Ch3_FavaA

=Lesson 1 - Vectors and Forces in Two Dimensions (Textbook Notes)= toc Vectors and Direction A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world, which can be divided into two categories - vectors and scalars. 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. Examples of vector quantities include displacement, velocity, acceleration, and force. A full description of the quantity demands that both a magnitude and a direction are listed. Vector quantities are often represented by scaled vector diagrams, which depict a vector by use of an arrow drawn to scale in a specific direction. Earlier diagrams that depict the forces acting upon an object are called free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of vector diagram:
 * 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.

**Conventions for Describing Directions of Vectors** Vectors can be directed due East, due West, due South, and due North. The two conventions for directions other than obvious four that will be discussed and used in this unit are described below: Two illustrations of the second convention (discussed above) for identifying the direction of a vector are shown below.
 * 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. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.

**Representing the Magnitude of a Vector** The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow in accordance with a chosen scale.



Vector Addition Two vectors can be added together to determine the result (or resultant). The task of summing vectors can be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. The //vector sum// will be determined as shown in the diagrams below.

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:
 * the Pythagorean theorem and trigonometric methods
 * the head-to-tail method using a scaled vector diagram

**The Pythagorean Theorem** A useful method for determining the result of adding two (and only two) vectors __that make a right angle__ to each other.

**Using Trigonometry to Determine a Vector's Direction** The direction of a //resultant// vector can often be determined by use of trigonometric functions. SOH CAH TOA. Must be two vectors that make right angle.

The measure of an angle as determined through use of SOH CAH TOA is __not__ always the direction of the vector.

**Use of Scaled Vector Diagrams to Determine a Resultant** 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 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.
 * 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.
 * 7) Measure the direction of the resultant using the counterclockwise convention discussed earlier.

Interestingly enough, the order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant. The resultant will still have the same magnitude and direction, therefore the order is insignificant.

Resultants The **resultant** is the vector sum of two or more vectors. As shown in the diagram, vector R can be determined by the use of an accurately drawn, scaled, vector addition diagram.

Displacement vector R gives the same //result// as displacement vectors A + B + C. That is why it can be said that **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 ... In all such cases, the resultant vector is the result of adding the individual vectors. "To do A + B + C is the same as to do R."

Vector Components A vector is a quantity that has both magnitude and direction. In the first couple of units, all vectors that we discussed were simply directed up, down, 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.

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 combined influence of the two components is equivalent to the influence of the single two-dimensional vector.

Vector Resolution 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 trigonometric method

**Parallelogram Method of Vector Resolution** The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. A step-by-step procedure for using the parallelogram method of vector resolution is: **Trigonometric Method of Vector Resolution** The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. The method of employing trigonometric functions to determine the components of a vector are as follows: Relative Velocity and Riverboat Problems On occasion objects move within a medium that is moving with respect to an observer. 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. To illustrate this principle, consider a plane flying amidst a **tailwind**. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane (that is, the result of the wind velocity contributing to the velocity due to the plane's motor) is the vector sum of the velocity of the plane and the velocity of the wind. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a **side wind** of 25 km/hr, West. The magnitude of the resultant velocity is determined using Pythagorean theorem. The direction of the resulting velocity can be determined using a trigonometric function. **Analysis of a Riverboat's Motion** The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat.
 * 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.
 * 4) Meaningfully label the components of the vectors with symbols to indicate which component represents which side.
 * 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.
 * 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.
 * 4) Meaningfully label the components of the vectors with symbols to indicate which component represents which side.
 * 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.

The resultant velocity of the motorboat can be determined in the same manner as was done for the plane. The resultant velocity of the boat is the vector sum of the boat velocity and the river velocity. Motorboat problems such as these are typically accompanied by three separate questions: 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 for both the x-component and y-component. **ave. speed = distance/time**
 * 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 difficulty of the problem is conceptual in nature; the difficulty lies in deciding which numbers to use in the equations. That decision emerges from one's conceptual understanding (or unfortunately, one's misunderstanding) of the complex motion that is occurring. The motion of the riverboat can be divided into two simultaneous parts - a motion in the direction straight across the river and a motion in the downstream direction. These two parts (or components) of the motion occur simultaneously for the same time duration. While the increased current may affect the resultant velocity - making the boat travel with a greater speed with respect to an observer on the ground - it does not increase the speed in the direction across the river. The component of the resultant velocity that is increased is the component that is in a direction pointing down the river. It is often said that "perpendicular components of motion are independent of each other." As applied to riverboat problems, this would mean that an across-the-river variable would be independent of (i.e., not be affected by) a downstream variable. The time to cross the river is dependent upon the velocity at which the boat crosses the river. Independence of Perpendicular Components of Motion 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 the horizontal component does not affect the vertical component. 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 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.

