Ch3_solomonk

=Chapter 3= Lesson 1atoc
 * Summary of Chapter 3, Lesson 1 (a,b) 10/12/11 **

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 are often represented by scaled [|vector diagrams]. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. These diagrams are commonly called as [|free-body diagrams]. Observe that there are several characteristics of a diagram that make it an appropriately drawn vector diagram: 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. The two conventions that will be discussed and used in this unit are described below:
 * 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. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction)
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its " [|tail] " from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east

Lesson 1b 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). // net force // experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object.

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 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. 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.

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. 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.

**Summary of Lesson 1 c,d 10/13/11**

 * Lesson 1c **

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.

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 all such cases, the resultant vector (whether a displacement vector, force vector, velocity vector, etc.) is the result of adding the individual vectors.


 * Lesson 1d **

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. A vector that is directed upward and rightward can be thought of as having two parts - an upward part and a rightward part.

Any vector directed in two dimensions can be thought of as having an influence in two different directions. 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.

**Summary of Chapter 3 Lesson 1e 10/17/11**
Any vector directed in two dimensions can be thought of as having two components.

The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. If one desires to determine the components as directed along the traditional x- and y-coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally. A step-by-step procedure for using the parallelogram method of vector resolution is:


 * 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 trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. As such, 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:


 * 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.

Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

** Summary of Chapter 3 lesson 1g,h 10/18/11 **
Lesson 1g 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. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. The direction of the resulting velocity can be determined using a [|trigonometric function]. 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. These two parts (or components) of the motion occur simultaneously for the same time duration (which was 20 seconds in the above problem). The decision as to which velocity value or distance value to use in the equation must be consistent with the diagram. 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.

Lesson 1h if you pull upon an object in an upward and rightward direction, then you are exerting an influence upon the object in two separate directions - an upward direction and a rightward direction. A component describes the affect of a single vector in a given direction. Any force vector that is exerted at an angle to the horizontal can be considered as having two parts or components. The vector sum of these two components is always equal to the force at the given angle. This is depicted in the diagram below. 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 independent of each other. A change in one component does not affect the other component. 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. 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.

**Summary of Lesson 2a,b 10/19/11**
Lesson 2a Questions: A projectile is an object upon which the only force acting is gravity. It is any object that once projected, or dropped, continues in motion by its own inertia and is influenced only by the downward force of gravity An object dropped from rest, an object that is thrown vertically upward, an object which is thrown upward at an angle A force is not required to keep an object in motion, just to maintain an acceleration. In the case of a projectile that is moving upward, there is a downward force and a downward acceleration - moving upward and slowing down. Gravity is the only force acting upon a projectile. It is the downward force that influences its vertical motion and causes the parabolic trajectory that is characteristic of projectiles. No, the only force needed is gravity.
 * 1) What is a projectile?
 * 1) What are the different types of projectiles?
 * 1) How do forces act on projectiles?
 * 1) How does gravity affect projectiles?
 * 1) Are forces in certain directions needed to keep an object moving?

Central ideas/themes
 * Gravity is the only force acting on projectiles
 * A projectile is an object that continues its motion by its own inertia
 * No forces but gravity act upon projectiles

Lesson 2b Projectiles undergo vertical and horizontal motion which are independent of each other. There is a horizontal and a vertical component If something is projected horizontally in the presence of gravity, then it would maintain the same horizontal motion as before – a constant horizontal velocity. The force of gravity is unable to alter the horizontal motion. There must be a horizontal force to cause a horizontal acceleration. For a horizontally launched projectile, the force of gravity will cause the same vertical motion as before – a downward acceleration. The vertical force acts perpendicular to the horizontal motion and will not affect it. for a non-horizontally launched projectile, the downward force of gravity would act upon the projectile to cause the same vertical motion as before – a downward acceleration. This is a projectile launched at an angle it would travel along a straight line, inertial path. The projectile would travel with a parabolic trajectory.
 * 1) What are the characteristics of a projectiles trajectory?
 * 1) What are the different components of a projectile’s motion?
 * 1) What is the horizontal component of a projectile?
 * 1) What is the vertical component of a projectile?
 * 1) What is a non-horizontally launched projectile?

central ideas/themes
 * gravity has no force on horizontal motion
 * a projectile has multiple components that are independent from each other
 * the vertical velocity of a projectile changes by 9.8m/s/s each second

**Summary of Lesson 2c 10/20/11**
Lesson 2c Horizontal velocity remains constant during the course of the trajectory. Vertical velocity changes by 9.8m/s/s each second. The horizontal displacement is only influenced by the speed at which it moves horizontally and the amount of time it has been moving horizontally. The vertical displacement is dependent only upon the acceleration of gravity and not dependent upon the horizontal velocity. It can be predicted using the same equation used to find the displacement of free-falling objects undergoing 1D motion? The gravity-free path is no longer a horizontal line. In the absence of gravity, a projectile would rise a vertical distance equivalent to the time multiplied by the vertical component of the initial velocity.
 * 1) How do the numerical values of the x- component of velocity change with time?
 * 1) How do the numerical values of the y- component of velocity change with time?
 * 1) How do the numerical values of the x- component of displacement change with time?
 * 1) How do the numerical values of the y- component of displacement change with time?
 * 1) How does the displacement of a projectile launched at an angle to the horizontal change with time?

