Ch3_VanMaldenJ

= **Chapter 3-** =

Homework- 10/12-
toc >>>>
 * All these quantities can by divided into two categories- vectors and scalors.
 * Distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc.
 * A vector quantity is a quantity that is fully described by both magnitude and direction.
 * Represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction
 * Commonly called as free body diagrams, EXAMPLE TO THE RIGHT -->[[image:http://www.physicsclassroom.com/Class/vectors/u3l1a3.gif width="179" height="184" align="right"]]
 * The vector diagram depicts a displacement vector.
 * Vectors can be directed due East, due West, due South, and due North. But some vectors are directed //northeast// (at a 45 degree angle); and some vectors are even directed //northeast//, yet more north than east.
 * The two conventions that will be discussed and used in this unit are described below:
 * 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.
 * 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.
 * 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.
 * A scalar quantity is a quantity that is fully described by its magnitude.
 * Vector Addition
 * A variety of mathematical operations can be performed with and upon vectors, one being the addition of vectors.
 * Two vectors can be added together to determine the result (or resultant).
 * In Newton's laws of motion, that 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 of adding up all the force vectors.
 * 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:
 * Pythagorean theorem and trigonometric methods-
 * 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 method is not applicable for adding more than two vectors or for adding vectors that are __not__ at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
 * Example problem- Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
 * Using Trigonomic ways-
 * The direction of a //resultant// vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle.


 * The measure of an angle as determined through use of SOH CAH TOA is __not__ always the direction of the vector.
 * Head to tail method using a scaled vector diagram-
 * 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 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. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.
 * 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) 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 lesson

Homework 10/13:

 * Resultants
 * The resultant is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R.
 * When displacement vectors are added, the result is a //resultant displacement//. 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.//
 * The resultant is the vector sum of all the individual vectors. The resultant is the result of combining the individual vectors together. The resultant can be determined by adding the individual forces together using vector addition methods.
 * Vectors-
 * A vector is a quantity that has both magnitude and direction. Displacement, velocity, acceleration, and force are the vector quantities..
 * When 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.
 * 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.
 * Any vector directed in two dimensions can be thought of as having two different components. The component of a single vector describes the influence of that vector in a given direction.

Homework 10/17:
>> How to: >>> Select a scale and accurately draw the vector to scale in the indicated direction. >>>
 * Vector Resolution
 * Any vector directed at an angle to the horizontal (or the vertical) can be thought of as having two parts (or components).
 * 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
 * the parallelogram method- 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) 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).
 * 2) 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).
 * 3) 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.
 * 4) 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- The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector.
 * How to:
 * 1) Construct a //rough// sketch (no scale needed) of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal.
 * 2) Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. 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.
 * 3) Draw the components of the vector. The components are the //sides// of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. 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 force velocity component might be labeled vx; etc.
 * 5) 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. Use some algebra to solve the equation for the length of the side opposite the indicated angle.
 * 6) Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

Vectors Lab-
Analysis Calculations Graphical Calcultions Percent Error-ANALYTICAL -Actual- 27.19 m - Calculated- 26.98 m -( l theoretical-experimental l / theoretical ) X 100 -( l 26.98- 27.19 l / 26.98 ) X 100 - Percent Error = .778% Percent Error- GRAPHICAL -Actual- 27.19 m -Measured- 26.8 m ( l theoretical-experimental l / theoretical ) X 100  -( l 26.8- 27.19 l / 26.8 ) X 100   - Percent Error = 1.46%

Homework 10/18-
Pythagorean (100 km/hr)2 + (25 km/hr)2 = R210 000 km2/hr2 + 625 km2/hr2 = R210 625 km2/hr2 = R2SQRT(10 625 km2/hr2) = R**103.1 km/hr = R**
 * ===Relative Velocity and Riverboat Problems-===
 * 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. As another example, a motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. 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.
 * Consider a plane flying amidst a tailwind. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity.
 * If the plane encounters a headwind, the resulting velocity will be less than 100 km/hr. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Suppose a plane traveling with a velocity of 100 km/hr with respect to the air meets a headwind with a velocity of 25 km/hr. I
 * Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a **side wind** of 25 km/hr, West. Pythagorean can be used to solve this.

