Ch3_WeissB

=__Chapter 3__= toc

Summarization (Version 1) of Vectors Lesson 1a-b
Date: 10/12/11 A vector quantity is a quantity that is fully described by both magnitude and direction. Understanding the fundamentals of this kind of value is key to understanding motion and forces that occur in two dimensions. Vector quantities demand both magnitude and direction. Vector quantities are often represented by scaled vector diagrams (also called free-body diagrams). These diagrams show a scale and a vector arrow showing a specified direction, so that magnitude and direction are clearly labeled. When a vector is a certain amount of degrees due in a direction, the vector is rotated counter-clockwise along its tail that same amount of degrees. The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow and is determined by a chosen scale. Two or more vectors can be added to find the total displacement if the displacement is only in one direction (up-down or left-right). If the displacement involves multiple directions, one can use the Pythagorean theorem to find length and trigonometry to find the angle (and therefore direction) **OR** the head-to-tail method, through which one draws out the change in a vector quantity (add up the total change to find displacement and find angle by using a protractor).

Summarization (Version 1) of Vectors Lesson 1e
Date: 10/17/11 Any vector directed in two dimensions can be thought of as having two components. There are two methods of determining the magnitude of a vector (or vector resolution): the parallelogram and trigonometric methods. The parallelogram method requires using a accurately drawn, scaled vector diagram and sketching a parallelogram around the vector (so that it is the diagonal) to determine the components of the vector (and their magnitudes). Apply the scale to the drawn sides of the parallelogram to determine the components. The trigonometric method requires using trig functions and the Pythagorean theorem to determine the components and their magnitudes.

Summarization (Version 1) of Vectors Lessons 1f-g
Date: 10/18/11 On occasion objects move within a medium that is moving with respect to an observer. Such examples include an airplane encountering wind and a motorboat moving against a river current. 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. The observed speed of the boat must always be described relative to who the observer is. Problems involving these scenarios involve vector addition in order to find the resultant velocity. Motorboat problems can also involve finding how long it takes for the motorboat to cover a distance (by using the “speed=distance/time” equation) and the distance travelled downstream (by plugging the calculated time into the above equation for the distance downstream). If a force is exerted on an object in two separate directions, then the forces are exerting an influence upon the object in two separate directions. These two parts of the two-dimensional vector are referred to as components, which are the effects of a single vector in a given direction. Any vector directed at an angle can be thought of as being composed of two perpendicular components, which can be considered parts of a right triangle. The two perpendicular parts or components of a vector are independent of each other.

Summarization (Version 3) of Vectors Lesson 2a-b
Date: 10/19/11 Main Idea: A projectile is an object that moves by its own inertia and is acted upon only by the force of gravity.

Questions: What is inertia? The resistance any object has to change in its motion.

How does a projectile move horizontally and vertically? A projectile moves at a constant horizontal rate, while it has a downward acceleration that eventually makes the object hit the ground.

How does gravity affect a projectile? Gravity affects the motion of a projectile by providing downward acceleration. In short, the object has parabolic motion and will eventually completely drop due to gravity.

What would happen to the motion of a projectile without gravity? A projectile would move in a straight line at constant speed.

What would be the result of a force affecting a projectile's horizontal motion? The object would no longer be a projectile.

Summarization (Version 3) of Vectors Lesson 2c
Date: 10/20/11

Main Idea: As a projectile moves, the horizontal component of velocity stays the same, while the vertical component of velocity decreases. At the same time, the horizontal component of displacement increases, while the vertical component of displacement decreases.

Questions: What would happen to the vertical component of velocity if there was no gravity? The vertical component of velocity would not change, instead of decreasing.

What would happen to the vertical component of velocity if the projectile started moving at an angle? The vertical component of velocity would still decrease over time (however, it would start with positive values instead of negative values).

Why exactly does the horizontal component of velocity never change? Because there are no horizontal forces acting on a projectile.

What would happen to the vertical component of displacement of velocity if there was no gravity? The vertical component of displacement would not change at all, instead of decreasing.

What would happen to the vertical component of displacement if the projectile started moving at an angle? The vertical component would still decrease over time (however, it would start with positive values before reaching negative values).

Velocity Addition Lab
Date: 10/19/11 Partners: Ryan Hall, Amanda Fava, Julia Sellman Purpose: To take another group's charted displacement and use it to determine whether or not the analytical and graphical methods of vector addition are practical in determining real-life displacement.

Starting at the water fountain in the cafeteria, my group displaced our position in the following manner: After we made our final displacement, we calculated our total displacement as being 21.78 with a measuring tape.

To determine if vector addition is practical in determining real-life situations like this, we performed the two methods of vector addition to see if we got the same (or at least an extremely close) displacement.

Our calculation of displacement through the analytical method is shown below: Based on our calculations, the total displacement is 22.02 m at about 157º. Using this calculation and our previous measurement, we then calculated the percent error of the result from the analytical method.

We found the percent error to be 1.1%. This means that the use of the analytical method is extremely practical in determining real-life displacement. We then attempted to do the same with the graphical method. Our calculation of displacement with the graphical method is below: According to the graphical method, the total displacement is 22.01 m @ 157º. The calculation of percent error of this calculation is below.