Key concepts/themes:
 * Horizontal velocity remains constant
 * Vertical velocity is affected by gravity, and so, changes by 9.8m/s/s each second
 * Each component is independent of the other.

**Vector La**b 10/19/11


Discussion of Results: After following the instructions from the other group, we ended up very close to their original ending position. The displacement we measured from the starting to ending position was 26.282m. We then analytically found the value of what it should be, and it was 27.29m, giving us a percent error of 3.84%. Using the graphical method, we calculated that the value was 26.98m, which gave us a percent error of 2.66%. The reason we may have been slightly off is due to the fact that it was difficult to get very precise measurements in such a relatively wide area. The centimeter values, could possibly have been slightly off.

** Ball in Cup Class Activity 10/25/11 **
media type="file" key="ball in cup video.m4v" width="300" height="300"

Discussion of Error and Results: For the first part of this activity, we calculated the initial velocity of the ball to be 7.238m/s. Since our test points were fairly spread out, we found the initial velocity using the average. For the second part of the activity, the data we collected calculated to be a .49% error. After factoring in the height of the cup, we had originally calculated that it needed to be placed 3.055 meters from its launch point. After testing this out, it came out that the actual placement of the cup was 3.04m, which is .015 meters away. At this location, the ball continued to hit the rim or the inside of the cup, making it in, but knocking the cup over in the process. This could have been because the cup was not taped down well enough, or it was very slightly out of place. It could also have been due to the fact that the ball, when launched, landed in a slightly different place most of the time.

Shoot Your Grade Lab 10/26/11
Kaila, Jake, Ali, Jessica Your objective is to launch a ball from the launcher at a given angle and speed setting so that the ball passes consecutively through five rings and lands in a cup on the floor. In this lab, we are trying to launch a ball from a 10degree angle at a speed of 6.92m/s through 5 rings and then land in a cup on the ground. After factoring in air resistance and the force of gravity, 9.8m/s/s, we calculated at what height and range to place each hoop and the cup. After calculating out where to place each ring and the cup, we believe that the ball will successfully travel through all 5 rings and then land in the cup on the floor. The materials we used were a ball, rings to shoot the ball through, the launcher, a cup, and carbon paper.
 * Objective**
 * Purpose and Rationale**
 * Hypothesis**
 * Materials and Methods**

The below calculations show the initial velocity of the launcher at a 10degree angle.
 * Observations and Date from Initial Velocity**

media type="file" key="4 hoops lol.m4v" width="270" height="270" The above video shows our best performance, where the ball went through 4 rings.
 * Observations and Data from Performance**


 * Physics Calculations**

The table below shows where each hoop is calculated to be placed, and a sample calculation is given.



The data table below shows the height we calculated for each ring and the actual height we measured. Next to each is the percent error we found. Below is a sample percent error calculation.
 * Error Analysis**


 * Conclusion**

Our hypothesis was not accurate. The heights we calculated were slightly different than the actual ones we measured after the ball successfully went through a hoop. We were not far off, but were also not fully correct on each hoop. For example, we calculated that the first hoop would be at a height of 1.2742m, which was slightly off, and was actually at 1.275m. Although different, these values are not far off. We calculated the percent error of each location, and they ranged from 0% to 3.4%. A chart of values and a sample calculation is shown above. There were a couple factors that led to experimental error. One very prominent one is that the launcher was not completely consistent throughout. The initial velocities varied slightly, and so we had to choose the average one. Without more consistent launchers, this is a very difficult source of error to correct. Finding the average, and making sure the string was pulled the same way each time and constant checking to make sure the angle was still set where it was supposed to be was the best way to deal with this. Additionally, when actually hanging the rings and measuring it is difficult to get them in the exact place and to be precise. Although we did calculate the range and height for each ring to by hung, and we marked off each spot, we maybe could have stood higher up to make it less strenuous to hang the attach the strings, or could have had more people holding the tape measure steady. Another source of error was that since many groups were setting up strings attached to the rings in a very close area, the strings sometimes moved when another group set theirs up. It was difficult for them to stay in the same place, causing us to have to spend a lot of time rearranging their location. A logical way to solve this if there was more room and supplies would be to have groups set up their rings farther apart, so one group would not be affecting the other when trying to hang their rings. There are many aspects of this lab that can be applied to real-life situations, making it very important to understand what we did. A simple relation is the idea of making sure measurements are as precise as possible, and doing multiple trials. Another application would be to show how objects move, and the paths that they follow. Whether attempting to kick a field goal, or calculating a dive or the speed and position at which to launch something, projectiles apply to multiple things. Having a general idea of what will happen to an object that is launched, and uderstanding why, can be very beneficial even outside of the physics classroom.

Gourdorama Project
Below shows the calculations for the initial velocity of the cart at the bottom of the ramp, as well as the acceleration of it.
 * Calculations**

Unfortunately, my project did not turn out exactly as I had planned. Because the width of the cart and wheels was very close to the width of the ramp, the cart kept hitting the ramp which pushed it off course and caused it to flip as it hit the ground. In order to solve this problem, I would make the dimensions smaller. In doing so, the cart would continue on its intended path, and the wheels would be going straight when they reached the ground, helping to avoid the cart flipping.
 * How to Make it Better**