Ch3_BehrensB

Chapter 3 - Vectors toc

= Vectors: Motion and Forces in Two Dimensions- Lesson 1; a & b = October 12, 2011

**Vectors and Direction** Examples of vector quantities include displacement, velocity , acceleration , and force. Each of these qualities demand that both a magnitude and a direction are listed.

Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. **Conventions for Describing Directions of Vectors**
 * 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.
 * 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.

**Representing the Magnitude of a Vector**



**Vector Addition** Two vectors can be added together to determine the result (or resultant). **The Pythagorean Theorem** The Pythagorean theorem is a useful method for determining the result of adding two vectors that make a right angle to each other.

**Using Trigonometry to Determine a Vector's Direction**
 * SOH CAH TOA



**Use of Scaled Vector Diagrams to Determine a Resultant** **Head-to-tail method**: Where the head of this first vector ends, the tail of the second vector begins. 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. 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.
 * Ex:

= Vectors: Motion and Forces in Two Dimensions - Lesson 1; c & d = October 13, 2011

**Resultants** The **resultant** is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R. 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//.

**Vector Components** 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.

= Vectors: Motion and Forces in Two Dimensions- Lesson 1; e = October 17, 2011

** Vector Resolution ** The process of determining the magnitude of a vector.


 * 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.



** Trigonometric Method of Vector Resolution ** The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector.

The method is illustrated below for determining the components of the force acting upon Fido. As the 60-Newton tension force acts upward and rightward on Fido at an angle of 40 degrees, the components of this force can be determined using trigonometric functions.



= **Orienteering Activity** = October 18, 2011 Lab Partners: Lerna Girgin & Molly Lambert

Classwork:

Analysis: The experimental measurement of the displacement for this activity resulted as 26.5 m, and our analytical and graphical calculations were relatively close to that numerical value. Analytically, the resultant was 26.3 m with a percent error of .76%, indicating that our measurements were off by .2 meters. Graphically, the displacement was 26 m exactly with a percent error of 1.92%. In comparison to the experimental measurement, the result of the analytical method was more accurate than that of the graphical method as shown by the percent errors.

=Vectors: Motion and Forces in Two Dimensions - Lesson 1; g & h = October 18, 2011

**Relative Velocity and Riverboat Problems** On occasion objects move within a medium that is moving with respect to an observer. The resultant velocity is the velocity relative to an observer on the ground.
 * Ex:

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. (100 km/hr) 2 + (25 km/hr) 2 = R 2 10,000 km 2 /hr 2 + 625 km 2 /hr 2 = R 2  10,625 km 2 /hr 2 = R 2  <span style="color: #7400cc; display: block; font-family: Arial,Helvetica,sans-serif; text-align: center;">SQRT(10,625 km 2 /hr 2 ) = R <span style="color: #7400cc; display: block; font-family: Arial,Helvetica,sans-serif; text-align: center;">**103.1 km/hr = R** <span style="color: #c300ff; font-family: Arial,Helvetica,sans-serif;">The angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions.

<span style="color: #7400cc; font-family: Arial,Helvetica,sans-serif;">tan (theta) = (opposite/adjacent) <span style="color: #7400cc; display: block; font-family: Arial,Helvetica,sans-serif; text-align: center;">tan (theta) = (25/100)  <span style="color: #7400cc; font-family: Arial,Helvetica,sans-serif;">theta = tan -1 (25/100)  <span style="color: #7400cc; display: block; font-family: Arial,Helvetica,sans-serif; text-align: center;">**theta = 14.0 degrees** <span style="color: #c300ff; font-family: Arial,Helvetica,sans-serif;">If the resultant velocity of the plane makes a 14.0 degree angle with the southward direction (theta in the above diagram), then the direction of the resultant is 256 degrees. Like any vector, the resultant's direction is measured as a counterclockwise angle of rotation from due East.

<span style="color: #7400cc; font-family: Arial,Helvetica,sans-serif; font-size: 90%;">**Independence of Perpendicular Components of Motion** <span style="color: #c300ff; font-family: Arial,Helvetica,sans-serif;">A force vector that is directed upward and rightward has two parts, referred to as **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.

<span style="color: #c300ff; font-family: Arial,Helvetica,sans-serif;">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.

