Sam,+Rachel,+Emily,+and+Navin+Projectile+Project

=Projectile Motion Project= Group Members: Sam, Rachel, Emily, and Navin Period 2 Due Date: 11.22.10

=**Introduction**=

The objective of this lab is to analyze the motion of an object moving in two-dimensions under the influence of gravity. We will use the natural parabolic path of a ball shot from a launcher to be able to determine the appropriate distance, height, and angle to correctly aim it through a hoop. Although air resistance is present, we will ignore it since it plays such a little role in the ball's motion because of the small size and weight of the ball. We know that the acceleration for the x component of the projectile is always 0 m/s and the acceleration for the y component is always -9.8 m/s due to gravitational pull. The heigh of our launcher from the base to the opening was 0.3 m. Therefore we always had to take that height into account when determining the y height by subtracting 0.3 m from the distance from the middle of the ring to the ground in order to get the correct y change. By calibrating our launcher, we can determine its initial velocity and therefore determine the angle and distance the launcher should be positioned at to reach certain heights. Taking into account the range, angle, and height of the launcher, we could calculate an initial velocity for each angle. Hopefully they are all very close to one another so we can average them out and find a good initial velocity. All recordings will be entered into excel. Using the launcher with this initial velocity at certain angles and distances, we could see if the ball does indeed go through the hoop during trial runs. We can adjust what we think is the initial velocity if the results do not hold up. An excel sheet will be made to determine what position and angle the launcher should be at given the height of the hoop and an initial velocity. Come presentation day, we will have determined which is the best initial velocity to use and then calculate, using excel, what our angle and position should be.
 * Theory and Rationale:**

For our projectile project, we tried to find the initial velocity of our launcher in order to get our ball through the hoop every single time, no matter how high, low, far, or close the hoop was. We first had to calibrate our launcher by doing a series of tests at different angles from 0 to 90 degrees. At each angle (we usually did it in intervals of ten), we shot a small, green ball (the ball is loaded into the launcher using a black, plastic stick) and measured how far the ball went by having it land on carbon paper on top of printer paper. By doing so, when the ball hit the carbon paper, a mark was made on the printer. This allowed us to measure the range of the shot with a tape measurer. A tape measurer was used to measure all of our distances. Tape was also used here to ensure that things such as the paper and the launcher would not get out of position. After we collected the data, we put it into a spreadsheet on excel, which gave us an average initial velocity. We then did a series of trials of getting the ball through the hoop. We used our initial velocity to determine the angle and distance away from the hoop the launcher should be positioned at. Sometimes, we used textbooks to raise and lower the height of the launcher. After doing these tests, new results were found and recorded. On presentation day, all of our recordings and numbers were taken into account to pick an angle and distance to get it into the hoop.
 * Method and Materials:**

=Calibration:= We shot the ball from the ground at eight different angles. We shot it from each angle three times and the range was found each time. These were then averaged and put into an equation to find the initial velocity of the ball. All of the velocities were then averaged.

This is a mathematical explanation of the formula. It shows how the initial velocity was found using each angle and range. Dy is equal to -.309 because the ball starts inside the launcher at .301 meters above the ground and then lands at ground level.
 * The formula:**
 * Excel Spreadsheet:**

This can be found here:


 * The Graph:**

The graph above shows the relationship between a specific angle we used during our trials, and the average distance we found it would cover after three attempts. It is important to note that the range peaked at about 43º, and then steadily went back down at 50º. This knowledge factored into our decisions to use different angles while adjusting. We would want to keep in mind that at certain angles, the range became lower.

I picked to show the spreadsheets with initial velocities of 7.18m/s, 7.06m/s, and 5.69m/s. 7.06m/s was chosen because it was the average initial velocity when finding out the range at each angle. The other two were used because they were our most consistent velocities on certain days. The Formula:
 * The Formula:**

This picture is a mathematical explanation of the formula. It shows why it works and how it gets us the x distance (how far away from the hoop the launcher should be positioned at).

