Tucker,+Siegel,+Hallowell,+Magda


 * PROJECTILE PROJECT**

Due: 11/22/10
INTRODUCTION/ PURPOSE: The objective of our project was to analyze the motion of an object moving in two-dimensions under the influence of gravity. Our goal was to use pieces of equipment and set equations in order to learn more about the physics of projectile motion. We were hoping to see projectile motion in the classroom, as well projectile motion on the computer during a sporting event. The goal was to analyze both scenarios using equations and the computer in order to strengthen our understanding of the physics behind this type of motion.

PART 1: CALIBRATION:
MATERIALS:

In order to complete the calibration step of our project we needed a designated launcher, a ball to launch, a loading stick, a measuring device (meter stick, ruler, tape, etc.), a roll of tape, printer paper, charcoal paper, several physics textbooks, a calculator, a pen or pencil, a computer with Microsoft Excel, a roll of string, and a roll of tape or a wooden hoop to launch the ball through. Once the materials are gathered, we are ready to begin the calibration stage of our experiment.

PROCEDURE:

There are many steps in the process to find out exactly how launch a ball through a hoop set at an unknown height. First, our group had to calibrate our designated launcher, set at a medium velocity launch. This task was completed by launching the small ball at a variety of different angles. For each angle, we would run a practice launch to estimate the range at white the ball would travel in the air. After the estimation was made, a piece of printer paper was taped firmly to the ground, to restrain from any movement. On top of the printer paper, we laid a single sheet of charcoal paper. At the instant that the ball hit the charcoal paper, it would leave a small mark on the printer paper, indicating a rough estimate of the horizontal distance traveled by the ball. For each angle, we launched the ball about five times, to achieve the most accurate results. Our group tested the following angles: 0 degrees, 10 degrees, 20 degrees 30 degrees, 45 degrees, and 60 degrees. After completing all 5 trials for each angle, we found the average horizontal distance traveled by the ball for each angle. This gave us a nearly precise range for each angle.

The next step in the process involved the computer and Microsoft Excel. Our group decided the make an Excel Spreadsheet to assist us in calculating some important information such as the average velocity, maximum height of the ball, and the time it takes for the ball to travel through the hoop. Once the spreadsheet was complete, we had to insert the information that was calibrated, or measured by hand such as the average range that the ball traveled per each angle that was tested, the vertical height of the ball to start, and the initial velocity of the ball being projected from the launcher. Once our Excel Spreadsheet was completed, we were ready to test out our results in preparation for the actual launching day.

In order to test out the accuracy of our results and calculations, we had to gather a measuring device (whether it be a ruler, measuring tape, or meter stick), a roll of string, a roll of masking tape, and our ball and launcher. We then found an open area to test. First we hung the roll of tape from the ceiling at a random height. Our group then worked together to measure the height of the ball at its starting point, the horizontal distance between the launcher and the hanging roll of tape, and the approximate vertical height of the roll of tape. Then, we plugged the measurements into our Excel Spreadsheet to calculate the angle at which the ball should be launched. Once all of the calculations were completed, we tested the accuracy of our ball launch at several different vertical and horizontal heights and distances.

GRAPHS: This chart was used to compare the maximum height to the angle. On test day, we first used this graph the calculate the angle we would need in order to achieve the desired max height. We plugged the angle in for y and then solved using the quadratic equation.

Once the angle was found, we used that calculation to find the range that we would need to make the ball go through the hoop at max height. In this graph, the angle is plugged in for x and then calculated to find the range.

EXCEL SPREADSHEETS: This is the excel chart that was used to create the graphs. put in the angle, radian, acceleration, and distances. We then used excel to calculate the initial velocity (both the final and each x and y components) and the time. We also used the same information to calculate the max height; however, we took one half of the time from the original equation because the max height will occur at half of the time, and final velocity is equal to zero.

EXAMPLE CALCULATIONS: Initial Velocity:

[[image:R=_equation_3.png width="69" height="33"]]
Time:

PART 2: PERFORMANCE OF LAUNCHER:
MATERIALS:

In order to perform our experiment on testing day we needed our designated launcher, our group's ball, a measuring device (ruler, measuring tape, meter stick), a calculator, a pen or pencil, a computer with Microsoft Excel, a roll of string, and a roll of tape or a wooden hoop. Once these few materials were gathered, we were ready to begin our final launches.

PROCEDURE:

The final step of our lab was to actually test the performance of our designated launcher by sending the ball through the middle of the wooden hoop or roll of tape as many times as possible out of five launches. First, each group was given a couple of minutes to gather their materials and set up their launcher in their desired distance from the already hung hoop or roll of tape. Once each group was set up, Mrs. Burns gave us twelve minutes to launch the ball five times. We had to launch the ball directly through the ring at least once out of the five tries to achieve success. Once the timer was started, we had to quickly work as a team to measure the initial height of the ball, the vertical distance of the hoop, and the horizontal distance (range) between the hoop and our launcher. Once the measurements were read, one team member recorded the information to remain organized. The measurements were then relayed to the person in charge of plugging the numbers into our Excel Spreadsheet to calculate the ball's maximum height, initial velocity, and most importantly, the correct launch angle. After this information was achieved, we set our launcher to the desired angle. The last step that was taken was making sure that our launcher was precisely in-line with the hoop. Finally, while one person held down our launcher to prevent any excessive movement, the ball was launched through the hoop.