= Vectors: Motion and Forces in Two Dimensions - Lesson 2; a & b = October 19, 2011

Preview;
 * Projectiles- objects moving solely with the influence of gravity, different types of projectiles
 * Projectile motion & inertia
 * Characteristics of a projectile's trajectory
 * Vertical & horizontal motion

Questions; //A dropped object and an object thrown upward, either vertically or horizontally, are examples of types of projectiles.// //Gravity has the same effect on each type of projectile because by definition, it is an object that moves only under the force of gravity. Therefore, regardless of whether the object is thrown upward or dropped, both types include an object moving at 9.8 m/s 2 .// //This law states that an object, if not acted upon by an unbalanced force such as gravity, would move in an unwavering horizontal path.// //Because inertia, in regard to a projectile's path of motion, is rather impossible, it does not really have a realistic relationship. Considering the two types of existing projectiles, those are the only possible situations in which an object would move without the influence of an outside force other than gravity. However, gravity is inevitable, thus deeming the law of inertia unfeasible.// //A parabolic trajectory is the path of a projectile in which acceleration due to gravity is present and velocity is changing. It is a realistic, vertically downward, gravity-influenced path in comparison to an inertial trajectory, which is basically a gravity-free horizontal line of motion.// //There is only an absence of gravity in Newton's law of inertia, and does not actually occur in life situations.// //Horizontal motion is the motion of an object that is unaffected by gravity and acceleration, and has a constant velocity. Vertical motion is the opposite, possessing a path influenced by the value of 9.8 m/s 2 .// // Horizontal, as previously mentioned, has no acceleration because there is no horizontal force to provide a horizontal acceleration. Vertical motion travels in a parabolic trajectory as a result of the acceleration due to gravity exerted upon it. //
 * What types of projectiles are in existence?
 * What effect does the force of gravity on each individual type?
 * What is Newton's law of inertia?
 * What relationship does inertia have with the motion of a projectile?
 * What is the difference between a parabolic trajectory and an inertial trajectory?
 * When is there an absence of gravity; why does this occur?
 * What is the difference between horizontal and vertical motion?
 * How does acceleration affect both horizontal and vertical motion?

State - the Main Idea;
 * A projectile is an object moving solely under the force of gravity, whose motion can be divided into two separate components - horizontal and vertical; the characteristics of force, acceleration, and velocity are affected differently in each component.

Test; (see question answers)

= Vectors: Motion and Forces in Two Dimensions - Lesson 2; c = October 20, 2011

Preview;
 * Horizontal and Vertical components of velocity
 * Numerical values for horizontal and vertical velocity
 * Horizontal and Vertical components of displacement
 * Finding vertical or horizontal displacement through mathematical equations

Questions; //Because horizontal and vertical components independent of each other, the values of velocity are as well.// //As a continuation of the answer to the previous question, the horizontal velocity stays constant while the vertical velocity changes at a rate of -9.8 m/s 2 .// //It is possible to find the vertical displacement of a projectile through the use of the equation y = v i t + 1 / 2 at 2, where y stands for the vertical displacement.// //The equation for horizontal displacement of a projectile is presented in a relatively different form than that for vertical displacement; x = v ix (t), where// //v ix is equivalent to the horizontal speed of the object and x stands for the horizontal displacement.// //An initial vertical component affects the inertial path on which the object would travel in a gravity-free situation, because it would no longer travel on a horizontal line. This changes the equation to y = viyt + 0.5at2//, where //viyt is equal to the initial velocity of the vertical component, multiplied by the time.//
 * What relationship is there between horizontal and vertical values of velocity?
 * Which component's velocity stays constant; which changes, and at what rate does it do so?
 * How is it possible to find the vertical displacement of a projectile?
 * What equation can be used to find the horizontal displacement of a projectile?
 * How does an initial vertical component affect the equation for vertical displacement, and vertical displacement itself?

State - the Main Idea;
 * Horizontal and vertical components are independent of each other, thus resulting in constant horizontal velocity and changing vertical velocity at the rate of acceleration of gravity. Horizontal and vertical displacements can also be described numerically and found through the generic displacement equations.

Test; (see question answers)

= Ball in Cup Activity; Part I = October 25, 2011 Lab Partners: Molly Lambert & Lerna Girgin

Classwork:

Percent Error: 3.05% The percent error calculated and shown above is based on the placement of the cup. During the process of calculating the cup's distance, we found that "d" = 2.62 meters, but when we tested this, the ball continually landed at 2.7 meters.

Video: media type="file" key="Movie on 2011-10-25 at 12.58.mov" width="330" height="330"

= "Shoot Your Grade" Lab = November 5, 2011 Lab Partners: Molly Lambert & Lerna Girgin

Purpose and Rationale: The purpose of this lab was to determine the coordinates of each hoop, through which the ball would theoretically be shot successfully and land in the cup. The angle of the launcher was 20 degrees, which during the process of verifying the initial velocity, allowed us to simplify each component to v i x = .94v i and v i y = .34v i. Because the ball was then going to be launched at an angle, we had already known that the velocity would no longer be the same, for either component. We planned our procedure by calculating the initial velocity for the x and y components, measuring the height of the counter top, and then determining the coordinates at which we would hang each hoop by dividing the entire horizontal distance into five segments and calculating for the vertical height.

