Ch3_SolimanoM

=toc Lesson 1 a,b (Summary 1)= A variety of quantities are used to describe the physical world. Examples include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. 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. A scalar quantity is a quantity that is fully described by its magnitude.

Vector quantities include [|displacement], [|velocity] , [|acceleration] , and [|force].

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. 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, some are directed //northeast// (at a 45 degree angle).
 * The direction of a vector is often expressed as an angle of rotation of the vector about its "__ tail __" from east, west, north, or south.
 * The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "__ tail __" from due East. 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.

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.
 * Representing the Magnitude of a Vector **

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.

One operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). The //net force// experienced by an object was the result (or [|resultant] ) of adding up all the force vectors. Examples:
 * Vector Addition **

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 Pythagorean Theorem **

The direction of a //resultant// vector can often be determined by use of trigonometric functions. Sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle.
 * Using Trigonometry to Determine a Vector's Direction **

The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the ** head-to-tail method ** is employed to determine the vector sum or resultant. A common Physics lab involves a //vector walk//. 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.
 * Use of Scaled Vector Diagrams to Determine a Resultant **
 * 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).

The order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant.

=Lesson 1 c,d Summary (method 1)= The ** resultant ** is the vector sum of two or more vectors. It is //the result// of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R.  ** A + B + C = R ** 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].



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. 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.

= Lesson 1 e Summary (Method 1) = Any vector directed in two dimensions can be thought of as having two components.The process of determining the magnitude of a vector is known as ** vector resolution. **
 * Vector Resolution **

The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. c.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).
 * Parallelogram Method of Vector Resolution **
 * 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).
 * 1) Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled F north.
 * 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.

Trigonometric functions can be used to determine the length of the sides of a right triangle if an angle measure and the length of one side are known. The method of employing trigonometric functions to determine the components of a vector are as follows: c.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).
 * Trigonometric Method of Vector Resolution **
 * 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.
 * 1) Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled F north . A rightward force velocity component might be labeled v x ; etc.
 * 2) 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.
 * 3) Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

In conclusion, a vector directed in two dimensions has two components - that is, an influence in two separate directions. The amount of influence in a given direction can be determined using methods of vector resolution. Two methods of vector resolution have been described here - __ a graphical method __ (parallelogram method) and a __ trigonometric method __.

= Lesson 1 g,h Summary (Method 1) = On occasion objects move within a medium that is moving with respect to an observer. In such instances the magnitude of the velocity of the moving object 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.
 * Velocity and Riverboat Problems **



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. To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity. Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other, the [|Pythagorean theorem] can be used.

** 103.1 km/hr = R **

The direction of the resulting velocity can be determined using a [|trigonometric function]. The tangent function can be used; this is shown below:

** theta = 14.0 degrees ** The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat.
 * Analysis of a Riverboat's Motion **

Motorboat problems such as these are typically accompanied by three separate questions: c.What distance downstream does the boat reach the opposite shore? The first of these three questions was answered above; the resultant velocity of the boat can be determined using the Pythagorean theorem (magnitude) and a trigonometric function (direction). The second and third of these questions can be answered using the [|average speed equation]. ** ave. speed = distance/time ** ** time = distance /(ave. speed) ** If one knew the ** distance C ** in the diagram below, then the ** average speed C ** could be used to calculate the time to reach the opposite shore. Similarly, if one knew the ** distance B ** in the diagram below, then the ** average speed B ** could be used to calculate the time to reach the opposite shore. And finally, if one knew the ** distance A ** in the diagram below, then the ** average speed A ** could be used to calculate the time to reach the opposite shore.
 * 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?

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.

A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. These two parts of the two-dimensional vector are referred to as [|components]. A ** component ** describes the affect of a single vector in a given direction. 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.
 * Independence of Perpendicular Components of Motion **

= Lesson 2 a,b Summary = Part a Questions: 1. What is a projectile? 2. What are examples of projectiles? 3. What is the force on an object thrown upward? 4. What type of trajectory does an object have when shot horizontally? 5. What is required to keep a projectile in motion?