=Vector Activity: Mapping= //Graphical Analysis:// //Analytical Analysis:// //Percent Error://
 * [[image:graphical.jpg]]
 * [[image:vector_calc.png]]
 * [[image:%_error_vector.png]]

Lesson 2 - Projectile Motion (Textbook Notes) //Lesson A// //Lesson B// //Lesson C// 
 * an object that is acting on only by gravity as acceleration is called a projectile
 * what is a projectile?
 * object dropped or thrown upwards at any angle to the horizontal
 * object with only gravity acting on it
 * continues in inertial motion
 * which forces act upon projectiles?
 * gravity is the only force to the vertical motion in downwards direction
 * what are some examples?
 * an object thrown vertically upward
 * an object dropped from rest
 * what is inertia?
 * inertia is the resistance of an object to change its state of motion; it would continue at constant speed if not for gravity causing acceleration
 * what is a free-body diagram?
 * diagram used for representing projectiles
 * shows parabolic trajectory which is a curve, similar to free fall
 * projectiles have two components and only the vertical is affected by gravity
 * what are the two components of a projectile?
 * horizontal motion
 * vertical motion
 * how does gravity affect these components?
 * only the vertical component is affected by gravity, the horizontal is never
 * how does the vertical component travel?
 * it is constantly acceleration at -9.8 m/s/s (acceleration of gravity)
 * how does the horizontal component travel?
 * at a constant rate since
 * horizontal displacement and overall motion remains constant throughout an entire projectile, whereas vertical is susceptible to change due to gravity
 * how does velocity differ during vertical and horizontal trajectories?
 * velocity during vertical changes by -9/8 m/s/s, while during horizontal it remains constant
 * are projectiles the same as free fall?
 * not necessarily, free fall is one type of projectile which has an initial velocity of zero, whereas many projectiles while have an actual initial velocity
 * what are the vectors of horizontal displacement of a projectile?
 * the x vector would remain constant, while the y vector would have an initial velocity, reach its maximum height, and then accelerate downwards
 * what equations can be used with projectiles?
 * y=1/2gt2 for vertical displacement
 * x=(Vix)t for horizontal displacement of horizontally launched
 * y = (Viy)t + 1/2gt2 for vertical displacement of angle-launched

=Projectile Class Notes=

//Projectile Bingo://
 * has to been a object
 * projected has an initial force, whereas dropped has 0
 * parabolic trajectory
 * no air resistance
 * projectile continue in straight line at constant speed without gravity
 * gravity only affects vertical component
 * acceleration of gravity = -9/8 m/s/s
 * only force acting is gravity
 * no horizontal forces
 * force produces acceleration not a velocity
 * free-body diagrams are same for all directions
 * horizontal component is constant speed
 * free fall is a type of projectile
 * perpendicular components are independent
 * inertia - want to keep doing what you're already doing
 * two components are vertical and horizontal
 * dx=vx(t)
 * dy=(-1/2)(g)(t2)+(vy)(t)

//Problem Solving://
 * three types
 * horizontally launched
 * use height and range (horizontal displacement)
 * "ground-to-ground" launch
 * "off-a-cliff" launch
 * use the same equations as Big Five, but not d=1/2(v1+vf)t
 * can't mix any of the x and y components, except for time since it is the same for both

=Projectile Activity: Ball in Cup=

//Procedure//:
 * media type="file" key="New Project 1.m4v" width="300" height="300"
 * media type="file" key="Movie on 2011-10-24 at 08.58.mov" width="300" height="300"

//Results//:
 * 1) initial horizontal velocity at medium range
 * [[image:horiz_veloc_ball_cup.jpg]]
 * 1) change initial height and calculate where to place cup so ball can land in it three times in a row
 * this includes taking into account the height of the cup when finding total height [height of counter (1.1085 m) -height of cup (.092m)]
 * [[image:ball_in_cup_distance.jpg]]
 * 1) calculate percent error
 * [[image:%_error_ball_cup.png]]

=Gourd-o-rama Project= Partner: Julia Sellman

//Final Project//:
 * [[image:Screen_shot_2011-11-08_at_12.17.22_PM.png width="379" height="273"]]
 * [[image:Screen_shot_2011-11-08_at_12.17.43_PM.png width="479" height="298"]]

//Calculations//:
 * [[image:Photo_on_2011-11-08_at_09.33.jpg]]

//Results//:
 * vehicle mass (g) - 1.5 kilograms
 * time (s) - 8.16 seconds
 * distance traveled (m) - 11 meters
 * acceleration value - -.33 m/s/s
 * velocity value - 2.696 m/s

//Improvements//:
 * To improve our project, we would make sure the axels were more perfectly aligned and that they didn't alter when the car crashed. Our biggest issue was that after the first run, the car crashed into the wall and the axels became slightly unaligned. Our car then continued to crash into the side walls, which caused it to not be able to reach its maximum distance. Everything else seemed to work perfectly, so we wouldn't really change any of it.

