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 * ** Chapter 3 **

** Physics Classroom Lesson 1 A+B **

 * Vectors Fundamentals and Operations **
 * Hw: 10/12 **

A ** vector quantity is a quantity that is fully described by both magnitude and direction **.
 * Vectors and Direction **
 * On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. **
 * Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. __ The vector diagram depicts a displacement vector. __**
 * ** 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). **
 * ** ex: **



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 **.

Assume it is second option if it does not give direction: Always assume from East if not given Northeast = North of 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.



Vectors are drawn to show magnitude of the movement of an object. However, they must include a scale, and either an angle according to a direction (North, South, East or West) or an angle according to the counter-clockwise measurement (referring to East)

** Vector Addition ** Follows Rules of Simple Addition
 * Two vectors can be added together to determine the result (or resultant) **.

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.....

There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors.
 * ** the Pythagorean theorem and trigonometric methods **
 * ** the head-to-tail method using a scaled vector diagram **

** The Pythagorean Theorem ** 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. **

** Using Trigonometry to Determine a Vector's Direction ** The direction of a r ** esultant **** 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. 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. 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. 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. O ** nce 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 **.


 * When the two vectors that are to be added do not make right angles to one another, or when there are more than two vectors to add together, we will employ a method known as the head-to-tail vector addition method **.

** Use of Scaled Vector Diagrams to Determine a Resultant ** Using a scaled diagram, the ** head-to-tail method ** is employed to determine the vector sum or resultant. ** Each time one measurement ended, the next measurement would begin. In essence, you would be using the head-to-tail method of vector addition. ** *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 this lesson] __.

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. For example, consider the addition of the same three vectors in a different order. Use drawing in head-tail method when you have more than 3 vectors because 3 vectors will always form a triangle and you solve that using pythagorean theorem (meet at right angle) or using trigonometry (not right angle)
 * Order does not matter at all when drawing vectors
 * Resultants are the same!!

Physics Classroom Lesson 1: C+D

 * Vectors: Motion and Forces in Two Dimensions **
 * HW 10/13 **

** Resultants **
 * The resultant is the vector sum of two or more vectors ** . ** It is the result of adding two or more vectors together. **** When displacement vectors are added, the result is a resultant displacement. ** An ** y two vectors can be added as long as they are the same vector quantity. **



In summary, the resultant is the vector sum of all the individual vectors. ** The resultant is the result of combining the individual vectors togethe ** r.

** Vector Components ** A __ [|vector] __ is a quantity that has both ** magnitude and direction. ** 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. **

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.

Measuring Angles Class Notes
10/13/11



Vector Class Notes
10/14/11

Activity: Orienteering
10/14/11 - 10/21/11 Nicole Tomasofsky, Hella Tallas, Noah Pardes

Objective
 * Get from location a to location b using only direction and distance

Materials
 * Tape Measure
 * Outdoor Lawn

Location
 * A: Tree in South East area of courtyard with the name "ray" on plaque
 * B: Largest tree in North West area of courtyard (Largest trunk)

Data:





Based off scale
 * Measured Resultant = 2646.0 cm
 * Calculated Resultant = 2695.5 cm



Percent Error
 * Theoretical (Measured based of scale) --> Actual (Outdoor measurement) = 1.63%
 * Theoretical (Pythagorean theorem calculated) --> Actual (Outdoor measurement) = .241%

Conclusion
 * These percent errors are very small and therefore very good. There was limited percent error because we personally did the measuring. Also our distances in the courtyard were smaller than the distances in the football field so we did not have to use multiple rolls of tape or switch over tapes which lessened percent error.

Other Group's Data:

Start: Bottom left corer of football field (at 0)



Based of Scale:
 * Measured Resultant= 7040 cm
 * Calculated Resultant = 7254.09 cm



Percent Error
 * Theoretical (Measured based of scale) --> Actual (Outdoor measurement) = 10.4%
 * Theoretical (Analytical) --> Actual (Outdoor measurement) = 13.0%

Our percent error was a little high for this type of experiment. Our actual resultant was much lower than our graphical and analytical answer. Most likely, we had a measuring error when measuring our resultant outside. This could come from reasons like not be able to measure the resultant with one tape measure because it was too long, also our tape measure not being at a straight. Finally the final point we measured from was probably not accurate either because we might have measured wrong, not have had our tape measure straight enough, and not been at directly 90 degree angles. If we had time, we should have measured the resultant again, with these errors in mind. and mostly likely our results would turn out better and our percent error would be smaller.

