Ch3_Ordover

toc
=Chapter 3=

Lesson 1a: Vectors and Direction
A vector quantity is a quantity that is fully described by both magnitude and direction Examples of vector quantities that have been previously discussed include displacement, velocity , acceleration , and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. Vector quantities are not fully described unless both magnitude and direction are listed. The vector diagram depicts a displacement vector. 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

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 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 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 and etc. This is the most commonly used convention. Assume it is this option if not given direction: Always assume from East if not given.

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow, which is drawn a precise length in accordance with a chosen scale.

Vectors are drawn to show magnitude of the movement of an object. They must include a scale and either an angle according to a Cardinal direction or an angle according to the counter-clockwise measurement from east.

TS: Vectors show both magnitude and direction and can be drawn using scaled vector diagrams.

Lesson 1b: Vector Addition
Two vectors can be added together to determine the result (or resultant). 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. To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Trigonometry 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 **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

Head to Tail 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.

TS: The three methods to add vectors are the pythagorean theorem method, the trigonometry method, and the head to tail method.

Lesson 1c: 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//. But any 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 together. The resultant can be determined by adding the individual forces together using vector addition methods.

TS: A resultant is the sum of two or more vectors.

Lesson 1d: Vector Components
[Vectors] can be thought of as having two parts. Each part of a two-dimensional vector is known as a component.

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.

TS: All two dimensional vectors have two different components.

Lesson 1e: Vector Resolution
The process of determining the magnitude of a vector is known as **vector resolution**. The two methods of vector resolution:
 * The parallelogram method
 * The trigonometric method

__Parallelogram 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.

__Trigonometry Method__
 * 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.

TS: There are two different methods for vector resolution, the parallelogram and the trigonometry methods.

Lesson 1g: 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. 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. 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.

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 Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a **side wind** of 25 km/hr, West. The resulting velocity of the plane is the vector sum of the two individual velocities.

The relative velocity in a motorboat problem can be found the same way as in a plane problem. 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 second and third of these questions can be answered using the average speed equation.

Lesson 1h: Independence of Perpendicular Components of Motion
A diagonal vector has two parts- an upward direction and a rightward direction. These two parts of the two-dimensional vector are components, which describe the affect of a single vector in a given direction. The vector sum of these two components is always equal to the force at the given angle.

Any vector 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. A change in the horizontal component does not affect the vertical component. This is what is meant by the phrase "perpendicular components of vectors are independent of each other." 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.

TS: Perpendicular components are independent of each other and therefore do not affect each other.

Lesson 2a: What is a Projectile?
1) What is a projectile? Any object that is only acted upon by the force of gravity. 2) What are some properties of projectiles? Projectiles are only acted up on by gravity. They move in a parabolic path. They move consistently in the horizontal direction. 3) What are the different types of projectiles? An object that is dropped from rest, an object that is thrown upwards, and an object that is thrown upwards with an angle to the horizontal are all different types of projectiles. 4) What is the difference between a projectile and a free falling object? A free falling object is a type of projectile, but not all projectiles are free falling objects because free falling objects only have motion in one direction and and projectiles can have motion in two directions. 5) How does inertia apply to projectiles? Inertia says that things have a tendency to continue their state of motion or rest. This applies to projectiles as they have a constant horizontal speed.

Main idea: Projectiles are objects in which only the force of gravity is acting upon them.

Lesson 2b: Characteristics of a Projectile's Trajectory
1) What affects horizontal motion of a projectile? There is not affect on the horizontal motion of a projectile. Projectiles move horizontally with a constant speed. 2) What affects vertical motion of a projectile? Gravity is the only thing that affects the vertical motion of a projectile. It affects it with an acceleration of -9.8 m/s^2. 3) Do the horizontal and vertical motion affect each other? No, horizontal and vertical motion are independent of each other. 4) Why is the path of a projectile parabolic? It is parabolic because gravity accelerates the projectile downward as opposed to the straight line trajectory it would follow if their was no gravity. 5) What is the only thing that affects the motion of a projectile? Projectiles are only acted upon by the force of gravity.

Main idea: Each component is independent of each other. The horizontal component is not affected by anything. The vertical component is only affected by gravity.

Lesson 2c: Describing Projectiles with Numbers (Horizontal and Vertical Velocity)
1) How does the velocity of the x and y components change over time? The velocity of the x component stays constant throughout. The y component slows down as it approaches its peak. It becomes 0 at the peak and then goes negative. As it goes down is becomes increasingly negative. 2) What equations are used for the x and y displacement? y=1/2gt^2 and x=vit 3) Does the force of gravity affect the horizontal motion? No gravity only affects the vertical motion. 4) How is horizontal and vertical displacement different? The horizontal speed is constant so the displacement is just velocity times time. However, the vertical motion has acceleration due to gravity and therefore speeds up as it goes. This causes the vertical displacement to change 9.8 meters more every second. 5) What is a vector diagram for? It shows the different velocities at certain times of a projectile.

Main Idea: The main idea is how to find the horizontal and vertical velocity and displacement of a projectile.

