Psi Vertical Air
What is the full equation for the horizontal displacement of a spherical projectile fired from an air cannon?
Specifically, a pumpkin like they do at Punkin Chunkin. Taking into account the initial force, the drag, the size and mass of the projectile, and the barrel length. I know an approximate drag coefficient, that a large air tank is needed to maintain a lot of pressure on the projectile until it exits the barrel, the barrel should be elevated to 45 degrees, and the time it takes for it to reach it’s maximum height is the time it has to get as far it can away from the cannon. I derived an equation for the vertical displacement but I am having a hard time solving for time. I really don’t know whether to factor in the velocity of the projectile just as it leaves the barrel or not. Apparently 300 psi yields a muzzle velocity just under the speed of sound (~720 mph) and a distance of about 4000 feet, depending on wind direction and speed.
Solving this kind of equations for displacement and velocity as a function of time is quite difficult. Question is: are you doing this because you’re interested in advanced calculus or you just want to know practical solution? In latter case, I’m advising you to try for numerical solution. Here are some links:
http://www.phy.davidson.edu/StuHome/jocampbell/projectile/projectile.ProjectileControl.html
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=145
http://math.fullerton.edu/mathews/N310/projects/e6.htm
http://wps.aw.com/wps/media/objects/877/898586/topics/topic01.pdf
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