Ch3_hwangt

= Chapter 3 toc= 10/12/11

Vectors - Lesson 1
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. For example, the diagram at the right shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is __1 cm = 5 miles__, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles Two or more vectors can be added together to determine the result (or resultant). The //net force//experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result (or resultant) of adding up all the force vectors.These rules for summing vectors were applied to free-body diagrams in order to determine the net force. In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The //vector sum// will be determined for the more complicated cases shown in the diagrams below. 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. The direction of a //resultant// vector can often be determined by use of trigonometric functions. These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. Once the measure of the angle is determined, the direction of the vector can be found. The measure of an angle as determined through use of SOH CAH TOA is __not__ always the direction of the vector. The following vector addition diagram is an example of such a situation. The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the **head-to-tail method** is employed to determine the vector sum or resultant. A common Physics lab involves a //vector walk//. Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, //head-to-tail// method). The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to //real// units using the given scale.
 * a)**
 * Vectors and Direction**
 * A vector quantity is a quantity that is fully described by both magnitude and direction.
 * On the other hand, a scalar quantity is a quantity that is fully described by its magnitude.
 * Vector quantities that have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed.
 * Vector quantities are not fully described unless both magnitude and direction are listed.
 * Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction.
 * [[image:http://www.physicsclassroom.com/Class/vectors/u3l1a3.gif]]The vector diagram depicts a displacement vector.
 * Vector diagrams need:
 * a clearly listed scale
 * 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)
 * [[image:http://www.physicsclassroom.com/Class/vectors/u3l1a6.gif]]
 * Conventions for Describing Directions of Vectors**
 * 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. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "[|tail]" from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit
 * b)**
 * Vector Addition**
 * The Pythagorean Theorem**
 * Using Trigonometry to Determine a Vector's Direction**
 * Use of Scaled Vector Diagrams to Determine a Resultant**

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below. The **resultant** is the vector sum of two or more vectors. It is //the result//of adding two or more vectors together. To say that vector R is the //resultant displacement//of displacement vectors A, B, and C is to say that a person who walked with displacements A, then B, and then C would be displaced by the same amount as a person who walked with displacement R. That is why it can be said as A+B+C=R. 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. If two or more velocity vectors are added, then the result is a //resultant velocity//. If two or more force vectors are added, then the result is a //resultant force//. If two or more momentum vectors are added, then the result is ... d)**Vector Components** When there was a free-body diagram depicting the forces acting upon an object, each individual force was directed in //one dimension//- either up or down or left or right. Now 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. 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. For example, a vector that is directed northwest can be thought of as having two parts - a northward part and a westward part. 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. 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:
 * 1) Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.
 * 2) 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 (e.g., SCALE: 1 cm = 20 m).
 * 3) 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) Repeat steps 2 and 3 for all vectors that are to be added
 * 5) Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as **Resultant** or simply **R**.
 * 6) Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).
 * 7) Measure the direction of the resultant using the counterclockwise convention discussed earlier in this lesson.
 * c)Resultants**
 * e)Vector Resolution**
 * the parallelogram method
 * the trigonometric method

The Parallelogram Method: The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the 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.
 * 1) Select a scale and accurately draw the vector to scale in the indicated direction.
 * 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 (a special case of a parallelogram).
 * 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. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right).
 * 4) Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled Fnorth. A rightward velocity component might be labeled vx; etc.
 * 5) Measure the length of the sides of the parallelogram and [|use the scale to determine the magnitude] of the components in //real// units. Label the magnitude on the diagram.

The Trigonometric Method: On occasion objects move within a medium that is moving with respect to an observer. For example, an airplane usually encounters a wind - air that is moving with respect to an observer on the ground below. In such instances as this, the magnitude of the velocity of the moving object (whether it be a plane or a motorboat) 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 observer on land, often named (or misnamed) the "stationary observer" would measure the speed to be different than that of the person on the boat. The observed speed of the boat must always be described relative to who the observer is. A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. These two parts of the two-dimensional vector are referred to as components. 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 parts or components. The vector sum of these two components is always equal to the force at the given angle.
 * g)Relative Velocity and Riverboat Problems**
 * h)Fundamentals and Operations**

Activity: Orienteering
Part 1:

Part 2:

Vectors: Lesson 2
The central idea of this section is that gravity is the only force acting upon a projectile. Central Idea: Vertical and Horizontal motion are independent from one another so gravity will not affect horizontal motion in anyway.
 * a)What is a projectile?**
 * 1) What is a projectile?
 * 2) How are projectiles and free-falling objects similar and how are they different?
 * 3) What does gravity do to projectiles?
 * 1) A projectile is an object upon which the only force acting is gravity.
 * 2) Projectiles and free-falling objects do not differ in any way except for the fact that projectiles move horizontally as well as vertically.
 * 3) Gravity alters the path that the projectile follows by making the object move upward and rightward for a certain amount of time, making it reach a maximum point, and fall for the same amount of time and distance as it did when it moved upward.
 * b)Characteristics of a Projectile's Trajectory**
 * 1) What kind of path does a projectile follow?
 * 1) A projectile moves both horizontally and vertically. A projectile always follows a parabolic path.