** Physics Classroom Lesson 1: E **
Vector Fundamentals & Operations Hw: 10/17/11

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
 * Vector Resolution **

** Parallelogram Method of Vector Resolution **
 * 1) Draw the components of the vector. The components are the ** sides ** of the parallelogram
 * 2) 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.

A velocity vector with a magnitude of 50 m/s and a direction of 60 degrees above the horizontal may be resolved into two components: ex:

** Trigonometric Method of Vector Resolution ** Trigonometric functions ** relate the ratio of the lengths of the sides of a right triangle to the measure of an acute angle within the right triangle **. The method of employing trigonometric functions to determine the components of a vector are as follows:
 * 1) Construct ** a rough sketch ** (no scale needed) of the vector in the indicated direction.
 * 2) Draw ** a rectangle about the vector ** such that the vector is the diagonal of the rectangle. ** The sketched lines will meet to form a rectangle. **
 * 3) ** Draw the components of the vector. ** The components are the ** sides ** of the rectangle.
 * 4) 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. **

After rearranging the order in which the three vectors are added, the resultant vector is now the hypotenuse of a right triangle. (**Adding North + South and West + East)** **The size of the resultant was not affected by this change in order.** This illustrates an important point about adding vectors: the resultant is **independent** by the order in which they are added. The above discussion explains the method for determining the magnitude of the resultant for three or more perpendicular vectors.The mnemonic ** SOH CAH TOA  ** is a helpful way of remembering which function to use. The problem is not over once the value of theta (Θ) has been calculated. **This angle measure must now be used to state the direction**.
 * Component Method of Vector Addition **
 * This Pythagorean approach is a useful approach for adding any two vectors that are directed at right angles to one another. **


 * In summary, the direction of a vector can be determined in the same way that it is always determined - by finding the angle of rotation counter-clockwise from due east. **

Now we will consider situations in which the two (or more) vectors that are being added are not at right angles to each other. T**he Pythagorean theorem is not applicable to such situations since it applies only to right triangles.**

So whenever we think of a northwest vector, we can think instead of two vectors - a north and a west vector. **The two components Ax + Ay can be substituted in for the single vector A in the problem.**

Physics Classroom Lesson 1: G+H
Vector Fundamental & Operations 10/18/11


 * Relative Velocity and Riverboat Problems **

To illustrate this principle, consider a plane flying amidst a ** tailwind **. If the plane is traveling at a velocity of 100 km/hr with respect to the air, and if the wind velocity is 25 km/hr, then what is the velocity of the plane relative to an observer on the ground below? The resultant velocity of the plane is the vector sum of the velocity of the plane and the velocity of the wind.

The resulting velocity of the plane is the ** vector sum ** of the two individual velocities. ** To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity **. The angle between the resultant vector and the southward vector can be determined using the ** sine, cosine, or tangent functions. **


 * Analysis of a Riverboat's Motion **

** 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? **


 * 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 i ** ndependent ** of each other.

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.

** Physics Classroom Lesson 2: A+B **
Projectile Motion Hw: 10/19/11

What is a Projectile?

P
 * Basic definition/idea behind Projectile Motion
 * Inertia and how it contributes to projectile motion
 * Newton's law and how projectile motion depends on it

Q
 * What is Projectile Motion?
 * How can projectile motion be found?
 * What forces contribute to projectile motion?
 * What is Newton's Law and what does it have to do to about Projectile Motion?
 * Is projectile motion evident in every day life?

R
 * Read the material

S
 * Projectile Motion is an object that only has the influence of gravity on it
 * A motions own inertia is the only aspect that affects it
 * Newton's law states that force is not required to keep an object moving
 * Force is only required to maintain an acceleration
 * ** Newton's Law of Inertia ** : In absence of gravity, an object in motion will continue in motion with the same speed in the same direction
 * Gravity causes vertical acceleration

T
 * Projectile motion is motion with only the influence of gravity acting upon it.
 * Projectile motion can be found knowing whether there is gravity or not
 * Gravity is the only force that contributes to projectile motion
 * Newton's law states that without gravity an object will continue to move in same direction, which applies to all objects moving in projectile motion
 * Projectile motion is not evident in everyday life because we life in a world where gravity can not be switched on and off

Characteristics of a Projectile's Trajectory

P
 * Projectiles with horizontal motion
 * Projectiles with diagnoal motion (not horizontal )
 * Finding the velocity, distance, and acceleration of both types ^^

Q
 * What happens to an object when it is released horizontally and there is no gravity?
 * Vice versa, what happens and it is released not horizontally (in a diagonal line)
 * Can the same kinematics Big 5 equations be applied to these types of problems?

