Jimmy,+Erica,+Steve,+Elena+Projectile+Project


 * Projectile Project**
 * Group Members**: Jimmy Ferrara, Erica Levine, Steve Thorwarth, and Elena Solis
 * Period 2**


 * Theory and Rationale:**

For our project we were trying to find the best angle that would get the balls through the hoops. To accomplish this we first calibrated our launcher. Calibrating the launcher involved testing the angles from 0 degrees to 90 degrees at 10 degree intervals. We measured how far the ball went by placing carbon paper over white paper so we could accurately measure how far each ball went. For every angle we launched 3 balls to make sure we had accurate measurements. This provided us with the range for all of the angles. Using this date we were able to calculate the maximum height for each angle. We plugged the equations into excel. On the presentation day we kept our launcher 1 meter from the hoop and then measure the height. Once we got the height we subtracted that from our initial y height. This was our ball’s maximum height so we worked backwards using the kinematic equations we had on excel. We knew the x distance and the maximum height so we found the interval that the angle would most likely be in (for example 45-55 degrees) by using the equations on excel, and then picked the angle we thought would be best. Then we set the launcher to that angle and launched it. The theoretical background we had was that at the maximum height the y-velocity would be 0 m/s. We also know that twice the x distance to maximum height it equal to the range (total x distance). And we knew what kinematic equations we needed to use. We then used this information to calculate which angle we needed to use.

The materials we used for our lab was a launcher and a ball. We always used the same ball and launcher so we would not have to account for any discrepancies between different launchers or balls. We also used meter stick and tape measures to measure the range of the ball and on the day of the performance to measure the height of the hoop. We stacked textbooks under the launcher because we knew from our calculations that our launcher can only reach a meter vertically and the hoop was greater than a meter so we needed the books to reach the hoop. Additionally, we used Excel to put our calculations on so that we could find the angle we needed more quickly.
 * Methods and Materials:**

After we calibrated the launcher (as described above) we manipulated the kinematic equations so that we could put them into Excel to generate the correct angle to use to get the ball through the hoop. We used the equations d=vit, d=vit+.5at2, vf2=vi2+2ad, and r=(vi2sin2 theta )/g (See calculations section for how we used them). We put these equations into excel (see spreadsheet) and for all the angles we tested we found their initial velocity, range, maximum height, and the time their trajectories took. We used this spread sheet on the launching day and then we worked backwards. We measured the distance to the maximum height (where the hoop was) which we know was half of the range. We also measured the height of the hoop (which was the maximum height of the trajectory) and made sure we accounted for the height of the books and the initial height of the ball (22.2 cm). Then by using the x distance and y distance we needed we matched it with an angle interval. We found the angle was in between 50 and 60 degrees. Then we picked what angle we thought would be the best and launched the ball.


 * Performance Day Results**:
 * Ball || Angle || Success ||
 * 1 || 55 degrees || The ball went too high; it went over the top of the hoop. ||
 * 2 || 54 degrees || The ball hit the edge of the hoop. ||
 * 3 || 53.5 degrees || The ball hit the edge of the hoop. ||
 * 4 || 53.5 degrees || The passed through the middle of the hoop. ||
 * 5 || 53 .5degrees || The passed through the middle of the hoop. ||

media type="file" key="Shoot Your Grade Trial(final).m4v" width="363" height="363"
 * Best Trial Video**

**Observations and Data:**

Range for Varying Angles The chart above shows the range that we measured for different launch angles. In making this chart, we were trying to decipher the effect that different angles have on range. We conducted several trials and then found the average of them to yield more accurate results. For this series of launches, the launcher was placed at ground level, making the position of the ball 22.2 cm above ground. After playing around with equations on excel we realized this information was invalid.We wanted to find initial velocity by manipulating the equation R=vi(squared)sin2(angle)/g. Technically, we weren't supposed to use this information because we hadn't conducted trials in which the starting height was the same as the ending height. Although this information was invalid, we were luckily still able to use it to find initial velocity. We were able to place this data into excel and calculate initial velocity for each of the angles shown above. We used the equation R=vi(squared)sin2(angle)/g (even though we knew were weren't supposed to) and manipulated it to solve for initial velocity, putting in the angles and ranges from the chart above. From this information we found out that our initial velocity went from 4.8 m/s to 6.8 m/s depending on the launch angle. The reason we were able to use these results even though they came from faulty data was because the initial velocity at different angles is constant. We knew that if the starting or ending height was changed, the initial velocity we calculated would still hold true.