Central idea: A projectile is any object on which gravity is the only force that is acting, whether thrown at rest or upwards.

Answers: 1. A projectile is an object on which only the force of gravity is acting. 2. Projectiles include objects that are dropped from rest, and an object that is thrown vertically upward. 3. Although thrown upward the force of on the object is still downward. 4. The object will have a parabolic trajectory in the downward direction. 5. Horizontal forces are not needed, only the force of gravity.

Part b Questions: 1. What is the motion of a horizontal fired object without gravity. 2. What is the effect of gravity on a horizontal motion. 3. What is the effect of gravity on a projectile fired upward? 4. How would a projectile fired upward move without gravity? 5. How is the overall displacement of an object affected by gravity?

Central idea: The force of gravity is negative and causes the motion of objects to move negatively.

Answers: 1. Without gravity, the object will have no downward force, and will move along a horizontal line. 2. Gravity has a negative effect which cause the motion to be parabolic in the downward direction. 3. Gravity causes the object to move downward from its original direction. 4. The object will move constantly in the direction that it was fired. 5. The overall displacement is the downward displacement from the position that the object would be if there were no gravity.

= Vector Cafeteria Activity = with Sammy Caspert, Nicole Kloorfain, and George Souflis Purpose: To use our understood methods of solving vectors in a real life scenario involving actual vectors. Procedure: Groups followed given measurements of the cafeteria to hopefully reach a final destination, with a certain displacement from the start point. We then calculated the error from our actual displacement and the given displacement value. Objective: Use another group's readings to follow a given path, to reach a point with a distinct displacement from the start point. Hypothesis: If we follow the given measurements, our distance from the start position will be equal to our theoretical displacement.

Clearer view of Measurement:

Conclusion: This error was rather large, and probably resulted from our group's execution of the coordinates. By not executing the coordinates correctly, our calculation data was off, and thus would lead to a large amount of error from the theoretical result. If we were to try this lab again, we could have tried to be more precise in our execution of the steps. Also, when we measured the vectors' displacement, we could have used a different means of measuring, rather than holding the tape measure in the air to measure the vector distance. By holding the tape up, its distance could have fluctuated when we measured the vector distance, and could have adversely affected our results.

= Lesson 2 Part c Summary (Method 3) =

Part a Questions:

1. How are vector diagrams used to express the motion of a projectile?

2. How do horizontal and vertical components change in a projectile?

3. How do these components change if the projectile is fired upwards?

4. If thrown upwards, what is the relationship of velocity.

5. What is the direction of a velocity vector?

Central idea: Vector diagrams can describe the motion of objects. The horizontal velocity stays the same and the vertical component may change.

Answers:

1. Vector diagrams may be drawn at the points on which a projectile falls, and their magnitudes represent the velocity.

2. The horizontal component stays the same while the vertical component changes by -9.8 m/s squared.

3. The horizontal component will still be the same, while the vertical component will still be acted on by gravity. the resulting graph will be

4. The velocities are symmetrical around the vertex.

5. On the way up the velocities are positive, and on the way down they are negative.

Part b Questions:

1. What is the equation of a projectile?

2. What do the components of the projectile equation mean?

3. What is the equation of horizontal velocity.

4. What is the displacement of horizontal and vertical velocities in a projectile fired upwards?

5. What is the equation for vertical displacement?

Central idea: Because horizontal components are constant and vertical components are usually affected negatively, the displacement between them will increase as time progresses.

Answers:

1. The equation for a projectile is y=.5xgxt^2

2. Y is the vertical displacement. G is /9.8, or gravity, and t is time, usually in seconds.

3. The equation is x=vix x t.

4. The displacement is constantly increasing, as the projectile horizontal velocity continues upward, while the vertical component falls quickly.

5. The equation is y = viy x t + .5 x g x t^2.

=Projectile Activity "Ball in cup"= with Sammy Caspert, George Souflis, and Nicole Kloorfain

Purpose: To utilize known physics calculations to estimate where a projectile will fall.