R
 * Read

S
 * Horizontal motion and vertical motion are perpendicular to each other and therefore independent
 * Gravity will push a projectile in a downward acceleration
 * Projectiles have a constant velocity since gravity does not act upon it since gravity is a vertical force

T
 * When an object is released without gravity it will continue on the same path at the same velocity
 * This will happen if it is released at an angle to the horizontal line
 * One can infer that velocity, distance, acceleration, and time can be found in a projectile, however it was not discussed in this reading.

Physics Classroom Lesson 2 C
Projectile Motion HW: 10/20/11

Describing Projectiles with Numbers

P
 * The horizontal and vertical velocity of a projectile
 * The horizontal and vertical displacement of a projectile

Q
 * How can the vertical/horizontal velocity/displacements be found in a projectile?
 * Are there any shortcuts that can be taken to find these steps (like symmetry)

R
 * Read

S T
 * Horizontal velocity stays the same in projectile because there is no horizontal force (constant velocity)
 * Vertical velocity is affected by the acceleration of gravity (-9.8)
 * Vertical Velocity = ** y = .5 * g * t(^2) **
 * Horizontal Displacement = ** x = vix * t **
 * Vertical Displacement: = ** y = viy * t + .5 * g * t^2 **
 * Horizontal velocity is constant
 * Vertical velocity and displacement is the same way you solve a problem in free fall (Using big 5 kinematic)
 * There is symmetry at opposite positions in parabola like free fall problems



Conceptual Practice HW
10/20/11





Projectile Motion Class Notes
10/21/11



Horizontal Launch Problems Class Notes
10/21/11


 * Procedure:**



** Launcher Class Activity **
Nicole, Noah, Hella 10/24

Procedure: media type="file" key="Movie on 2011-10-25 at 10.53.mov" width="300" height="300" media type="file" key="Movie on 2011-10-25 at 10.57.mov" width="300" height="300" After Ball hits paper (We found average of these distances)


 * Initial Velocity = 7.10 m/s **

Part II Ball in Cup




 * Percent Error **

Theoretical - experimental/ theoretical x 100

(2.91-2.65)/2.91 x 100 = 8.93%

Our percent error is relatively good, it is less than 10% and on an activity like this that is good. There were several problems with the launchers, they were not precise, meaning they went different distances every time. We could have eliminated some of this inprecieness by being more acurate on which way we pulled the string, and making sure the ball was all the way at the back of the shooter. Also, our ball only went into the cup once because we ran out of time. For a more accurate experimental value we should have run the test more time and averaged out the distances if they were different.
 * Conclusion **

** Ground-to-Ground Launch Class Notes **
10/24/11





** Off-A-Cliff Problems Class Notes **
10/26/11



Guard-o-Rama Project
Nicole Tomasofsky & Maddie Marguiles 10/31/11



Materials:
 * Plastic wheels (taken off a toy monster car)
 * Cardboard
 * -Duct Tape
 * -Stickers
 * -Pumpkin

Design
 * Cardboard that is reinforced with duct tape so it will not bend with pumpkin on it
 * Light plastic wheels
 * PUmpkin with carved circle smaller and then opposite side larger
 * Gives air a place to enter and then spread out as it goes through
 * Making it more aerodynamic
 * Pumpkin is just duct taped to car so it doesn't fall off

Results


 * Conclusion **
 * Our biggest problem was that our cart moved to the side a little when it went down, it would have gone farther except it ran into the wall. We could have fixed this by using different wheels that were straight and not able to move side to side like ours were. Looking back, the carving of our pumpkin might not have actually done anything to help it move quicker. Other parts of our design were good though, because our cart did not collapse or fall over, because it was balanced well.

Lab: Shoot Your Grade
Nicole Tomasofsky, Hella Tallas, Noah Pardes 10/28-11/8

**Objective**: Use your knowledge about projectiles to launch, using a launcher, a plastic ball through strategically placed rings and finally into a cup, with basically no background knowledge except what angle your launcher is at.


