Pneumatic Ballistics help

[ninja57]

Ok. Me and my friend have a pneumatic potato cannon and want a fair prediction of what the distance will be once we increase the pressure. Here are the results from previous firing:

29psi - 313ft.(199.9479614kPa - 95.4024 meters)

60psi - 467ft.(413.6854374kPa - 142.3416 meters)

80psi - 582ft.(551.5805832kPa - 177.3936 meters)

90psi - 630ft.(620.5281561kPa - 192.024 meters)

Assuming that all tests were fired at a 43 degree angle and the air temperature was 70 fahrenheit(21.1 celsius) and that all potatos fired were 2.74oz(77.678grams) and 1.5in(38.1mm) in diameter. The barrel is 1.5in(38.1mm) in diameter and 10.2916ft(3.13687968meters) in length. The potatoes fit tightly in the barrel so that's important too. Leave out the aerodynamics as that is too hard for me to determine. Can anyone figure out what the distance will be at 100psi(689.475729kPa) and 115psi(792.897088716kPa)? I have been working on this for weeks with no luck and I would be very grateful if someone can get the answer.

[Schrödinger's hat]

That's not a lot of info to go on.

From this data I can't tell if you'd be limited by energy stored (in which case force goes down significantly as the potato moved down the barrel), or more in the domain of constant force, or somewhere in between.

How big is the reservoir?

 

Getting into the theory, the distance will be roughly proportional to the initial energy (wiki if you want derivation)

Now the initial energy is either going to be proportional to the pressure (in the limiting case where you have an infinite reservoir of pressure), ie. [math]P\propto F[/math] and [math] F\times distance = energy[/math].(distance down the barrel is constant) In which case you want to extrapolate linearly.

 

In the limiting case where you have no excess gas (it reaches 1 atmosphere before ejecting the potato) your energy will instead be proportional to the log of the pressure.

 

A quick plot of the data seems to indicate that it's roughly linear (although over a small enough range that a log plot doesn't look much different).

 

Graph of distance in metres vs pressure in kPa

 

 

Acting on an assumption of linearity, a quick mental calculation yields 680 feet for 100psi, and 700 for 115.

 

If you care about getting more precision and some error bounds (assuming the linearity assumption of linearity holds), then a linear regression would be a good idea.

 

 

 

Without more data, it's hard to tell how accurate this model is. The only thing I can think of if you can't take more data is to do the calculations from the pressures (maybe including air resistance or an approximation thereof), and see if those distances pop out.

It would help to know exactly why you wanted to know this and whether or not you could take any more data.

The constant pressure assumption would also need to be justified (what is the volume of the section of the gun which the potato does not pass through?)

Edited December 13, 2011 by Schrödinger's hat

[ewmon]

I threw the Range (ft) vs Pressure (psi) data into Excel, and assuming a linear relationship, obtained Range = 5.24∙Pressure + 158.7, with an R² of 0.999. Extrapolating, I obtained 683 feet at 100 psi and 761 feet at 115 psi.

 

BTW, I also solved for muzzle velocity from the classic ballistic equation R = V²∙sin2α/g (which required knowing that α was 43°), and even at 29 psi, it's 100 fps (68 mph), so don't stand in front of the muzzle. At 115 psi, it's 157 fps (107 mph), and almost 2½ times the kinetic energy at 29 psi.

[DrRocket]

Given that this is a physics forum, rather than offering estimates of distance based on questionable assumptions paltry data, perhaps one ought to identify the physics at work.

 

First, neglecting aerodynamic effects (which may be very important with an irregular body like a potato) the distance traveled over a flat surface is easily calculated if the initial velocity and launch angle are known. Without the launch angle, essentially nothing can be determined. Incidentally in this idealization it can be shown that maximum distance occurs with a launch angle of 45 degrees.

 

Second the muzzle velocity will be dependent on the volume of propelling gas available, and the length of the barrel. If you assume an ideal gas, an infinite volume of available gas (approximated by a reservoir that is large with respect to the volume in the barrel at the time of exit of the potato), and reversible adiabatic expansion then the pressure will be constant and the work done on the potato during travel in the barrel is just pressure x barrel cross-sectional area x barrel length which determines the velocity if the mass of the potato is known. From that and launch angle you can calculate distance.

 

If the available gas volume cannot be approximated as infinite, then the situation is more complicated and you need to know what the gas is, particularly the ratio of specific heats. If the physical situation precludes significant heat transfer during the shot, you can then do an isentropic expansion calculation and again determine the initial velocity. If there is significant heat transfer going on then the situation is much more complicated and you have to deal with coupled equations -- computer simulation time.

[Schrödinger's hat]

761 feet at 115 psi.

Apologies, I misread 115 as 105 somehow.

As Dr Rocket said, this extrapolation is based on some assumptions. Namely:

Distance linear with launch energy, which only holds with no air resistance and,

Launch energy linear with pressure, which only holds with some questionable assumptions about the gas reservoir.

 

 

You also haven't really told us what you wanted this for. If you just wanted a ball-park 'about there' and then adjust once you are in the field, then I'd go with my earlier (rather lazy) approach. If you want to understand how and why it went that far, then going into more detail is a good idea.

 

 

Given that this is a physics forum, rather than offering estimates of distance based on questionable assumptions paltry data, perhaps one ought to identify the physics at work.

Indeed. As you said, the assumption about the amount of gas available needs to be validated/replaced before this is useful.

 

First, neglecting aerodynamic effects (which may be very important with an irregular body like a potato) the distance traveled over a flat surface is easily calculated if the initial velocity and launch angle are known. Without the launch angle, essentially nothing can be determined. Incidentally in this idealization it can be shown that maximum distance occurs with a launch angle of 45 degrees.

 

A simple way of checking how significant aerodynamic effects are would be to compare the kinetic energy of the potato to an estimate of the air resistance force at launch velocity * distance travelled

If this number is small then it can reasonably safely be ignored. Else it should be included in the model.

[ninja57]

Ok guys I won't be able to do ant tests at the moment, but I should have the volume of the air tank soon. I plan to fill it with water and measure that. Also in response to DrRocket, we use a small piece of pvc with the same thickness and diameter to cut the potato down so its circumference is perfectly round so it's sides are streamlined, however the front is mostly flattened. Also, Schrodinger's hat, a ball-park estimate was all I really wanted and thanks for the calculations. I didn't know how to do it without it being linear graph, and since the actual distances weren't linear, it was off of Schrodinger's hat's predictions by at least 45-50ft. When I get the results from the air tank, I'll immediately post them (ft³ and cm³ forms).

[Schrödinger's hat]

When I get the results from the air tank, I'll immediately post them (ft³ and cm³ forms).

 

Even an estimate is enough to know if the assumption is any good (ie. is it bigger than the barrel? what would be the estimate for the ratio of its radius to the barrel radius, and length to barrel length?)