to use this velocity in part two to help us. However, at first, we were not able to calculate the correct result. We did not take the height of the cup into consideration and calculated that it would be 2.19 meters away from the starting point. Then, we realized that we needed to consider the height of the cup, and after that we calculated that we should place it 2.08 meters away, which was accurate. There also were sources of error in this lab that could be fixed to find even more precise results. For example, if we did not have to estimate so much while measuring distances and where to place the cup, we would have had more accurate numbers. Also, it was difficult to directly line up the cup with the launcher so the ball often landed to either to the right or to the left of the cup.
 * Conclusion**

__**Gourdorama Contest**__
**Pictures**
 * Partner: Danielle Bonnett**

**Calculations** The total mass of our Gourd-o-Rama was 1.31 kg. After we rolled our project down the incline, we found that the highest distance it went was 4.5 meters and that took 3.2 seconds. After we used the basic formulas and found our calculations, we learned that the initial velocity of our vehicle was 2.81 m/s and had an acceleration of -.88 m/s/s.
 * Results**

**Conclusion** In order to improve our cart, Danielle and I should have tried using different wheels that could carry the weight of the pumpkin further than it went. If we used skate wheels, for example, we probably would have achieved a much better result. In addition, because our pumpkin veered off to the side and crashed into the wall, we would have made sure that our wheels were lined up straight and that we dropped it directly straight down the incline, without pointing it in a certain direction

__**Shoot Your Grade Lab**__

 * Partners: Maddy Weinfeld, John Chiavelli, Jenna Malley**


 * Purpose and Rationale:** The purpose of the Shoot Your Grade Lab was to set the launcher to a specific angle (our group had 30 degrees) and use the initial velocity (which we had to find) to calculate where to place 5 hoops and a cup so the ball would go through all of them. The goal was to find the calculations of where to theoretically place the hoops/cup and then we had to use those calculations to hang the hoops so that the ball would go through all five of them and then land inside the cup.

Our rationale for the lab was based on information we learned previously in the unit. For example, we knew that we had to use the carbon paper to find the distance of where to ball was landing, measure the vertical distance of the launcher to the ground, and use the initial velocity which we found along with our given angle measurement to find out where to place the five different hoops and the cup. We then knew that we had to divide the horizontal distance by six in order to find the placement of each and distance between each hoop.


 * Materials and Methods:** The main materials in this lab include the launcher, which was set at 30 degrees, the ball, the five hoops, and the cup. However, there were also other additional materials used that were able to help us find out results. We used measuring tape to find the exact distances of where all the materials needed to be placed, and string and tape to mark where our hoops needed to be held so we were able to reset them after the next class adjusted them. After finding the initial velocity, the horizontal distance, the vertical distance, and the hang time, we were able to use all of these things to calculate the six exact placements of the objects and began to set up. We used both vertical and horizontal distances to place all of the hoops/cup, and continually launched the ball in order to see if we were placing them correctly. We often had to adjust our measurements, but evidently we were able to find where the ideal placements should be.

Procedure Pictures: Procedure Video: media type="file" key="6+test+run+for+procedure.mov" width="300" height="300"

These calculations show how we were able to calculate the initial velocity of the launcher for both the x and the y components. We used carbon paper while launching the ball the mark the exact spot where the ball was falling. We then recorded five different trials of this measurement and averaged them together to find the horizontal distance, which was 3.342 meters away from the launcher. Then, we drew a diagram to show what the situation looked like and that helped us to figure out what else we needed to calculate. We made a chart of all the information we needed, and then solved for the hang-time, which was .7955 seconds. Finally, we plugged all our information into the formula to find the initial velocity for both the x and the y components. The initial velocity for the y was 2.43 m/s and the initial velocity for the x was 4.20 m/s.
 * Calculations for Initial Velocity:**

Also, we had to calculate the horizontal distance between each hoop/cup so in order to do that we took the total horizontal distance and divided it by six. Each ring had to be 0.577 meters away from the previous one.
 * Physics Calculations**

Calculations of each point's horizontal and vertical distance Calculations of each point's horizontal and vertical distance in chart form:
 * Vertical distance is relevant to the height of the launcher

media type="file" key="FInal Video.m4v" width="300" height="300"
 * Measurements of When Rings Actually Worked**
 * Final Video (Four Rings)**


 * Percent Error Calculations**

In conclusion, we were able to get our ball to go through all five hoops, however, on the presentation day we were only able to get it to go through four. We also were slightly off because based on our calculations, we hypothesized that we would be able to get the ball to go through all five hoops and land in the cup. Although our hypothesis was not exactly correct, our percent error values showed that our results were somewhat precise. For example, we had a percent error of 0% on the second hoop and all the percent errors were all less than 3%, which shows that they all were pretty accurate. However, by looking at the actual measurements of the final presentation, we can see how we were slightly off. For example, according to our calculations we found that each horizontal distance would be .557 m away from the previous one, but we can see that the actual distances were often either forward or behind that point. The same situation happened with the vertical distances. We found our theoretical distances, but during the actual set up we often had to shift the hoops either up or down from the point that we calculated. Because our hypothesis was not exactly correct, we were able to look into different sources of error that could have contributed to this issue. First, basic human error was a huge factor while placing our rings. The rings would often shift from their original position after being placed, and it was difficult to measure the exact spot as to where the ideal placement should have been. In addition, the launcher was often sporadic and did not shoot the ball in the same exact spot every time that it was utilized. Also, the angle on the launcher often accidentally shifted, altering the trajectory of the launched ball. To fix these sources of error we could have done a number of things. First, we could have took longer to measure the exact position of where we wanted the hoops to be rather than just testing it out and adjusting it after. If we did this from the beginning, we could have had a better chance of getting all five hoops straight form the start. Also, we did not start using the weight until halfway through setting up, so some of the earlier hoops were not as even as most of the later hoops. If we used the weight/string tool the entire time, the rings could have been better positioned than we had them to be. Finally, we know that we can fix the fact that the launcher did not have consistent launches, however, we could used a clamp of some sort to keep the angle on the launcher from adjusting throughout the lab. In real life, this situation can be applied in many aspects. For example, since I play tennis, I understand that this general concept can be used during the game. Although there is no way to calculate the exact measurements and speeds while playing, I know that if I have a general idea of what the trajectory will look like then I will have a better clue as to where, what angle, and how hard to hit the ball. Although this concept can be applied to all sports, I find it rather helpful in this situation because it is one that applies personally to my life.
 * Conclusion**