 * Rationale: ** As previously found in our launcher activity, it is possible to get a relativly accurate vertical initial velocity for a launcher. Therefore, after we know the initial velocity, if we set the horizontal distances between the tapes to be about equal distances apart, we can use the equation d=vit + 1/2at^2 and find out the time it takes for the ball to hit the center of each tape in addition to the vertical height needed for the ball to go through each tape hoop.


 * Materials and Methods **

First, we had to determine the initial velocity of the ball leaving our launcher at a 20 degree angle. To do so, we used carbon paper and regular paper, letting the ball hit both sheets with a mark on the regular paper. After five test dots, we found the distance from the launcher to the dots, and averaged out these distances. This average then becomes our horizontal distance. In addition we also measured the vertical distance from the launcher to the ground. With both horizontal and vertical distances we were able to calculate the initial velocity. We then used our knowledge about projectiles to help solve the problem. Knowing this, we began setting up our tape rings, using string to loop them through the ceiling tiles making it possible to move up and down. The distance between this was basically evenly split, with a few exceptions, due to the ceiling tile placement. Then we continued doing calculations, finding vertically, how high or low each ring should be. After trial and error, we moved our tape rings a little until the ball went through the rings. Then, we measured again because we had to move some our tape rings. Although our group never got to it, we would typically place a cup at the end of the range, which would be the exact horizontal distance for the cup to also enter into the cup. After our runs, we calculated percent error based off our theoretical and experimental values.


 * Data & Observations (Initial Velocity) **


 * We had the horizontal distance, from the average dots on carbon paper, and we had the vertical distance, measuring from the launcher to the ground. We used our knowledge of projectiles to solve this. We separated it into an x component and y component, and preset the acceleration of the horizontal component to zero and the vertical component to -9.8m/s/s


 * ** vi = 6.73 m/s **


 * This compares because when we first did the laucnher activity, we found that our initial velocity with the same launcher was 7.10 m/s. This time however we shot it out at angle so it changed, slowing down, because of the force of gravity. However, there was also aspects of percent error in each initial velocity calculation. Therefore, since our answers were close anyway we knew we had a valid initial velocity.


 * Performance **

media type="file" key="physics best trial.m4v" width="300" height="300"


 * (Best Trial: 4 Rings) **




 * Physics Calculations **






 * *(1.27m = distance from launcher to start of cabinets) *needed to be added to all horizontal distances **

**Data**

Our percent error w as relatively small, under 10% which is good. There are a lot of possible reasons for this which will be discussed in the conclusion below.


 * Conclusion **

Although we did not completely fulfill our hypothesis, it is still correct. We only made it through 4 hoops, yet our approach was the proven successful. The only aspect that stopped us from completely the lab through 5 hoops and into the cup was lack of time, due to the fact that we had to put up each tape hoop ourselves and change and remeasure the vertical height every time we can into class because they would be moved. Therefore, our approach was not wrong, we did not complete the lab completely because of time constraints.

We were successfuly able to find the vertical heights of tape rings through our knowledge of projectiles. We had a small percent error, all under 10%, which shows that our theoretical calculations were correct. The reason we had percent error and our experimental calculations were different from our theoretical was due to different reasons. First, the launcher was inconsistent. The launcher worked because of springs being pulled which means if the springs were hot or cold, if the person pulled the stinger harder or softer, or if the person did not pull the string precisely straight up, the launch would be altered. The best way to deal with this problem, would be to ensure we were being as consistent as possible with the launcher, and using multiple trials before changing the tape at all. Another issue would be the angle of our launcher, when the ball was launched it sometimes would be changed up to a 2 degree angle change. This again is a problem with consistency of the launcher that could not really be fixed, but by making sure it was clamped down as hard it was possible and try to be as humanly precise as possible. Finally, we ran into issues with our tape rings falling when other groups moved their tape which caused our tape rings to change. We could have definitely used a better system to attach our tape rings to the ceiling. If we had multiple clips we could have rigged a pulley system instead of just looping the string through. This would have made our tape basically stay in the same place and not move which would have lowered our percent error.

Projectile motion as discussed through this lab is applicable to many sports. For example, golf or basketball. When hitting a golf ball you have to hit it at an angle to get into a hole basically similar to how we shot a ball to get it into a cup. To get a hole in one you have to have the right angle and the right initial velocity to get it into the exact horizontal distance (hole). Also in basketball to get the ball into a net you would also need to know what angle and initial velocity to throw the ball at to get it into a hoop that is a certain horizontal and vertical distance away. Both golf balls and basketballs travel in a parabolic trajectory.