=Lab: Shoot Your Grade= Group: Julia Sellman, Ben Weiss, Ryan Hall

//Hypothesis//:
 * The ball should go through all five hoops and land inside the cup following a parabolic trajectory if our calculations are correct and the five hoops are hung accurately.

//Available Materials//:
 * data studio
 * plumb bob
 * ramp
 * masking tape
 * yellow ball
 * photogate timer
 * meter stick or measuring tape
 * target
 * carbon paper
 * two right-angle clamps
 * newsprint
 * calipers

//Procedure//:
 * media type="file" key="Shoot for Your Grade.mov"
 * The launcher was set to 25 degrees and the hoops were hung from the ceiling and secured using the ceiling tiles and binder clips. The hoops were placed at the calculated distances and the cup was placed at the horizontal distance we calculated (2.953 meters). The ball was placed in the shooter at medium speed and shot. We adjusted the hoops the necessary amount for the ball to go through and repeated the launching until we made it through as many hoops as possible.

//Calculations//:
 * [[image:photo.jpg width="480" height="360"]]
 * Above are the calculations for the initial velocity of the ball at a 25 degree angle. In addition, the average horizontal distance was found to be 2.953 m, so we divided that into six subgroups to get the horizontal distances of each hoop and then the cup.
 * [[image:photo-1.jpg width="720" height="1005"]]
 * Above are the calculations for the vertical distances of each hoop and the time for each, with a total hang time of .652 seconds. The vertical distance from the shooter was found, so negatives were below the shooter and positives were above, and then added to the total height of the shooter (1.112 m) to find the height of each hoop from the ground (which is what h represents in my work).

//Data://
 * [[image:Screen_shot_2011-11-13_at_9.30.38_AM.png]]
 * We were not able to successfully get the ball through the Hoop 5 and the cup and therefore, the two do not have an experimental height.

//Analysis//:
 * [[image:Screen_shot_2011-11-13_at_9.30.52_AM.png]]
 * Above are our percent errors for each of the four hoops that we were able to get the ball through. Since there is no experimental height for the fifth hoop and cup, percent error could not be found for these. The percent error ranged from a very low 0.61% to a pretty high 16.99%.

//Conclusion//:
 * After hanging the hoops at the theoretical heights and conducting the experiment, we found our hypothesis to be partially correct. Some of the theoretical heights were very close to the experimental, yet others were pretty far or too far to be found. These theoretical values were based off of an initial velocity of the ball (4.83 m/s) that we had calculated. Therefore, if this initial velocity was off or varied, then the heights would subsequently be slightly off as well. The most accurate part of our hypothesis was the ball following a parabolic trajectory. Our theoretical values showed this trajectory, with the vertical distance increasing and then decreasing as the ball go farther away from the shooter. Our experimental values also showed this, as the hoops increased height from the shooter to Hoop 1 and then to Hoop 2, and then continued to decrease distance from Hoop 2 to 3 to 4, and would most likely follow this pattern had we found the other heights. If we had more time, then we would have found these other hoops and the cup and therefore proved this portion of the hypothesis accurate. Our percent error was lowest at 0.61% and highest at 16.99%. The lowest percent error was at the highest hoop (Hoop 2) and therefore the one closest to maximum height. The highest percent error was at Hoop 4, which was the farthest from maximum height that we had found. If we had also found Hoop 5 and the Cup, percent highest would probably be at the cup, as it is the absolute farthest from maximum height. Since the projectile accelerates at -9.8 m/s/s, each second that the ball falls from maximum height, the more inaccurate our theoretical values become.There were many sources of error in this lab that could also contribute to the high percent error. One major source is the inconsistency of the shooter. In finding the initial velocty, an average horizontal distance was used, which shows that the shooter didn't always reach the same distance and therefore varied in its initial velocty. The angle of the shooter wasn't secured and therefore could have been slightly off from 25 degrees on some launches. The spring inside the shooter also heats up after being used a lot and causes the ball to not be launched as fast/far. Also, the hoops we used were also used by two other classes and therefore were inevitably adjusted, causing another source of error. Our hoops were also very close to a vent so the air blowing out of it had moved our hoops from the exact theoretical placement. After evaluating all of these sources of error, many of them could be avoided if we were to redo the lab. Having our own space to hang the hoops would eliminate other groups from adjusting their placement. Also, being able to better secure the strings with a clamp would prevent them from slightly moving due to air pressure. Eliminating vents from the area would also help prevent the hoops from moving. To address the problems with the shooter, a different shooter could be used that doesn't rely on a spring that overheats or a cooling system to prevent overheating could be used. To eliminate any slight changes in the angle, the shooter could be secured at 25 degrees by using a clamp of some sort. By correcting all of these errors, we would probably be able to get the ball through all five hoops and into the cup with very little percent error.