Range for Varying Initial Heights This chart displays the range based on different initial heights. Through these launches, we were trying to see the effect of of varying heights on the range. We conducted three trials and then took an average, in order to produce more accurate results. We ended up not using this data when we realized it would be easier to solve for maximum height of the projectile. If we made the height of the ring the maximum height of the projectile, we knew it would be a lot easier to generate a useful equation.

Range for Varying Angles (same starting and ending heights)

After forming our first chart, we were able to reform our methods to produce helpful results. In these trials, we used text books to make the starting and ending height equivalent, taking the height of the ball in the launcher into account. Although the initial velocity from the first chart was valid, the ranges that we calculated were not. In a second spread sheet we plugged in the angle from this chart, half of the range, and the initial velocity from the previous chart to solve maximum height and time to maximum height. This was the most helpful data we collected because it allowed us to generate an equation where we could make educated guesses of correct launch angles based on the height and range of the ring on presentation day.

This graph was created by the data from the first trial. Inputting the angles as x values and the x distances or ranges as y values formed the graph. The best-fit line used was polynomial with the R2 value being relatively close to zero. The graph resembles a parable because the x distances increase detrimentally from 0 degrees to 45 degrees then decrease exponentially from 45 degrees to 90 degrees. This graph was used to interpret data and extrapolate angles from known ranges. For example, a known distance, such as the range from the target, can be used to estimate the angle by extrapolating from the graph. If the target were 2.25cm away the angle to achieve that distance would be about 52 degrees according to the graph. In the actual performance, the graph was not used because it is not accurate enough, but it was used to help analyze the big picture and cross check with other methods.
 * Graphs**



This graph is similar to the first graph, but was created for trial two. The difference between this graph and the last is that meters were used instead of centimeters, radians were used instead of degrees, and the initial displacement height (not shown on the graph) is 0m. The line of best fit was once again polynomial with a precise R2 value. Just like the previous graph, this one can be used to extrapolate angles from known x distances, but it is more accurate. Although this graph too was not used in the final performance, it served as an aide to the excel spreadsheet and a backup plan.


 * Excel Spreadsheets:**



We used a combination of two spreadsheets to help us estimate a launch angle based on the range and height of the hoop that we measured on presentation day. This is the procedure for using our excel sheets that we performed on presentation day:
 * 1) Measure the range
 * 2) See what angles on the second chart have similar ranges to the one measured in step 1
 * 3) Make an educated guess for an angle based off the second chart
 * 4) Plug the angle into the first sheet
 * 5) Plug double the distance into the first sheet
 * 6) Get initial velocity and plug it into the second sheet
 * 7) Look to make sure the max height matches what we measured for maximum height

The first spreadsheet uses the equation R=vi(squared)sin2(angle)/g, and we manipulated it to solve for initial velocity. Plugging in angle and range from the first table, we were able to produce the initial velocity for varying launch angles. We found that depending on the angle, our initial velocity can range from 4.3 m/s to 6.8 m/s. This was important to know because it changes our results drastically compared to if we simply used one constant value for initial velocity at all launch angles.

The second spreadsheet uses two equations. First we manipulated the equation vf = vi + at to solve for time. By plugging in the angle from the third table, half the range from the third table, and the appropriate initial velocity from the first spreadsheet, we were able to solve for time to maximum height. The second equation we used was d= vit +1/2 at(squared). We manipulated this equation to solve for distance to maximum height. We already had the angle, half the range, and the initial velocity in this spreadsheet, so it was easy to solve for this unknown. Solving for maximum height proved extremely useful because we centered our presentation off of making the height of the hoop the maximum height. Although our spreadsheet didn't solve for an angle, it allowed us to work backwards to achieve our goal. By looking at values on the second chart for range and maximum height, we were able to compare them to our experimental range and height and make an educated guess for our experimental angle.

On presentation day we measured the range to be 1.101 m. We compared this to data on the second spreadsheet and saw that our angle must fall between 50 and 60 degrees based on the chart data for range. We made an educated guess that the angle that would put the ball through the hoop would be 56 degrees. We plugged 56 degrees and 2.202 cm into the first spreadsheet to find the appropriate initial velocity. We double the the range for this calculation because 1.101 cm represents the range to maximum height, or half of the projectile. We found initial velocity to be 4.82 m/s and plugged this value into our second spreadsheet. The yield value for maximum height and our measured maximum height were a little different, but we still launched using the angle believing it was just error in our calculations. The ball went over the hoop, but we were able to adjust our calculations for the second shot and get it through the hoop. Our excel spreadsheets were extremely helpful in guessing the correct launch angle on presentation day.