tan (theta) = (opposite/adjacent)tan (theta) = (25/100)theta = invtan (25/100)**theta = 14.0 degrees** RIverboat's-
 * Motorboat problems such as these are typically accompanied by three separate questions:
 * 1) What is the resultant velocity (both magnitude and direction) of the boat?
 * 2) 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) What distance downstream does the boat reach the opposite shore?
 * The first of these three questions was answered above; the resultant velocity of the boat can be determined using the Pythagorean theorem (magnitude) and a trigonometric function (direction). The second and third of these questions can be answered using the average speed equation (and a lot of logic).
 * Independence of a Perpendicular Components of Motion-
 * A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. That is to say, if you pull upon an object in an upward and rightward direction, then you are exerting an influence upon the object in two separate directions - an upward direction and a rightward direction. 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. This is depicted in the diagram below.
 * Any vector - whether it is a force vector, displacement vector, velocity vector, etc. - directed at an angle can be thought of as being composed of two perpendicular components. These 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.
 * All vectors can be thought of as having perpendicular components. In fact, any motion that is at an angle to the horizontal or the vertical can be thought of as having two perpendicular motions occurring simultaneously. These perpendicular components of motion occur independently of each other. Any component of motion occurring strictly in the horizontal direction will have no affect upon the motion in the vertical direction. Any alteration in one set of these components will have no affect on the other set.

Homework 10/19-

 * What is a projectile?
 * A projectile is an object dropped from rest or thrown vertically upwards at an angle to the horizontal. It is any object projected or dropped that continues in motion by its own inertia.
 * Does vertical motion increase? decrease?
 * Yes, the diagram should most of the time look like a parabola (parabolic trajectory). Therefore, the velocities will increase (while going down) and decrease(on the way up).
 * What are the many different projectiles?
 * [[image:http://www.physicsclassroom.com/Class/vectors/u3l2a2.gif width="298" height="152"]]
 * Does the horizontal or vertical velocity change?
 * The horizontal velocity of a projectile remains constant throughout, while the vertical velocity is constantly changing.
 * What force acts upon the vertical motion?
 * Gravity acts upon the vertical motion so that the acceleration will always be -9.8 m/s^2.
 * Central Theme- Horizontal and Vertical motion are independent from each other. Vertical motion is acted upon by the force of gravity.
 * Gravity acts upon the vertical motion so that the acceleration will always be -9.8 m/s^2.
 * Central Theme- Horizontal and Vertical motion are independent from each other. Vertical motion is acted upon by the force of gravity.

Homework 10/20-
>>> This diagram shows us that the horizontal motion is remaining constant while the vertical is not.
 * Questions-
 * Why doesn't the horizontal velocity undergo acceleration due to gravity?
 * Because horizontal motion is independent from vertical motion, and horizontal motion remains constant throughout, the horizontal velocity is unaffected by the acceleration due to gravity.
 * Can vertical velocity ever be zero?
 * Because it undergoes adg when the object reaches its maximum height, the velocity is zero. Also, we know from the time of launching every second the velocity changes by 9.8 m/s.
 * Which independent component's displacement is affected by gravity?
 * The vertical displacement is affected by the adg, while the horizontal displacement is not affected at all.
 * What would a projectile diagram look like?
 * [[image:http://www.physicsclassroom.com/Class/vectors/u3l2c11.gif width="257" height="271" align="bottom"]]
 * How are projectiles measured?
 * By how the numerical values of the x- and y-components of the velocity and displacement change with time (or remain constant).
 * Central Idea/ Theme-
 * Horizontal motion and displacement are unaffected by the acceleration due to gravity and remain constant the whole time, while the vertical motion and displacement are affected by it and do not remain constant throughout.

Ball In Cup Activity-
Procedure- media type="file" key="Movie on 2011-10-24 at 12.42.mov" width="300" height="300" media type="file" key="Movie on 2011-10-25 at 13.07.mov" width="300" height="300"

2 3 4 5 Average || 2.8 2.75 2.81 2.83 2.85 2.82 ||
 * Run Number || Distance Traveled (m) ||
 * 1

Conclusion- The reason for the percent error of 8.3% could have been due to multiple factors. One of the factors was the inconsistency of the launchers. Other factors could have been as a result of incorrect measurements and etc. Also, because it is so easily to move and twist the launches without realizing, they could have not been angled completely straight, also leading to a chance of having a percent error.

