CH3_braunsteinc

=lesson 1: A & B= toc All these quantities can by divided into two categories - __ [|vectors and scalars] __. 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. Examples of vector quantities include __ [|displacement] __, __ [|velocity] __, __ [|acceleration] __, and __ [|force] __. 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.Observe that there are several characteristics of this diagram that make it an appropriately drawn 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. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

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. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is __not__ due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions that will be discussed and used in this unit are described 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. 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.

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.



In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). 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 (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 (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram 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 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 direction of a //resultant// vector can often be determined by use of trigonometric 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. 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. Observe that the angle within the triangle is determined to be 26.6 degrees using SOH CAH TOA. This angle is the southward angle of rotation that the vector R makes with respect to West. Yet the direction of the vector as expressed with the CCW (counterclockwise from East) convention is 206.6 degrees.

In the above problems, the magnitude and direction of the sum of two vectors is determined using the Pythagorean theorem and trigonometric methods (SOH CAH TOA). The procedure is restricted to the addition of __two vectors that make right angles to each other__. 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. This method is described below.

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. 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. The __ [|direction] __ of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.

Topic sentence: A vector has magnitude and direction; they can be added using several methods: the pythagorean theorem and the head-tail method.

=lesson 1- C & D= The **resultant** is the vector sum of two or more vectors. 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//. In all such cases, the resultant vector (whether a displacement vector, force vector, velocity vector, etc.) is the result of adding the individual vectors.

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

topic sentence: a resultant is a vector sum of two or more vectors and each part of a vector in two dimensions is considered a component. =lesson 1- E= That is, any vector directed in two dimensions can be thought of as having two components.

 A step-by-step procedure for using the parallelogram method of vector resolution is:
 * the parallelogram method
 * __ [|the trigonometric method] __
 * 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 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. 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.

 There are two methods discussed to solve vectors with two components: the parallelogram method which involves a to-scale sketch and the trigonometric method which involves trigonometry equations.

=lesson 1- G and H= On occasion objects move within a medium that is moving with respect to an observer. The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. 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?

he boat's motor is what carries the boat across the river the **Distance A** ; and so any calculation involving the **Distance A** must involve the speed value labeled as **Speed A** (the boat speed relative to the water). Similarly, it is the current of the river that carries the boat downstream for the **Distance B** ; and so any calculation involving the **Distance B** must involve the speed value labeled as **Speed B** (the river speed). Together, these two parts (or components) add up to give the resulting motion of the boat. That is, the across-the-river component of displacement adds to the downstream displacement to equal the resulting displacement. And likewise, the boat velocity (across the river) adds to the river velocity (down the river) to equal the resulting velocity. And so any calculation of the Distance C or the Average Speed C ("Resultant Velocity") can be performed using the Pythagorean theorem.

Velocity is relative to observer; when solving riverboat problems the different forces must be considered to determine the speed/distance the boat travels across the river.

H: A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. the vector sum of these two components is always equal to the force at a given angle. 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 components are independent of each other.

the two vector components can be thought of as legs of a right triangle and the hypotenuse in the resultant when solving these problems.

=lesson 2-- A & B= questions: what is a projectile? a projectile is an object upon which the only acting force is gravity what types of projectiles are there? anything thrown upwards, dropped from rest, or thrown at a horizontal angle (provided there is no air resistance) what is a horizontal projectile? a projectile thrown at a horizontal angle what other kinds of projectiles are there? things thrown upwards things thrown at angles what types of problems do you do with projectiles? dropping, throwing, projecting things and seeing how long it takes to reach the ground, or things like that

the central idea: a projectile is an object upon which the only acting force is gravity,

=lesson 2-- C= preview: its about projectile velocities questions: what velocities are there? how do they differ? how do they change? central idea: Horizontal and vertical velocities impact a projectile; horizontal velocity is how it moves on the x axis and vertical is how it moves on the y axis. test: horizontal and vertical velocities horizontal is how it moves on the x axis, the range, vertical is how it moves on the y axis. horizontal velocity is always the same, vertical velocity changes due to gravity.

