What is the math behind the physics of an air cannon?

[Dragonrider99]

Greetings all. My son has asked me to help him build a small air cannon. He is very interested in the sciences, especially physics, and always wants to know as much as possible about things of that nature. The cannon we are building has the following specifications:

It will be made out of sch 40 3 inch pressure rated PVC as follows:

pressure chamber: 3" x 8"
Barrel : 3" x 24"

We plan on using approximately 40 psi in the chamber (or less)

His questions to me, which I cannot answer simply because I have no idea, is the math behind the physics of the air cannon. Based on the dimensions and psi that I have provided, and assuming we will be shooting a tennis ball 2.7" diameter, can anyone explain, as simply as possible and show the formulas, for how we can derive initial velocity, final velocity, acceleration out of the barrel, etc. If I need to provide any other info please ask. I am totally lost when it comes to the math, but my son loves this stuff and really wants to understand how it will all work together. It is tough not being able to explain to your own child when he needs help. At least we will build it together and have fun doing that lol

Thanks in advance for your kind help.

 

[DocZaius]

A good starting point for simple physics would be to take a video of the projectile motion of the tennis ball, download video tracking analysis software, and see how different the trajectory is from the one that might be expected in a vacuum given similar initial velocity and angle. I suspect aspects of the problem relating pressure in the chamber to velocity of the ball would best be approached empirically (by how much does a change in psi change the range of the projectile?)

 

[Dr. Courtney]

The physics is well described in these two papers:

http://arxiv.org/pdf/1106.2803.pdf

http://www.ewp.rpi.edu/hartford/~ernesto/S2012/EP/MaterialsforStudents/LeBlanc/Mungan2009.PDF

The math is beyond most high school students.

I like the idea of using video to determine muzzle velocities and relating muzzle velocities to trajectory. The math is much easier and forms a foundation for a lot of interesting physics.

Almost all cannon designs (pneumatic and combustion) benefit from a chamber that is 30-60% larger in diameter than the barrel. We've tried a few designs with the chamber and barrel having the same diameter, and they all have been disappointing.

Let me also add, combustion cannons are much easier (and cheaper) to build, and also tend to be more satisfying for students. Operating pressures will likely be below 20 PSI with a tennis ball projectile and common fuels.

 

[samisbored]

Cool man! sounds like a fun project to do with your son, my father and I made several potato cannons it was awesome.

biggest thing that helps air cannons and all projectile based weapons successful is a proper seal of the intended propelled object and chamber/barrel. Too tight = energy is lost due to friction also too loose = energy is lost due to propellent not getting enough time to actually propel simply put. I usually remedy this situation in two ways when using potato cannons.

Too lose: line the projectile with a dry rag to keep gas from slipping past.
Too tight: lube the barrel with WD-40/water/whatever really.

but back to the questions:

initial velocity:
automatically 0 (the tennis ball remains stationary in the barrel prior to launch)
(not trying to be sarcastic)

final velocity:
automatically 0 (the tennis ball will not be moving once it's done with its trip)
(not trying to be sarcastic)

acceleration out of the barrel:
the maximum velocity of the tennis ball will be 1/1,000,000 of a second before the seal is broken between the tennis ball and barrel. the gas expanding when it leaves the barrel behind the tennis ball will give it a tad more power but mostly its negligible. the tennis ball is accelerating the entire length of the barrel IF the pressure in the chamber/barrel is not equalized before it reaches the end of the barrel. if the pressure in the chamber/barrel equalizes before it reaches the tip of the barrel the tennis ball will be stuck inside, so barrel length to propellent ratio is rather important I'd automatically assume. remember more potential energy = more kinetic energy in firearms, more powder/air = more boom.

hmmmmm. with no research into the topic at all and if it was just me sitting in my backyard with my son, here is how I would figure out what you're asking i think.
know these things:
1. as soon as a projectile leaves the barrel of a weapon it stops accelerating.
2. wind resistance is the main slower of projectiles.
3. things like elevation come into play (how far above sea level). the flight path of bullets is completely different when you're on top of a mountain or sitting on a beach (different amounts of air to come in contact with). most weapons are initially tested as close to sea level as they can possibly get for this reason (that's why you see a lot of naval testing in open water for firearms).

this method uses the least amount of math possible (this method will tell you max. velocity).
1. position the cannon at 0 degrees. flat
2. set up a target (piece of wood or better yet something that will catch it because getting hit with it would suck) roughly 10 feet from the chamber of the cannon. you said that your barrel is 24 inches so 8 feet from the tip of the barrel. this will give the tennis ball exactly 10 feet to travel.
3. record the entire thing with a decent camera, a cellphone will do.
4. fire it and make sure its recording... do a few of these so you have a few clips to study.
5. slow the video down so you can time exactly how long it takes from ignition to impact.
6. since 5280 feet is a mile all you have to do now is multiple the time from ignition to impact by 528 since 10 rounds into it exactly.
- being that 8 feet from the barrel is rather close so things like wind resistance wouldn't really come into play yet so you'd have a SOMEWHAT accurate top speed of this system.
- you can test several different PSI in chamber using this method to see where your cannon tops out at.If you're trying to figure out flight path... math will be needed
https://en.wikipedia.org/wiki/Trajectory_of_a_projectile

Good luck man and have fun!