Gordarama Activity-
Distance Traveled- 6.45 m Change in Time- 5.2 s

Change in distance= 1/2(vf+vi)t 6.45= 1/2 (5.2)(vi) vi=2.5 m/s

Change in distnace= vit + 1/2at^2 6.45=2.4(5.2) + 1/2(5.2^2)a -6.03=13.52 a a= -.45 m/s^2

If I were to redo this experiment and change it so that the cart would have traveled a farther distance in a shorter amount of time I would change a few things. First of all I would use different wheels that would move faster and glide better, like the wheels of a skateboard or a scooter. Also, I would hollow out the pumpkin and make it the minimum weight allowed and create a whole in the middle so that the pumpkin wouldn't be as affected by air resistance. Finally, I would have attached the pumpkin to the front of the cart so that when it was rolled off the ramp it would not have tipped over because the weight would have been centered around the front.

Shoot Your Grade-

 * Purpose**- The purpose of this lab is to find the positions of a ball will be after shot from a launcher at a certain angle. Once given the angle of a projectile, you can find the horizontal and vertical displacements of the ball and the initial velocity. GIven this information, you can plug in numbers into the projectile equations which you will use to calculate the trajectory of the ball and positions at which you would like to place your hoops. With the calculated positions, you would hang up the hoops and launch the projectile, hoping it will pass through the calculated positions. However because the launcher are not completely consistent, the launches may not always make it through every hoops, so the position may need to be altered higher or lower depending on where the ball passes the horizontal position.


 * Materials**- The materials that were used in this lab are as followed: masking tape, launchers, string, ball, carbon paper, normal paper, and measuring tapes. The masking tape was used as the ring from the projectile to pass through and as a way of making sure that the strings attached to the ceiling didn't move throughout the experiment. We able to attach the ring to the ceiling by attaching to strings onto in and weaving it through the ceiling tiles. The carbon paper allowed us to find out the horizontal distance traveled while testing for the initial velocity by marking the exact spot where the projectile came in contact with the ground. Finally, we used the measuring tapes to measure the exact vertical and horizontal distances traveled by the projectile.

media type="file" key="physics vid2.mov" width="300" height="300" After shooting the launcher through the hoops multiple times, our projectile made it through fours consecutive hoops three times in a row.
 * Video Procedure-**

Angle- 15 degrees Vertical Distance- -1.16m Average Horizontal Distance- 2.74m

Initial Velocity- The data in the table above was collected from multiple different sources. We found the y-distance by measuring the distance from the launcher to the floor and we found the x-distance by using the average horizontal distance calculated after testing the launcher a few times. Because this was a projectile problem, we know that the x- acceleration will always be 0 m/s^2 while the y-acceleration will be -9.8 m/s^2. We didn't have to calculate the angle of the launcher because it was given to us, angling it at 15 degrees. From the already known information, we were able to solve for the total time of the projectile and with the total time able to find the initial velocity.
 * Calculations-**

**Actual Ring Positions-** Ring 1- (.45m, 1.3m) Ring 2- (.81m, 1.24m) Ring 3- (1.38m, 1.09m ) Ring 4- (1.67m, .99m ) Ring 5- (2.24m, .52m ) **Percent Error-**Percent Error- ( l theoretical-experimental l / theoretical ) X 100 ( l 1.23 - 1.3 l / 1.23) X 100 Percent Error- 5.7%
 * Calculations for Hoops and Cup-**

**Conclusion-**No our hypothesis or theoretical positions were not correct. Our theoretical guesses as for where the hoops should be placed were not correct because we had to move each ring vertically up so that the projectile would go through them, instead of hitting them like they did for our theoretical calculations. We can see that this proves true by looking at the percent error and the differences between the actual y positions and the theoretical y positions. The error occurred through the inconsistency of the launcher and the fact that the launcher tended to shift every time we shot it so that the ball wasn't being shot straight ahead, but instead to one side. Also, error may have occurred in the calculation of the initial velocity, we may have mis-measured the horizontal distance, making every one of our calculated positions a little to low for the actual positions. If we were to do this lab again I would do multiple things differently. Firstly, I would use a different launcher because the angle of the launcher was altered every time we shot the projectile. Also, if possible I would have used a tool that would have allowed us to see where the launcher was aimed, so we could line the rings up perfectly with it. This can be applied to many real-life situations, for example any sport involving aiming a ball. For example, a pitcher can use this concept when pitching. It will allow them to know at what angle and with what velocity up they should aim the ball so that it will be in the strike zone.