The quadratic formula is used to figure out how far away the launcher should be from the hoop. The only two things that need to be inserted are the angle and the initial velocity. When “#NUM!” appears as the solution, it means that using those specific numbers, the equation spits out unreal numbers. This is because in real life, the ball (with that speed and at that angle) will never reach the height of the hoop. The quadratic formula gives you two answers, one where there is a plus in the original equation and one where there is minus in the original equation. The one with a plus is used because this gives us a smaller y distance. Also, sometimes the equation with the minus is negative, which cannot be done in real life.
 * About these spreadsheets:**

Plug in any initial velocity and angle on one line and it will give you how far away your launcher should be from the hoop. You can also change the dy. This is how high the hoop is. You need to remember to subtract how high your launcher is and make it negative. This is what we need on presentation day.
 * How to use:**





You can find all three of these here:

7.062m/s was our initial-initial velocity. We found this after we performed the test trials. This information, however, was thrown out due to the decreasing velocity each time we calculated our launcher's velocity. The problem we faced with 7.062 was that it was not repeatable. It only worked for one angle, but did not seem to mathematically work out for other angles.
 * The Graphs:**

7.18m/s was the next velocity we settled upon for presentation day, and was actually used our first time on presentation day. However, as is seen later, it was a huge miss, and came well under when set to 65º, as represented by the x-distance. Each of these lab graphs was made when the y distance of 1.2m.

We eventually came to rest on 5.69m/s since it came the closest to the entire x-distance on our angle of 58º. It is important to note, however, that much of this information was not initially incorporated on presentation day. We shot based on our previous encounters, and then derived our knowledge from the previous shot and past experiences. The above graphs put into perspective what we had determined originally. Due to the fact that we found our launcher's velocity became progressively weaker, we found that these representations of the data may be misleading. They are dependent on the launcher staying at a certain velocity, which we found does not happen.


 * Calculations with Calibration:**

After calibrating our launcher and solving through excel, we got our velocity to be 7.062 m/s. However, we noticed that this velocity rarely worked. We decided to see what would happen if we increased the velocity and test them using the equation below. We went up by 0.1 m/s, narrowed it down and then found the velocity that worked best to the hundredth decimal place. The velocity that worked best for us originally was 7.18 m/s. We first started off by calculating our x-distance using the given height of the ring, a comfortable angle of 65 °, and the velocity that seemed to work best for us of 7.18 m/s.

From there, we would set up the launcher and shoot the ball. In the beginning it would usually work, sometimes slight adjustments to the angle (1 degree either way) would need to be done. However, as we started to launch more and more times, we noticed the ball would gradually start to stray from the ring and eventually was shot nowhere close to it. After noticing this flaw, we would continue to adjust x-distances until we were successful. We liked to stay in the range of 65 ° to 69 ° for our angle and 1.4m to 1.8m for our height that way we knew it wasn’t the angle or height to blame for our various velocities. We recorded a few trials (as seen below in the table) that were successful and used the height, distance and angle to solve for our initial velocity of that launch. We noticed a pattern as time progressed for our launcher, the longer and longer we used it, the lower the initial velocity was. We used the equations also shown below to solve for our velocity using the correct information we were given through trials. These three initial velocity values were very consistent on certain days. One period, 7.8m/s worked for everything while the next day 6.63m/s worked well. Of all initial velocity values, those two were the most consistent.
 * Dy (Given) || Dx (Given) || Angle (Given) || Initial Velocity (Solved) ||
 * 1.7 m || 1.08 m || 65 ° || 7.18 m/s ||
 * 1.698 m || 1.02 m || 68 ° || 6.629 m/s ||
 * 1.43 m || 1.1 m || 68.5 ° || 5.692 m/s ||



For the actual presentation day, we used the first equation shown to find the x-distance. We started with the velocity of 7.18 m/s (we hadn’t really tested so the velocity wouldn’t have been warn down), our ring height was 1.42 m but the launcher is 0.3 m so we ended with a height of 1.12 m and used a 50 ° angle. Unfortunately, our comfortable area with our trials was not possible when it came to launch day, the height was very low, the angle was very low (anything higher didn’t allow a distance of at least 1 m). Using our first equation and the numbers given, we got an x-distance of 1.23 m.