TABLE (RESULTS): As seen in the data table, we successfully made it through the target only one time without contact. That trial is shown below. We were able to hit the edge of the target and go through on two out of the other four trials. We were not able to make contact with the target on the other two trials.

BEST LAUNCH CLIP: trial 3: media type="file" key="projectile lab.m4v" width="300" height="300"

CALCULATIONS: For example, let's say that the height from launch point to the center of the target was 2 meters. First, we used the equation from the "max height vs. angle" graph from the calibration part of this project. We plugged in 2 meters for our "y" as seen below. The resulting "x" was the angle for the launcher.

After we achieved our angle, we then used the equation from the "range vs. angle" graph from the calibration part of this project. We plugged in our angle for our "x" as seen below. The resulting "y" was the range of the ball at that angle. We then divided that number by two in order to get the distance to place the launcher from the target.

ERROR ANALYSIS AND CONCLUSION: On the launch day, our group missed the target twice, hit the rim twice, and shot it through the center once. Our launches were not perfect because we had some error to deal with. The first source of error was in the launcher itself. The initial velocity varied depending on the angle due to the gravitational pull on only on the ball, but also on the springs in the launcher. Our calculations were based on an average initial velocity, which was an average of all our gathered data; therefore it meant that the velocity could be a little less or more depending on the angle. Based on numerous tests, it was determined that our calculations were producing an angle that was about 4 degrees too low, however this number decreased with lower targets. This was determined qualitatively, and therefore was subject to human error and judgment. On the launch day, our target was relatively low compared to our tests, so we sent the angle 3 degrees above what out calculations predicted the angle should be. This was too high an estimate, and our ball went over the target. We lowered the angle slightly and then the ball touched the rim and went through, but not cleanly. By the third try the angle was correct and the ball went through the hoop.

Another source of error was evident in our fourth and fifth launches on launch day. Although we tried to keep the launcher still, it still jerked after each launch, slightly altering its position. The angle and range were perfect (as shown by our perfect third shot) but because after each launch the launcher moved. Also, after reviewing the video taken on launch day we discovered that the hoop was still moving after our second shot. This is obviously a source of error because we were aiming for a stationary target and were not accounting for any motion by the target.

If ever attempting this type of project again, we would make the following changes: Calculate quantitatively the different velocities at different angles and account for them while launching; possibly tape the launcher to the table to ensure no movement after each launch; and make sure the target is absolutely stationary. This project increased our knowledge of projectiles by forcing us to examine the relationships between angle, velocity and range. We had to figure out max height also in order to be successful in this lab. This knowledge of projectiles could be useful in other areas of life outside of the classroom. For example, if I were to throw a ball to someone standing at a higher elevation (on a ladder, for example), then I would know to throw the ball harder and at a higher angle. Really, a solid background in projectiles can help when throwing any type of ball or object, and therefore be useful in sports. Even though a basketball player is not going to realistically be doing calculations on the sideline, it is still helpful to have a general background of the effects of different velocities and angles on the range and max height. In addition to increasing our knowledge of projectiles, this project as increased our ability to use Microsoft Excel, which is important in science, but also in life because excel can be used in many job fields and with personal finance.


 * PART 3: SPORTS BROADCAST**:

CALCULATIONS:
 * These assumptions were used for all three parts of the calculations. Lebron's height, the Height of the Rim, and the Distance from LeBron to Rim were acquired from the internet. The time was acquired by using iMovie to time how long it took from the time the ball left Lebron's hands to the time it passed the rim. All of these assumptions were needed to solve for initial velocity, maximum height, and maximum range.


 * These calculations were needed in order to get our x-component and y-component of the velocity of the ball when it left Lebron's hand.




 * Notice that a new row was added to this data table. The new column is the final velocity of the maximum height. The y-component column has a zero because at maximum height, the ball is not rising or falling. This maximum height is the height above the ground.


 * Notice that in rows "d" and "t", the information has changed. We are trying to find the distance of the "x" column. The distance of the "y" column is now -2.051 because the ball's overall change in height from Lebron to the ground is 2.051 in the negative direction. We are also trying to find the time of this projectile in order to find out the maximum range.

CONCLUSION: In making this video, our group learned many important things about projectiles. We chose to go with Lebron shooting a free throw for our video because it was a clear projectile and a perfect showcase for projectiles in real life. We had the following information available to us: Lebron’s height, the height of the basket, the range and the time. We were able to calculate the initial velocity and angle using this information. This was a practical application to the material we had learned in class. This aspect of the project also challenged us to figure out other parts of projectiles, such as the maximum range. Calculating the range of the ball, as if the hoop had disappeared, forced us into thinking differently about projectiles and really apply and synthesize the information about projectile we had learned. Obviously our calculations regarding the angle and velocity are not perfect because we had to make certain assumptions about his height and the time. However considering the information we were able to collect, our data and results represent a realistic representation of the velocity, max height, and angle of the ball when Lebron’s shooting a free throw.