An Analysis of the Relationship between the Air Pressure and Muzzle Velocity of an Air Cannon

Lyubomir Gankov

Introduction

Method

Results

Conclusion

In the twenty-first century, the term ‘air cannon’ most commonly refers to a hobbyist-build pneumatic weapon, which fires small projectiles for the amusement of its builders. A colloquial term for an air cannon is “spud gun”, in reference to the popularity of potatoes as a projectile in such devices. Some variants of the spud gun feature a combustion chamber rather than a pneumatic mechanism for launch of potatoes (and other projectiles)[1]. While mostly used for entertainment, there are commercial applications of the air cannon, such as the cleaning of storage silos, bins and hoppers in cement processing plants. This is exemplified by the AirBoost® Air Blaster[2], which is used to “clean the jamming or coating in any wall... and assist in a resistance free flow of the material”. No matter the application, be it amateur or commercial, the principles underlying the air cannon remain the same.

The concept of the air cannon is based upon the principle of conservation of energy, which states that energy is conserved. In the case of the air cannon, the potential energy of the pressurized air is converted into kinetic energy upon the firing of the cannon, with this kinetic energy propelling the projectile out of the cannon. This can be expressed through the following equations:

Ek = ½(m)(v2)               (1)

Ep = ½(V)(P2)              (2)

Ek = Ep                         (3a)

½(m)(v2) = ½(V)(P2)    (3b)

Where Ep is potential energy (J), Ek is kinetic energy (J), m is mass (kg), and P is pressure (Pa)

The work done to compress air into the barrel can be expressed through the follow equation:

ΔW = PΔV

W = ∫PΔV

W = ½(V)(P2)

Where W is work done (J), P is pressure (Pa), and V is volume (m3)

Pressure is defined through Boyle’s Law, which correlates pressure and volume inversely at a constant temperature:

P = (nRT)/(V)

Where P is pressure (Pa), n is moles of gas (mol), R is the characteristic gas constant (8.31 J mol−1 K−1), T is temperature (K), and V is volume (m3)

In this case, Boyle’s Law is used to demonstrate that as moles of gas (air) are added into the tank, which is of a fixed volume, pressure in the system increases. Thus, the work done in compressing air into the barrel is exponential, and is stored in the air cannon’s tank, which serves as a pneumatic capacitor, in that it stores energy based on a differential of pressure (between the inside and outside of the tank). The work done to compress the air into the tank is equal to the potential energy of the air inside the tank due to the law of conservation of energy. The kinetic energy transferred into the projectile is therefore fired with the energy put into the system with the work done to compress the gas.

There are several potential variables to be adjusted in the design of an air cannon, including barrel diameter, barrel length, air tank pressure, air tank volume, and projectile weight[3]. While of these variables have the potential to affect the projectile velocity, this study will analyze the effect of air tank pressure on muzzle velocity exclusively. However, even in this more narrow approach, there is still a broad range of issues to be addressed. One is the composition of the projectile itself; the density of the projectile affects the distance it has the potential to travel based upon the kinetic energy it can absorb[4]. Furthermore, the shape of the projectile affects the airflow through the barrel, as if it is not uniform (such as in the case of the potato, a popular projectile), not all of the potential energy that is released will be imparted as kinetic energy into the projectile as some of the air will escape around the object[5].

The purpose of this investigation is to determine the effect that air pressure has on the initial muzzle velocity of a projectile fired from an air cannon.

The relationship between air pressure, the independent variable, and resultant muzzle velocity, the dependent variable, will be a linear positive correlation. This will occur due to the underlying conservation of energy inherent to the pneumatic system of an air cannon: as air is compressed, the exponential amount of work required to pressurize it results in the potential energy in the system, from which velocity is quadratically derived.

Air pressure is defined as the gauge pressure of a tank (a closed volume), and muzzle velocity is defined as the initial velocity of a fired projectile. The control variables include, but are not limited to, the size and mass of the projectile, ambient air pressure and temperature, volume of the air tank, and the length and diameter of the cannon’s barrel.

(Diagram not to scale)

The cannon is placed flat on a table, with the photogates in front of the barrel, in order to minimize external sources of error and isolate the effects of the relationship between pressure and muzzle velocity.