=Ball in cup:= lab partners: Sarah Malley, Danielle Bonnet, Andrew Chung


 * 1) If you were to drop a ball, releasing it from rest, what information would be needed to predict how much time it would take for the ball to hit the floor? What assumptions must you make?
 * 2) to determine the time to the floor, you would need to know from what height the ball is dropped.
 * 3) you would need to assume that there is no air resistance because that would influence the effect of gravity on the ball


 * 1) If the ball in Question 1 is traveling at a known horizontal velocity when it starts to fall, explain how you would calculate how far it will travel before it hits the ground.


 * 1) A single Photogate can be used to accurately measure the time interval for an object to break the beam of the Photogate. If you wanted to know the velocity of the object, what additional information would you need?
 * 2) you will need to know the horizontal range between the object and the photogate


 * 1) Write your procedure and get approval from Mrs. Burns before you proceed any further!
 * 2) What data will you need to collect? Remember that you must run multiple trials. Keep in mind your end goal!
 * 3) we'll need to figure out the vertical distance between the launcher and the floor. we also will need to measure the horizontal distance between the launcher and where the ball strikes the floor. to measure this we used carbon paper and did 5 trials of where the ball would land, we then measured them all and averaged all of them together.


 * 1) How will you analyze your results in terms of precision and/or in terms of accuracy?
 * 2) our measurements should be pretty precise because we ran multiple trials to determine the horizontal distance. to test accuracy we can calculate percent error after we do the lab.

calculations: we calculated our initial velocity at 7.11 m/s we calculated that we should place the cup 3.26 meters away from the launcher we than figured out that we had to adjust our calculations according to the height of the cup, so here are the adjusted calculations: *i was out for this day* percent error between the two calculations:

=Vector Displacement:= graphical: analytical: percent error:

=Shoot your grade:= lab partners: Andrew, Sarah, Danielle materials: launcher, ball, cup, tape, string, carbon paper hypothesis: the height of the rings will go up slightly with the first two, then the last three will be decreasing in height, finally landing in the cup. Objective: With a given angle and speed, launch a ball from the launcher and successfully get it to travel through five rings hanging from the ceiling and then land in a cup on the floor. given angle: 20 degrees

calculations: this is the distance the ball traveled horizontally and the vertical distance from the shooter to the ground. this is the time and initial velocity of the ball this is the calculation of where to place the cup this is where to place the rings percent error for the rings we got it through

results video: media type="file" key="Movie on 2011-11-09 at 09.13.mov" width="300" height="300" we got the ball through four rings by the end of the project. results analysis: we calculated where the cup and fifth ring should be placed, but never had a chance to test them.

conclusion: At the end of the lab we got the ball through 4 rings, meaning that were able to trace the path of a projectile. Several things could have gone wrong in this lab. 1. The math portion of this had to be very exact, so we had to keep checking over our work to make sure it was correct. In the end our results were very accurate. Our lowest percent error was 0%, so we couldn't have been any more accurate than that and our highest was 1.47% which is still not high at all. Another source of error was the way we had to suspend the rings. It was nearly impossible to get the rings to stay where we wanted them to because of the other kids moving ceiling tiles and the other groups using our rings moving them during their class. Another source of error were the shooters themselves. They would shoot the balls differently each time due to the temperature making the spring either stiffer or less stiff. Also, getting them to stay at the angle we needed them to be at was difficult as they kept shifting. If I were to change this lab I would have done it all in one day as opposed to over several periods.

=Gourdorama:= side view:

front view:

calculations: weight: 1.8 kg distance traveled: 8 meters time: 2.92 seconds velocity at bottom of ramp: 5.5 m/s acceleration: -1.89 m/s/s

results: My car traveled down the ramp and left the ramp with a velocity of 5.5 m/s. It then traveled 8 meters in 2.92 seconds. I calculated the acceleration to be -1.89 m/s/s. The total mass of the car was 1.8 kg.

conclusion: If I were to change anything about my project I would make sure that the axels wouldn't move as much because that made the car go on an angle and crash into the wall. I would also make it so that there would be less friction on the wheels because it caused the car to slow down quicker, making it travel less distance. Other than that, there wasn't much I would change.