Activity: Orienteering
Part One: Actual Resultant: 26.40 m
 * Leg A: || North || 8.24 m ||
 * Leg B: || West || 9.26 m ||
 * Leg C: || North || 4.28 m ||
 * Leg D: || West || 10.17 m ||
 * Leg E: || North || 5.20 m ||



Part Two:

Actual Displacement: 67.16 m
 * Leg A: || North || 9.22 m ||
 * Leg B: || East || 24.52 m ||
 * Leg C: || North || 18.31 m ||
 * Leg D: || East || 24.23 m ||
 * Leg E: || North || 18.33 m ||





Activity: Ball in Cup
Partners: Robert Kwark, Ryan Luo

media type="file" key="ball in cup.mov" width="300" height="300"

Data:

Part 1) What is the initial velocity at medium speed? The initial velocity was 7.07 m/s.

Part 2) Change the initial height of the launcher and calculate where to put a cup on the floor so the projectile goes in the cup. The cup should be 2.694 m away from the launch point.

Calculate the percent error: The percent error is 0% because the theoretical distance, 2.694m, was actually the exact position where the cup needed to be for the ball to land in the cup. Despite this, the ball did not go in every time because the launcher is inconsistent. Sometimes is launched it to far or too short, but we still had a fairly high success rate of about 60% or 6 out of 10 shots.

Activity: Gordorama
Picture: Materials: Cardboard Box, Roller Skate Wheels, Metal Rod, Duct Tape, Washers, (Dont know the name of the piece but its metal and it holds the wheels on the rod)

Design: We used two metal rods for the axels. We put the washers on the axels first so the wheels wouldn't create friction with the box. Then we put the wheels on. Next, we put the box on top of the axels and moved the wheels so they were in the right place. We then put stoppers on the end so the wheels wouldn't move. Finally, we duct taped the cardboard box to the axels.

Calculations for Initial Velocity and Acceleration: Distance: 18.4 m Time: 7.58s Initial Velocity: 4.85 m/s Acceleration: -.604 m/s^2

Conclusion: Our project did very well, except there were two things that hindered its performance. We merely eyeballed the axels when we taped them to the box, so they were not exactly straight. This caused the vehicle to go slightly to the left which caused it to hit the wall. Next time, we could measure the axels and make sure they are straight. The second thing is that our body was a cardboard box and it was not very sturdy. It broke after the first run, so we could only have one run. Next time, we could use a sturdier material for our car.

Lab: Shoot Your Grade
Lab Partners: Ryan Luo, Robert Kwark

11/4/11

Purpose: Our purpose was to calculate the trajectory of a launcher with a set velocity and angle. Our goal was to set up rings and a cup so the projectile goes through all the rings and lands in the cup.

Hypothesis: My hypothesis is that the projectile's trajectory will be a parabolic shape and that we will be able to find the height and time of the projectile at certain distances away from the projectile. I think we will be able to do this and the projectile will go through all our rings and into the cup.

Materials Used: Tape, Measuring Tape, Yard Stick, Paper, Carbon Paper, String, Launchers, Projectiles (green balls), Binder Clips, Duct Tape, Black Tube (for loading the launcher), Clamps

Method: First we clamped the launcher down using the clamps and set the angle to 15º. We proceeded to shoot the projectile at medium speed. We taped the carbon paper in the area where the ball landed. Then we shot it 6 more times and measured how far each shot was from the launcher. We took the average of this and used it to calculate the initial velocity. We then measured the the height of the cup and the height of the launcher and calculated how far the cup needed to be placed from the launcher. Next, we measured the distances of the tape rings that were already there from the other class. We used these distances to calculate the height each ring needed to be. Then, we adjusted the rings so they were at the heights that we calculated. Finally, we placed the cup in its position and started shooting our projectile. After about 5 tries we got the ball through all the rings and into the cup.

Horizontal Range:

Vertical Height of Launcher: 1.172 m

Initial Velocity Calculations:

Cup Distance Calculations:

Ring Calculation Example:

All Ring Calculations:

Percent Error Example Calculation:

All Percent Error Calculations:

Performance: Our ball went through all the rings and into the cup.

Video: media type="file" key="Shoot Your Grade.mov" width="300" height="300"

Conclusion: My hypothesis was correct as we were able to find the height and time of a projectile at certain distances. Due to this we were able to set up rings and a cup and the projectile went through all five rings and into the cup. In addition, the ball seemed to have a parabolic trajectory, but we did not calculate it to be sure. We got a 0% percent error because the ball went through the rings at our theoretical spots and we did not have to adjust the rings at all. There were however two problems that affected the consistency. One was that we had to make sure the rings and cup were all lined up so it did not hit the sides. This was also made more difficult by the fact that the launcher moved slightly during every shot. We fixed this by using two clamps, taping it down, and holding it down with our hands. The second problem was that the launcher was not consistent. Sometimes it would shoot it a little farther and sometimes it would shoot it a little shorter. Unfortunately there is not way to fix this. A real life application of this would be shooting a cannon through rings and trying to hit a target on the ground. This would be extremely similar to our experiment and it would probably have similar problems to our experiment.