Hypothesis: If our calculations are correct, we will be able to estimate where the ball will finally land.

Procedure: Preliminary tests were fired. Based on these, calculations were generated for where the ball will land. A cup was then placed at this point to test the accuracy of our calculations. Percent Error=10.98%

This percent error was rather large. This is probably because of problems with our launcher. When launching the projectile, the launcher often moved its position, and even sometimes its angle of launch. These would have drastically affected the ball making it into the cup, and would have led to a large amount of error. If we were to do this lab again, we would have used something to keep the launcher in one position, which could have reduced the error in our lab. media type="file" key="Projectile Activity.mov" width="300" height="300" =Gourdo-rama Project=

Purpose: To use physics concepts to create a cart to hold and carry a pumpkin a large distance.

Procedure: Pumpkin projects were rolled down a ramp, and their distance traveled, as well as their masses were measured. The most successful projects were the lightest and the ones that rolled the farthest.

Hypothesis: Our project's large wheels and tennis ball wheels will reduce the shock of the ramp, and the pumpkin holder weight will aid in it rolling a long distance. Mass: 2.0 kg

Distance: 1.0 meters in .35 seconds

Velocity and Acceleration Calculations: Discussion: Our hypothesis was not accurate, as our project did not roll a significant distance. If I were to attempt this lab again, I would have tried new wheels. While our body that held the pumpkin was technically sound, the wheels fell apart when they went down the ramp. If I were to do this again, I would have used proper, metallic wheels, as our hand made "tape roll" wheels, were not fit to move down the ramp. This would have increased our distance by letting our project progress past the ramp. Had the project made it past the ramp, I believe the weight and shape of the pumpkin holder could have led to a significant roll.

=Shoot Your Grade= with Sammy Caspert, George Souflis, Nicole Kloorfain

Purpose: To see if analytical calculations of a projectiles path will be an accurate representation of its actual traveled path.

Hypothesis: Through calculating the height of a projectile at certain intervals, with a given initial velocity. we will be able to have our projectile pass through five hoops at these heights, and into a cup at its final position.

Procedure: After calculations of expected vertical heights, hoops were hung at these heights from the ceiling. The projectile was then fired through the hoops to see if our calculations were accurate.

Calculations for each time interval: Note: Distances above are from the reference point (point of launch)

Percent Error:

Note: Distances in percent error are from the ground Average Percent Error = %5.81

Sample Percent Error: % Error = ((I theoretical - experimental I)/(theoretical)) x 100 % Error = (I 1.359 - 1.28 I)/(1.359) x 100 % Error = %5.81

Video: media type="file" key="Movie on 2011-11-09 at 09.26.mov" width="300" height="300"

Conclusion: Our average percent error was %5.81. This shows that our calculations were accurate, and that our execution of the hanging of rings was good. Many sources of error, which will be discussed below, could have contributed to this percent error though. This also shows that both our vertical and horizontal measurements were accurate, and that our launcher generally had consistent launches.

There were many sources of error in this lab. Included in the errors were the shooting of the projectile, and the hanging of our hoops. When shooting the projectile, it did not always fire exactly the same, and its angles may have changed slightly, and the ball may not have been fired at uniform initial velocities. Also, the strings were difficult to keep hung in exact positions, and movements in these may have led to error. If we were to do this lab again, I would have used a more secure means of hanging the strings, and also would have found a way to keep the launcher from moving horizontally. Also, we should have let the launcher cool down after each use to assure that our launches were uniform.

This lab had many real life implications. Projectiles are seen in many areas of our lives, whether it be sports, throwing a rock off a ledge, or doing forensic analysis of a projectile or bullet, these calculations can all be used. By doing these calculations we can find theoretical heights and velocities at any point in its path, which can be very useful in the previously stated scenarios, especially in forensic analysis of a gunshot. Overall, this lab used calculations that have a wide use in all of our lives, which can be practically applied to many things that we do everyday. The concept of trial and error when hanging the rings also taught us patience and we gained teamwork within our group.