Evan,+Dylan,+Ryan,+Emily+Projectile


 * Group Members:** Evan Bloom, Dylan Waldman, Emily Van Malden, and Ryan Listro

__**Introduction** __

Theory/Rationale: In this project, we are studying the trajectory of a projectile. A projectile is split up into two components, the x and the y. A projectile’s path can be illustrated on a graph by a parabola. The x component of acceleration always equals 0 because theoretically there is no horizontal force acting on it. In reality, air resistance plays a small part, but considering the size and weight of the ball, not a big one. The y component of acceleration always equals -9.8 m/s/s, because it is the acceleration due to gravity (a purely vertical force). Air resistance would also have a small effect here as well. Our goal in this project was to find the relationship between the x distance traveled and the projection angle, to further calculate the y max, and other vital information in making sure the ball would go through the hoop at any height given to us. We tested a bunch of different angles and the distances they traveled (three times each to be as accurate as possible) and found the initial velocity of our launcher. Using this information, we were able to calculate the x distance and angle needed to hit the target at all heights.


 * Purpose: to calibrate a launcher and be able to direct it to send a projectile through a small target hoop using naught but the pure strength of physics.
 * Materials/ Method: During our data collection period, we used a pretty effective method. We lined up the launcher with the end of the measuring tape and placed carbon paper in line with the launcher. By using carbon paper, our measurements could be extremely precise. At each angle we tested, we launched and recorded the range of the projectile three different times to determine an average range for the said angle. With carbon paper, the projectile made a mark that we precisely measured to the nearest hundredth. We used this to calibrate the launcher and find it's initial velocity. Then, we created an excel workbook that would allow us to input an average initial velocity and a max height to get the proper angle and range that the launcher must have in order to get the projectile through the hoop. Our projectile was a small, green, plastic ball. The materials we used to calibrate were a launcher, the projectile, a baton for pushing the projectile in to Medium Range, a measuring tape to measure the range, carbon paper/regular paper to mark how far the projectile flew, tape to hold down the carbon paper, and patience. On presentation day (and the trials before it), we used a roll of masking tape as a target, textbooks to decrease the distance to max tape, measuring tape to, well, measure range and max height, launcher, projectile, baton, a computer/video recorder to catch the final result, and slot more patience.
 * Procedure
 * Collecting the Data
 * Place a tape measure on the ground with the projectile launcher at the beginning of it
 * Set the projectile launcher to a certain angle
 * Load the ball to the "Medium" setting on the launcher
 * Pull the string to launch the ball, and take note of where the ball lands
 * Set the piece of carbon paper taped to a piece of printer paper around the predicted landing location of the ball based on your observations from Step 4
 * Launch the ball three times using the same launch angle
 * Find the distance that each ball traveled by seeing where the mark was made by the carbon paper; calculate the average of the three launches
 * Repeat previous steps for multiple launch angles
 * Test day
 * Measure how high the center of the tape (max height) is from the launching surface (table)
 * Plug the max height into the formula created on excel to get the range (how far away the launcher should be set from the hoop) and the launch angle
 * (optional step) If max height can not be reached because the velocity of the ball from the launcher is too slow, shorten the max height by adding textbooks underneath the projectile launcher. Recalculate the max height, and then plug the new value into the formula
 * Launch the ball, and hope that your calculations were correct!

__ **﻿Data Collection** __
 * = Observations (from calibrating): =
 * While calibrating our projectiles, a group member also wrote down the launch angles and their corresponding ranges. We shot the projectile from various angles, and from each angle, we recorded three different ranges. From the three ranges, we found the average and input the number into a spreadsheet. See the tables and graphs below for the average ranges of each angle. Once we calibrated the launcher and all of our data was recorded, we noticed a few observations. The range increased as the angle increased. This was only true, however, for angle increasing from 0˚ to 45˚. Once the angle rose over 45˚, the range decreased. This shows that 45˚ is the optimal angle of our launcher. Also, we noticed that the higher the angle, the more it fought straight against gravity and the slower initial velocity was, while the closer the angle got to zero, the less it fought against gravity and the faster the initial velocity was.