Central Idea: Due to the fact that horizontal and vertical motion are independent from one another, gravity has no affect on horizontal motion which means that the horizontal velocity will stay the same throughout the entire flight of the object.
 * c)Describing Projectiles With Numbers**
 * 1) How does one calculate horizontal and vertical motion for projectiles?
 * 1) They can be found by using the equation ∆d = vit +1/2at2.

Activity: Ball in Cup
Procedure: media type="file" key="My First Project.mov"

Part 1:

Part 2:

Shoot Your Grade Lab
Partners: Jonathon Itskovitch, Lauren Barins

Purpose: The purpose of this lab is to basically shoot your grade. Each group needs to shoot a ball through must find the initial velocity of the launchers they are using and find the distance that it travels. The point is to use the information gathered and hang up 5 rings on the ceiling and a cup on the floor. We find 5 different points, find their x and y displacements and hang them on the ceiling. Then put the cup where the ball should land. Point are given by how many rings and the ball goes through and if it goes into the cup. If it goes through one ring, then the group gets a 75; if it goes through 2 rings, then the group gets an 80. If a group get it through all five rings and into the cup than the group receives a 100. Materials: Launcher, ball, paper, carbon paper, 5 rolls of masking tape, string, cup, measuring tape

Methods: First, since we did not know the the initial velocity of the launcher, we set up the launcher at a 20 degree angle, taped a piece of paper on the ground, put carbon paper over it and shot the ball at medium power to see where the ball landed. We then found the average of the points. We then made a table and calculated the initial velocity of the ball when launched. After we found the initial velocity, we then calculated the x and y displacements of 5 different points since we already know where the ball was going to land. We then positioned all the rings so that they were right in front of the launcher and would go through the rings and into a cup on the floor. We adjusted each ring and the cup so that the ball would sail smoothly every time through each ring and fall into the cup. We did this until we got the ball through 5 rings but just missed getting it into the cup.

Procedure: media type="file" key="ballthroughrings-1.mov" width="300" height="300"

Data:

Analysis: Point to show Percent Error: Point 1 (t=0.1 seconds)
 * Time (seconds) || X (meters) || Theoretical Y (meters) || Actual Y (meters) || Percent Error ||
 * 0.1 || 0.44 || 1.28 || 1.26 || 1.56% ||
 * 0.2 || 0.87 || 1.29 || 1.27 || 0.78% ||
 * 0.3 || 1.3 || 1.20 || 1.21 || 0.67% ||
 * 0.4 || 1.73 || 1.02 || 1.02 || 0% ||
 * 0.6 || 2.17 || 0.73 || 0.7 || 4.11% ||

Sample Calc for Percent Error: Percent Error = {(theoretical-experimental)/theoretical} * 100 Percent Error = {(1.28-1.26)/1.28} * 100 Percent Error = 1.56%

Conclusion: In the end, the lab went well. my group was able to get the ball through 5 rings but sadly did not go into the cup. It was close though. It hit the side of the cup and knocked the cup over. The lab was fun to do even though it was tedious moving the rings where you wanted them to be. There were many sources of error for this lab. One of which is that we did not measure the first ring correctly. We put it much farther away than it was originally suppose to be. Another source of error was that the launcher kept on changing positions. When we pulled the string to launch the ball, the launcher would sometimes move so that the angle of the launcher was no longer at 20 degrees or the base of the launcher moved from side to side. To correct these errors, we could hold the launcher down so that the base does not move from side to side, pull the string to the launcher more gently, and measure correctly. This concept is very important when a person is shooting a bow and arrow. When shooting a bow and arrow, a person must accurately shoot an arrow at the target so that the arrow hits the center of the target. If the arrow is not shot correctly, the arrow could hit a different part of target, sail above, far right, or far left of the target, or not reach the target at all. So the person must angle the bow and arrow so that the arrow hits the center of the target.