Hypothesis: The ball will be launched through all five hoops and land in the cup successfully, traveling on a parabolic trajectory.

Materials and Methods: The materials that we used in order to obtain our results and complete the lab include a launcher set to 20 degrees at medium range, a ball, five tape rings, black string with which we hung the hoops, a tape measurer, right-angle clamps, and a plumb bob. We constantly altered the methods that we used to successfully "shoot our grade," originally basing horizontal and vertical placements on calculations. However, this proved to be inaccurate because of the previous group's positions, which caused us to proceed on a trial-and-error basis. The inconsistency of the launcher along with the surrounding effects of the classroom, including the air vent, provided evidence that we could not use our calculations. Ultimately, the procedure of trial-and-error proved to be, at most, eighty-percent successful after editing the height of each ring multiple times.

Observations and Data from Initial Velocity: We were able to calculate the initial velocity of the launcher by finding an expression for "v i t", inserting the value of "v i t" into an equation to find the time, and then replacing "t" in the expression with 0.78 s. The initial velocity was 6.67 m/s.

Observations and Data from Performance: media type="file" key="shoot your grade clip.mov" width="330" height="330"

Physics Calculations: Data Table of Theoretical Measurements: The theoretical measurements above show our calculated horizontal positions, found by dividing the entire distance into five equal segments. The vertical positions were calculated by using the equation d = v i t + 1 / 2 at 2, with which we solved for time and inserted that value back into the equation for the y-position. The time for each segment is based on our original calculations, that each x-position would be equally distanced. These measurements were found in order to help us during the procedure of hanging the rings.

Error Analysis: Actual Measurements (from Performance Day) Because we were unable to hang our rings before the other groups, the actual measurements shown in the table above are results of their calculations. We measured these positions on performance day after working with the launcher and editing the vertical position of each hoop.

Percent Errors for x & y Positions:

Conclusion: <span style="color: #ce84f6; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Our hypothesis was not entirely correct because by the end of performance day, we had not been able to successfully launch the ball through all five hoops and into the cup. As previously explained, the inconsistency of the launcher contributed to lengthening the time it took to find accurate positions; this did not give us enough time to work on and finish the placement of the fifth hoop and the cup. As a result, our hypothesis was partially, but not fully, correct because the positions were not based on our own work, though we were able to launch the ball through four of five hoops. <span style="color: #ce84f6; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Because our calculations differed from the other groups’ work, our horizontal and vertical points did not overlap, with percent errors for both x and y coordinates. For example, the first hoop had an x-percent error of 5.13% and a y-percent error of 2.37%. The largest x-percent error was for the fifth hoop, which was 28.05%, and the smallest x-percent was for the third hoop, which was 0.81%. The largest y-percent error was for the fifth hoop, which was 64.56%, and the smallest y-percent error was for the first hoop, which was 2.37%. These values are understandable because we only spent a fraction of the time editing the fifth hoop’s measurements in comparison to the time spent on the first and second hoops. The errors occurred during the procedure of the lab, and can be attributed to the launcher’s inability to consistently shoot the ball in the same parabolic trajectory each trial, as mentioned above. Also, the air vent effected the hoops’ positioning by shaking the strings that held them and possibly altering the height. In addition, the estimation of the ball’s trajectory in order to find the y-positions of the hoops in correspondence with the new x-positions was not as accurate as our calculations. We had to estimate the y-position of each hoop because of the fact that we were beginning with new x-positions, rather than those we originally calculated. <span style="color: #ce84f6; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">To address these errors, it would be more practical for each group to do the lab separately, so that every group would be able to use the launchers in full possession rather than share them and compromise individual calculations. However, because this alteration is improbable, we would edit this lab in other ways to be more successful. First we would relocate our launcher to minimize the effects of the surrounding environment. We would originally measure the horizontal distance more than once to verify each position. The launcher’s inconsistency cannot be completely fixed, though we would double-check the angle and the settings of it before each launch. <span style="color: #ce84f6; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">This is an important concept to understand in case of dire situations, such as rescuing or sending supplies to stranded climbers or hikers. Perhaps the only option for sending supplies was to launch them from a cliff. This lab demonstrates how to find the positions at which we should hang the hoops and place the cup to meet the purpose of the lab.

=<span style="color: #9d00ff; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">"Gourd-o-rama" Project = <span style="color: #9d00ff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">November 1, 2011

<span style="color: #9d00ff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"> Results: Our project, with a mass of 2 kg, travelled 13.75 meters and had a time of 6.59 seconds.

Calculations:

Over all, our project was successful and provided good results. If one aspect of the cart could be changed, we could have made it a little lighter in weight by replacing some of the heavier materials, like wood, with a thinner material. Also, we could have decorated the cart a little more and added decorations to coincide with the theme of Halloween.