=**__Results (observations from Performance Day) __** =
 * On test day, we really had limited work to do thanks to our automated formula created on Excel. Once we figured out the max height, we could plug that in to Excel to find the range and angle needed for a hopefully perfect result.
 * Table with trials:
 * [[image:ijcjddsu.png]]
 * This is the table we used on presentation day. The final line shows the angle and range we were launching from.
 * [[image:duqhsdpoasgpfd.png]]
 * This chart shows the results of our trials on presentation day.


 * Trial 1: Right through the center without touching anything. Nothing we can do to improve this repetition. This video is not shown because somebody was covering the camera lens during this shot.
 * Trial 2: Hit the edge of the tape, but still went through. After our first trial, we might have moved the launcher a little bit or slightly changed the angle. If we were to redo this, we would have to be more conscious that we keep everything the exact same.
 * Trial 3: The ball did not go in. This time, we made shifts to the launcher to try to get the same result as Trial 1. However, we adjusted the wrong way, so we should make sure to make the proper adjustments next time, mainly by raising the launcher a tad.
 * Trial 4: The ball did not go in again. This trial was closer than the last one, but again not enough to go in. We must still make the proper adjustments, to make sure we are lined up horizontally.
 * Trial 5: The ball hit the side, but went through. Our last trial was better than our 3rd and 4th, but not quite what we were looking for. We just have to pay more attention to detail to set up the proper angle, range, and calibration so that the ball just goes straight through the hoop.
 * media type="file" key="Shoot your grade dw evm rl eb.mov" width="300" height="300"here is a video of our second shot, because in our best shot, someone walked in front of the camera.


 * =<span style="font-family: Arial,Helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">Excel Workbook with Graphs and Data:[[file:projectiles project.xls]] =
 * =<span style="font-family: Arial,Helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">Graphs and Table with Written Analysis: =
 * <span style="font-family: Arial,Helvetica,sans-serif;">Table A:[[image:uhgyutrs.png]]
 * <span style="font-family: Arial,Helvetica,sans-serif;">This table displays the angles that we tested and other information including range, max height, initial vertical height, and initial velocity. We ended up using this table when we tried automating our spreadsheet. By automating, I mean setting up the excel document to determine what the angle and range would be according to what was plugged in for the y-distance. Calculations used to automate this table are shown below:
 * <span style="font-family: Arial,Helvetica,sans-serif;">Initial Velocity: [[image:oadnsf.png]]
 * <span style="font-family: Arial,Helvetica,sans-serif;">Max Height:[[image:oiajshdfsa.png]]
 * <span style="font-family: Arial,Helvetica,sans-serif;">[[image:pddfk.png]]
 * <span style="font-family: Arial,Helvetica,sans-serif;">We did not end up using this graph because it did not serve a purpose. However, it does show us where the range maximizes and minimizes. The x-distance increases as the angle increases from 0˚ to 45˚. Angles greater than 45˚ will display a lower range than 45˚.
 * <span style="font-family: Arial,Helvetica,sans-serif;">Table B:[[image:ijcjddsu.png]]
 * <span style="font-family: Arial,Helvetica,sans-serif;">This is the table that we found to be most helpful. It was automated in the excel spreadsheet to tell us what the launch angle, time, and range would be if we plugged in a certain max height. We used this on presentation day to help us launch our projectile through the target.
 * <span style="font-family: Arial,Helvetica,sans-serif;">Calculations: Any calculations you performed, along with an explanation for how/why you used them.

**CALCULATIONS TO CALIBRATE THE LAUNCHER:** **Calculations Used to Find the Proper Initial Velocity, given the angle:** In order to shoot our projectile through a hoop successfully and consistently, we had to calibrate it, to make sure that we knew with what speed the projectile shot out of the launcher. We set up a number of trials, and recorded the range (R), y-displacement (Dy) and angle (theta) of each. This we recorded in Table A above. To calculate the initial velocity for each angle (in m/s) we derived the following equation. Then we plugged in the values we knew would be constant for each trial. For instance, acceleration of the y component (due to gravity) will always be -9.8. And in g’s acceleration is positive 9.8. Then, in an excel spreadsheet, we calculated the initial velocity for each trial. The way we ran our experiment, we were able to record the specific angle and the range of the projectile each time, and these were plugged in the excel equation each time. For example: Trial 2: angle = 17 degrees, R = 1.81 meters **Calculations Used to Find the Maximum, given the angle and calculated initial velocity:** To calculate the maximum height each angle would give us we used: Then we plugged in the values we knew would be constant for each trial. For instance, at max height the final velocity should be zero while the acceleration o the y component will always be -9.81. Then using an automated excel spreadsheet, we did this for each of our trials – using the specific angle we tested and the initial velocity we had calculated for said angle before we found the max height. In reality we had to convert the angle which was input in radians (multiplied by pi/180) in order to use it in excel; the answer was still the same, but here is an example: Trial 2: angle = 17 degrees, Vi = 5.629 m/s On presentation day though, we did not end up using these max heights to help us shoot the ball accurately because we had found another system, in which we consciously chose what max height to use. This worked better because on presentation day we wouldn’t know for sure what angle to use in the first place. However, these equations and subsequently, Table A were helpful for us to see in general, how high our projectile could be launched.