=**Presentation Day:**=

We came together on presentation day and incorporated all of what we learned to make the ball straight through the hoop twice. Here are our results to start off with...

Notice how we adjusted our angle accordingly to gain more accuracy. Along with doing this, we constantly made slight adjustments to the alignment of the launcher with the hoop depending on where the launcher hit. For example, the launcher hit too high on launch 3, so we adjusted the angle down to make it straight through. Then we repeated that to have a perfect launch 5. Here is the procedure that we undertook to reach the perfect shots...
 * Results:**
 * Trial || Angle (degrees) || Distance of "x" (m) || Distance of "y" (m) || Through hoop? ||
 * 1 || 65º || 1.23m || 1.12m || Miss ||
 * 2 || 59º || 1.23m || 1.12m || Hit Edge ||
 * 3 || 59º || 1.23m || 1.12m || Hit Top ||
 * 4 || 58º || 1.23m || 1.12m || Straight Through ||
 * 5 || 58º || 1.23m || 1.12m || Straight Through ||


 * Procedure:**

Once the hoop is set in place, we measured the distance from the center of the hoop to the floor. In order to find the dy value, we had to subtract .3m (height of launcher from ground) from 1.42m (height of hoop from ground); this created 1.12m as the dy. The next step was to find the dx by measuring the distance from the center of the launcher to directly below the hoop; this was 1.23m. Our launcher has a very erratic velocity, but we calculated ours to be 5.58m/s on launch day, and found that 65º was a comfortable angle to start with. However, as the launches continued, that was adjusted. Another important final step was the alignment. By placing yardsticks from the exact center of the hoop to the ground, we found out to what line the launcher should be aligned with. This too, however, was slightly adjusted until it became perfect enough to go clean through.

=Error:=

We found out that the exact initial velocity of our launcher that would have reached the exact distance, height, and angle was 5.58m/s. We hypothesized that the initial velocity was 5.69m/s through our calculations.

Percent Error= __|theoretical-experimental|__ X100
 * experimental |

Percent Error= __|5.69-5.58|__ X100 | 5.58 |

Percent Error= __|0.11|__ X100 |5.69|

Percent Error= |0.193| X100

//__Percent Error= 1.93% error__//
We see that we had a 1.93% error in our initial velocity, which explains why it took extra adjusting of the angle and alignment to get it just right. =**Conclusion:**=

A relevant real-life application of this concept is the launching of missiles and rockets. A missile is an object that is forcibly propelled at a target, either by hand or from a mechanical weapon, and a rocket is a cylindrical projectile that can be propelled to a great height or distance by the combustion of its contents. Missiles and rockets have to be made so that they hit the right spot at the right time. Missiles are used in combat. It is imperative to find all of the data before launching a missile. A computer calculates the distance between you and the enemy, the height difference (if necessary), the initial velocity of the missile, etc. This information is needed in order to hit the enemy right on target. Rockets require the same knowledge, but they usually go straight up as a flair or go into space. You need to know the projectile concept so launches of rockets and missiles can be precise.

There are many errors that occurred in this lab. Measuring was a big source of error because it is impossible to be accurate on 100% of one's measurements, especially with the amount of times we had to measure distances. Also, air resistance was not taken into account and this definitely had a small influence on the flight of the ball. These were minuscule errors compared to the inconsistency of our launcher. At the beginning of the project, our launcher was shooting around a consistent 7.06m/s. By the end, on presentation day, the launcher shot at 5.58m/s. This was probably due to the spring in the launcher. When we first started, the launcher had not been used in a while and the spring was stiff and strong. After using it many times, the spring became softer and weaker, thus causing the initial velocity to lower.

We were able to successfully experience the objective of this lab. We became experts at calculating a ball's projectile motion given different data. This lab also allowed us to truly put our educated guessing skills into play. As talked about previously, we had some unavoidable difficulties pertaining to the launcher's velocity, therefore we had to be able to quickly fix any slight error for presentation day that were unattainable by calculations. Therefore, we were able to successfully calculate the position and angle the launcher should be placed at in order to make the ball go through the hoop, however, due to material imperfections, our math success was not always able to be seen through demonstration. But through some adjustments, the ultimate goal was eventually achieved.