The percent error here is 1.06%. This means that the graphical method is a practical method of determining real-life displacement.

In conclusion, both the analytical and graphical methods are good ways of finding real-life displacement, as evidenced by the small percents of error obtained from each in this activity.

Ball in a Cup Lab
Date: 10/24/11 Partners: Ryan Hall, Amanda Fava, Julia Sellman Purpose: To calculate the different facets of the projectile motion of a ball and use them to determine the distance at which a cup should be placed so that the ball will land in the cup when shot.

__Part 1__ Calculations: During Part 1 of this lab, my group attempted to calculate the horizontal velocity of a ball shot out of a ball shooter. To do this, we first calculated the height the ball shooter was above the ground by using a tape measure. We found that the total height was .8464 m. We then shot the ball out a total of seven times onto a piece of carbon paper to determine the horizontal distance the ball traveled. The imprint that the point of impact on the carbon paper left on a different piece of paper underneath allowed us to measure these distances. The distances were:
 * = Trial1 ||= 1.9462 m ||
 * = Trial 2 ||= 1.9367 m ||
 * = Trial 3 ||= 1.9545 m ||
 * = Trial 4 ||= 1.9667 m ||
 * = Trial 5 ||= 1.9632 m ||
 * = Trial 6 ||= 1.9788 m ||
 * = Trial 7 ||= 1.9112 m ||

Using these calculations, we calculated the average distance As shown above, the average distance was calculated as being 1.9539 m. We used this information, along with other information we already knew (acceleration due to gravity is -9.8 m/s^2, the height of the shooter, etc.) to find the initial velocity of the ball.

The initial velocity was calculated as being 4.70 m/s.

__Part 2__ For Part 2 of this lab, my group was supposed to determine the distance at which a cup would have to be positioned from the ball shooter so that the ball would consistently land inside the cup. Because the shooter was moved to a different counter than the one before, we had to measure a new height. This height was 1.1085 m. Using this new data, as well as the data collected from Part 1, we then calculated what would have been the ideal distance of the cup.

Using our calculations, we found that the ideal distance would be 2.2372 m (the theoretical distance). However, as one can see from the picture above, this distance did not work out as planned. The ball would hit the rim of the cup, but would never go in. When we moved the cup slightly towards the shooter however, the ball went into the cup consistently. We measured this new distance and found it to be a distance of 2.204 m (the experimental distance). Faced with this new value, we had to then find the percent error of this experiment calculation.

The percent error was found to be .0148%.

After this calculation, we attempted to minimize the percent error by using a total height that took into account the height of the cup. After measuring the height of the cup as being .092 m, we subtracted this from the height of counter to find a new total height of 1.0165 m. Using this height, we then calculated a new horizontal distance for the cup to be placed at.

As seen before, the horizontal distance for the ball to travel in order to land in the cup was calculated as 2.1385 m. After positioning the cup appropriately, we found that the ball landed in the cup every time it was shot out. Using this information, we calculated a new percent error. The new percent error was 0%! This meant that we had completely erased all potential error from the experiment.

__Videos__ The ball was shot out of the shooter in the manner shown in the below video. media type="file" key="Movie on 2011-10-24 at 08.58.mov" width="300" height="300"

For each trial, the ball landed in the cup, as shown below.

media type="file" key="New Project 1.m4v" width="300" height="300"

__Discussion__ In Part 2 of this activity, my group and I had to perform several trials and use different calculations for vertical and horizontal distance before we have zero percent error. For the first set of trials, we used a vertical distance of 1.1085 m, which led to a theoretical horizontal distance of 2.2372 m. After we found that we had to move the cup slightly for the ball to land in (giving us an experimental value different than the theoretical value), we calculated percent error as being .0148%. While extremely small, this value prevented us from fulfilling our task. Ultimately, we reasoned that this percent error existed because we had found the vertical distance the ball would have to travel to land //on// //the floor//, not land //inside the cup//. Therefore, it made sense that the ball would be hitting the floor with the calculated theoretical horizontal distance. As a result, we were required to take in account the height of the cup (.092 m) and subtract that from the vertical distance we had used before, which gave us a vertical distance of 1.0165 m. This new value was the vertical distance the ball would have to travel to land inside the cup, instead of hitting the ground. With this new value, we found that the corresponding horizontal distance was 2.1385 m. After different trials using this distance, we found that the ball landed inside the cup every time. Therefore, the theoretical and experimental values were the same. As a result, when calculated percent error, we found that there was 0% percent value! This would make sense as the ball landed in the cup without fail.

Shoot Your Grade Lab
Date: 10/26/11 Partners: Ryan Hall, Amanda Fava, Julia Sellman

__Purpose w/ Rationale__ Purpose: To launch a ball at a given angle and position five rings and a cup on the floor, so that the ball will pass through the rings hung in the air and ultimately will land in the cup. To do this, we must establish the theoretical positions of these rings and the cup through calculations, then set them at these positions, and determine whether or not the ball will move through the cup and land in the ball. If it does not, we must find new experimental positions that will work. This should be done in order to provide a definite position for where the rings and cup will be, instead of organizing them randomly. //For a more complete depiction of this process, see Procedure//.