 * Calculations:**

Our calculations were completed on a spreadsheet in Microsoft excel with three main equations.

First we plugged in angle and distance to find the initial velocity. We did this with an equation that range= initial velocity times the cosign of the angle divided by gravity, or R = Vi (squared) cos**θ**/ g. This equation gave us the initial velocity of the ball at a specific angle, which is important because the initial velocity varied with each angle.

After finding the initial velocity we calculated the time it took for the ball to travel. We did this by using the equation Vf = Vi + at, where Vf = 0, a = -9.8(gravity), and Vi was found from the previous chart.

With what we found for time, we are now able to calculate the maximum height. Our goal was to have the ball go through the ring at the shot's maximum height. This would cause the ball to go through the hoop without hitting any of the edges and the ball would easily move through the hoop horizontally without moving up or down due to gravity. For maximum height, we used the equation d = Vi t + 1/2 a t^2. With this equation we find the maximum height and the horizontal distance the ball travels to reach this height. Because we are able to determine how far away the canon is horizontally from the hoop, we can position it so the ball will reach the hoop when it is at maximum height, when shooting so that the ball's maximum height is equal to the height of the center of the hoop.




 * Error Analysis**

__Trial 1__ Theoretical Maximum Height: 0.633m Actual Max-Height: 0.713m

% Error = |Theoretical - Experimental| / |Theoretical| x 100

% Error = |.633m - .713m| / |.633m| x 100

% Error = .08m/.633m x 100

% Error = 12.64%

__Trial 2__ Theoretical Maximum Height: 0.633m Actual Max-Height: 0.663m*

% Error = |Theoretical - Experimental| / |Theoretical| x 100

% Error = |.633m - .663m| / |.633m| x 100

% Error = .03m/.633m x 100

% Error = 4.739% *note: the ball went though, but it hit the top of the hoop and bounced in so it did not really achieve the desired hieght

__Trial 3__ Theoretical Maximum Height: 0.633m Actual Max-Height: 0.633 (middle of the hoop)*

% Error = |Theoretical - Experimental| / |Theoretical| x 100

% Error = |.633m - .633m| / |.633m| x 100

% Error = 0% *note: even though in this trial the ball did not go in, it was to the side of the center so thier is actually no error except the fact that the ball missed laterally

__Trial 4__ (same as Trial 3)

% Error =0%

__Trial 5__ (same as Trial 3)

% Error = 0%

In the "Shoot Your Grade" lab we had to shoot a ball through a hoop that is 5 cm in diameter and is hanging from the ceiling. In order to do this we had to calculate the velocity(s) of the canon when firing at different angles. Then we had to calculate the amount of time it takes for the ball to travel, and use it to help calculate the maximum height that the ball will travel at a certain angle. By adjusting the position of the canon, we set the height of the hoop above the canon equal to the maximum height we calculated for the angle we wanted to shoot from. We also knew the horizontal distance the ball would travel to reach maximum height, and set the canon that horizontal distance away from the hoop. Our only restriction was that the canon's horizontal distance away from the hoop had to be at least one meter. With the help of Microsoft excel we were able to find the angle most appropriate to fire from, and we made three of our five shots.
 * Conclusion**



The error from performance day is mostly due to the z-axis that we did not account for in the preparations. The z-axis is the 3rd dimension or the lateral movement of the projectile. The projectile did not move significantly left or right, but when aiming at the hoop, the shooter was not perfectly lined up. This is partly due to human error and the fact that as a group there was no preparation for this aspect of the shot. To improve the method a ruler or string could have been used to perfectly line up the shooter with the hoop.

Besides this source of error, the first shot was not at the ideal angle for the range and height the hoop. This is due to the preciseness of the calculations, which almost resulted in the angle (only off by 2 degrees). By observing that the ball went over the hoop, it was realized that the angle needed to be adjusted downwards. For the next trial the angle was decreased by 2 degrees. This resulted in a score, but it hit the top of the hoop. The rest of the trials the angle was adjusted by half a degree down and it resulted in two scores. One trial missed to the side because of the previous explanation of error, lateral movement.