**CALCULATIONS USED ON PRESENTATION DAY:** As a group, we realized that on presentation day, there would be a number of factors that we could control – the height of the target, the distance between the launcher and target (the range) and the angle. The only thing that was somewhat constant was the initial velocity. We averaged the initial velocities we calculated above, and found a mean velocity of 5.058 m/s. However, whenever we tried to put use this velocity in our trials for presentation day, the projectile would always go below the ring, suggesting that we were overestimating the actual speed of the projectile. So we instead started to use the absolute minimum velocity of our projectile, the one we got when launching it almost straight up into the air: 4.8 m/s. We used this velocity because we figured that this would be the slowest the projectile could travel, and so we figured we would adjust the angle and range based on wherever the projectile lands. We decided to use the minimum initial velocity and the desired max height as constants we would input into our equations. We knew that the launcher had to be at least one meters away from the target and we also knew (based on the Table A) that our launcher could not shoot higher then about 1.3 meters. So using a lab table and a number of 5 cm thick science textbooks, we adjusted the vertical distance between target and launcher so that it would fit into these standards.

**Calculations used to find the launch angle, given initial velocity and max height:** On presentation day, we automated this equation in the excel Table B shown above to find the proper angle: Then we plugged in the constants we knew to be true for every trial. Then, on excel, we inputted the minimum initial velocity for each trial (4.8 m/s) and the max height we decided to use. Here is an example with a target that we made .83 meters taller then the launcher. In reality, the angle excel popped out was in radians because it was calculated by excel, but to switch it to degrees, we multiplied it by (180/pi).

**Calculations used to find the Proper Range, given Initial Velocity, and Angle:** However, we also needed to know how far away we should place the launcher from the target so that the projectile's max height occurred in the right place. To do this we needed to know one more variable – the time it took for the projectile to reach max height. Once we knew that, we could plug time (t) in to find the range.

__ To find time: __ Then we plugged in the values we knew to be true for every trial. Acceleration = -9.81 while velocity at max height = 0: Then we plugged in our minimum initial velocity and the specific angle we were using for each trial to find time. For example: __ To find the range: __ Once we knew time, we could then finish and find the range (Dx): Then, excel would plug in the constant (angle/velocity/time to max height) for that specific trial. For example: So for this specific trial, to get a projectile traveling a minimum of 4.8 m/s into a hoop .83 meters above the ground, the angle of launch should be 57.22 degrees and the launcher should be placed 1.069 meters away. The culmination of these calculations can be seen in Table B.

For our experiment, we assumed a few estimates to be true.
 * Assumptions**

__Initial velocity:__ At the time of our trials, we assumed that because our average velocity had always sent the ball below the hoop, that the spring on our launcher was less springy then before, and that it had started to send the projectiles out at a slower speed then we first calculated. Also, we thought that air resistance might have played a small part in the projectile’s path, by slowing down how much vertical and horizontal distance the projectile covered thereby lowering its maximum height and causing the projectile to peak too early. So to compensate for that, we used the absolute slowest calculated velocity our launcher had (4.8 m/s), because we thought this was the most reasonable estimate of how fast the launcher was actually sending the projectile. But then the projectile was continually being projected underneath the hoop, with its max height occurring way too early in its path to reach the hoop. So we figured that either a.) there was an error in our calculations or b.) that we weren’t compensating enough. However, a quick check lead us to believe that it was b.) rather then a.). We adjusted the angle, making it bigger, in order to make the projectile reach the center of the hoop each time. So we developed a system. We found that if we added 7 degrees, to whatever angle we got out of Table B, the projectile would always make it through the target. It was this system that we used on presentation day, because it continually worked.