__Hypotheses__ If our calculations of vertical distance of the ball are correct and we place the rings and the cup accordingly, then the ball should be able to pass through all five rings and land in the cup. If the ball is shot out of the shooter at a 25º angle, then the ball will move upward for a period of time, then descend downwards towards the cup.

__Materials__
 * data studio
 * plumb bob
 * ramp
 * masking tape
 * yellow ball
 * photogate timer
 * meter stick or measuring tape
 * target
 * carbon paper
 * two right-angle clamps
 * newsprint
 * calipers
 * cup

Ultimately, the materials ought to be assembled in this fashion:

__Procedure__ Before we began to determine the vertical distance at which the rings would have to be hung, we had to first measure the total horizontal range, as well as the initial velocity of this shooter. To do this, we used what was the initial velocity for the shooter at 0º (4.7 m/s) and the vertical distance of the ground from the shooter (-.8464 m). Using these values, along with vertical and horizontal acceleration, we calculated that the horizontal distance would be 2.85 m.
 * Theoretical:**



To test whether or not this was the actual horizontal range, we shot the ball out four times, marked the places of impact on carbon paper, and measured the distance. Each time, the distance was greater than 2.85 m; therefore, this value did not work. However, all the distances were different values (2.951 m. 2.995 m, 2.951 m, 2.915 m), so we had to average them. By doing so, we found that the average horizontal distance was 2.953 m.



With this value, we could find the total traveling time, as well as the initial velocity times time.



As seen above, the total time was .67 s, while the initial velocity was 4.84 m/s. To find the vertical distance the rings should be hung out (compared to the top of the counter the shooter was on), we set up different intervals of .5 m horizontally at which the rings would be hung up. There are intervals going up to 2.5 m, along with a position at 2.953 m horizontally for the cup to be at. With these values in mind, the time of travel to each interval and the y-position at each interval could be found.





As seen above, the positions at each interval (as well as the times at these positions) were calculated as:



While testing these theoretical values, we found that they often did not work and that the ball would not be able to pass through a ring. As a result, we had to adjust the rings, so that the ball would (preferably) pass through all of the rings and land in the rings. After attempting to reposition the rings, we managed to have the ball pass through four rings, without passing through the fifth or landing in the cup. This is depicted in the video below:
 * Experimental:**

media type="file" key="Shoot for Your Grade.mov" width="300" height="300" The measured positions of these rings (along with the percent error of each vertical distance) is shown below:

__Error Analysis__ The percent error for each ring's vertical is shown in the table above. An example calculation (for the first ring) is shown below.



__Conclusion__ Part of my hypothesis was correct; the ball did relatively travel in the fashion that I stated it would. As one can see from the video clip of the performance, the ball initially traveled upon and then made a descent due to gravity. However, at the theoretical values I found for the heights of the rings and cup, the ball could not travel through all the rings and land in the cup. This became clear when while testing these values, the ball would not travel through any. To be able to have the ball pass through some/all rings and/or the cup, we had to change the vertical height of the rings. Ultimately, after analyzing the trajectory of the ball and attempting to position the rings accordingly, we only managed to get the ball to pass through four rings, representing a significant amount of error. Specifically, the percent error between each ring ranged between 100%-145%. This error occurred as a result of the fact that the positions of the rings needed to be shifted. A potential source of error could have been shifts in angle that the shooter made. We noticed that after several attempts at a 25º angle, the shooter would reposition itself to a different angle over time. Though we attempted to fix this as often as possible, this may have led to some results that were off. To avoid in this in the future, we would use a shooter that remains static at the same angle, unless changed. Another potential source of error could have been the movement of the rings. We noticed that with each test the rings would shake slightly due to the air vent. At the same time, there were a wide variety of different groups working on the same lab as us in an enclosed area, leading to the possibility that people may have banged into our rings and thrown off their position. To avoid this in the future, we could use sturdier support for the rings and perform this experiment separately from others in order to avoid the rings being shaken.

This lab can be applied in many ways. In direct relation to physics, this lab could be used to predict the trajectory of any object that is shot up at any angle. On a more personal level, this lab taught me to be patient and calm during labs. This particular lab took a very long time and positioning the rings so that the ball would be able to pass through some of them was a very tedious, arduous effort. It was helpful to remain level-headed because I was able to remain calm and not get emotions get in the way of the lab.

Gourdorama Contest
Date: 11/1/11

__Picture__

__Calculations and Results__ Distance- 4.5 m Time- 2.63 s Mass- 1.94 kg (w/ the pumpkin)

As shown above, the total acceleration of the cart was calculated as -1.30 m/s^2, while the initial velocity was calculated as 3.42 m/s.

__Possible Improvements__ If given the time to improve my cart, I would decrease its mass. As mentioned above, the original cart had a mass of about 1.94 kg with the pumpkin, a greater mass than I would have liked. I believe this was the case because I used wood to create part of the cart, which no one else in my class did. Therefore, I would make a vehicle of the same design, but instead making it from lighter material like cardboard or with the use of less wood.