__Acceleration:__ We assumed an acceleration of (a) -9.81 m/s2 and (g’s) 9.8 m/s/s.

The method we used, while successful, made no physical sense, and even though we didn’t have time to before, we thoroughly went back and checked all the equations we used and we found a possible source of error. The equation we used when calibrating for our initial velocities was. This equation can be used //only if the y displacement is zero//. Unlike other groups, in our trials to calibrate the initial velocity, the projectile was shot from an initial height of about 27 cm and landed at ground level. The y-displacement is about 27 cm each time! So it didn’t make sense for us to use this equation to calculate the initial velocity because the y-displacement of our trials is not zero. Next, we tried a method that we used originally, before we used the R one in excel. We used Trial 2 as an example to see how different our results were. We automated this process on an excel sheet, on the top right part of our original one and calculated the initial velocity for each trial angle this way. Not only were the individual velocities much different, but the average velocity was as well. We found that 6.3 m/s was in reality our average velocity not 5.058 m/s. It makes sense that the first average initial velocity of 5.058 m/s went under the hoop consistently because the ball was actually traveling faster (covering more x distance) then we thought it would, and so it was peaking after the hoop, instead of right on the hoop, as it should have had it actually had the slower velocity we initially calculated. Because we were using a velocity that was in fact, slower then the actual velocity, it was peaking at its max height too late in its path. To compensate for this, we had to consistently add 7 degrees to the angle to force it to reach max height earlier in its path – just when it passed through the hoop. This error was a systematic error, not human error, so that is why our error is always a constant 7 degrees off each time. If we could run some more trials, this time using an average velocity of 6.3 m/s, our projectile might have made it accurately through the hoop each time. Unfortunately, we didn’t calculate a margin of error for our previous velocities because we couldn’t get the average to work in our trials so we didn’t see any purpose in having a margin of error based on the average. We weren’t sure at first what part of minimum velocity played in the launch and thus we did not use it on Presentation Day. As it turns out, our velocities were flawed anyway, so the margin of error would not have done us much good.
 * ERROR ANALYSIS:[[file:physicsprojecttaketwo.xls]]**

=__Sports Broadcast__=

Our video was of a girl shooting a half court shot. We knew that she was 1.3732 m tall and since she was at the half court, she was 14.32 meters away from the basket. We also knew that it took two seconds, from the moment the ball was released, to hit the hoop. We plugged this information into the equation, dy=viyt+1/2(at^2) and found that 10.4966 m/s = viy. we also plugged this information into dy = vix+1/2(at^2) and found that 7.16 m/s = vix. Then we found that initial velocity = the square root of ((7.16)^2+(10.4966)^2) which is equal to 12.706 m/s. We then used the equation dy=viyt+1/2at^2, plugged in our new initial velocity, her height, -9.8 m/s and found that it took 2.2914 seconds to reach max range and using that information plugged into dx=7.16(2.2914) we found that the max range was 16.406 meters. To find the max height, we used the equation (vfy)^2=(viy)^2 + 2ad, plugged in 10.4966 for initial velocity, -9.8 for acceleration and 0 m/s for final velocity and found that the max height was 5.6156 m.


 * Overall Conclusion:**

Through this experiment we discovered the it’s possible based on only the knowledge of an angle and range to calculate a launcher’s power – the speed at which it is able to launch a projectile as well as the maximum height of the projectile and the time it would spend in the air. Then we discovered that we could reverse it, that we could take an initial velocity and a maximum height and calculate the angle and range needed to reach it. The most important thing we discovered was the relationship between initial and final velocity, time, range, height, and launch angle of a projectile. We learned through physics, how with only 2 or 3 pieces of information, so much more can be figured out.

Because of our careful observations and calculations, we were able to get the ball through the hoop on the first shot cleanly. Even though our initial equation was used incorrectly and the consequential velocities were wrong, we were still able to use our knowledge of projectiles (air resistance and assumed changing or constant component velocities) to come up with a system that worked and make in 3 out of 5 shots.

To avoid this source of error in the future, we could use the equation above (the one that accounts for a y-displacement) to find the average velocity (6.3 m/s). We could use this velocity in trials and test whether or not by inputting it along with a fixed max height into the excel sheet, we would successfully shoot the ball through that ole’ hoop.