1. Cannon

a.       (1) 2” (diameter) x 10’ length of PVC pipe (for tank/ barrel)

b.      (2) 2” to 1” PVC reducers

c.       (1) 1”(diameter) x 6” length of PVC pipe (for connecting solenoid with reducers)

d.      (1) 1” Anti-Siphon Sprinkler Valve

e.       (1) 2” PVC End Cap

f.       (1) Car Tire Valve Stem

g.      (1) Air Gauge (not pictured above)

h.      PVC Primer and Glue

Cannon Wiring*

.        (3) 9V Batteries

a.       (3) 9V Battery Snap Connectors

b.      (1) Rotary Button

c.       Electrical Tape**

Tools

.        Drill

a.       Drill Bits

b.      Jigsaw (for cutting PVC pipe)

Data Collection

.        (1) ping pong ball

a.       (2) Vernier photogate sensors

b.      (1) Vernier Sensor Hub

c.       (1) Computer

d.      (1) Air compressor

Safety Equipment

.        (1) Ball-catching device***

a.       (1) Pair of Safety glasses

b.      (1) Set of earplugs

* No additional wires were necessary, as the snap connectors, solenoid, and button all came with adequate lengths of wire.

** A soldering iron could be substituted in order to establish and maintain connections between wires.

*** In this experiment, a small cardboard box, stood up to face the cannon, was used to catch the fired projectiles.

Set up the working environment and have the air cannon assembled. Prepare the box for ball-catching, and put on earplugs and safety glasses.

First, fill the air tank with the desired air pressure, starting at 10 PSI. Then, load the ping pong into the barrel. Turn on the data collection in Logger Pro, and launch the ball by toggling the solenoid. Repeat this process for a total of 10 trials at each given PSI. Upon the completion of 10 trials at a PSI, increase the PSI by an increment of 10 PSI (up to and including 100 PSI). The end result will be a 10x10 data table of time values for PSIs between 10 and 100.

Table 1: Average Velocities at PSI (in increments of 10 PSI)

 PSI (± 1 PSI) Average Velocity (m/s) Uncertainty (m/s) 10 8.59 0.0095 20 21.78 0.0167 30 34.28 0.0253 40 48.26 0.0378 50 63.77 0.0501 60 74.70 0.0624 70 89.22 0.0732 80.00 101.23 0.0913 90.00 111.99 0.102 100.00 122.30 0.117

Figure 1: Graph depicting average velocities (m/s) versus tank pressure (PSI) with line of best fit (generated in Google Spreadsheets)

Raw Data: Text .:. Excel

Data file: Text .:. Excel

The hypothesis, which predicted that the relationship between the air pressure and muzzle velocity of an air cannon would be a positive linear correlation, was validated by the data gathered. After a total of 100 trials, with 10 individual trials performed for each desired PSI (from 10 to 100, in integer increments of 10), the average muzzle velocities correlated in a positively linear fashion to the air tank pressure values. The R2 value for the data set is 0.998 (as calculated through Google Spreadsheets), which shows that the correlation is almost entirely linear. The justification for the hypothesis, based on the law of conservation of energy, was that since the energy into the system should equal the energy out of the system, the velocity and pressure should be correlated linearly. The following equations show the mathematical derivation of this principle:

Ek = ½(m)(v2)                           (1)

Ep = ½(V)(P2)                           (2)

Ek = Ep                                     (3a)

½(m)(v2) = ½(V)(P2)                 (3b)

v = (P)(V/m)½                           (4)

Given that pressure and resultant velocity are correlated linearly with a constant of volume and mass through the law of conservation of energy, this experiment could be conducted with other volumes of tank or alternate projectiles and would yield very similar results.

As the experiment progressed, with trials being conducted at higher pressures, the uncertainties grow larger. However, for the range of pressures tested (10 PSI to 100 PSI), the uncertainties do not significantly affect the data gathered, and thus has little bearing on the outcome of the experiment.

While this lab made use of accurate sensors for the gathering of time data (for velocities), there are nonetheless points that potentially merit improvement. One such area is the use of a more accurate pressure gauge. The gauge used had an uncertainty of ± 1 PSI, which is very large relative to the ± 0.001 m/s of the Vernier photogates. Instead of using an analogue pressure gauge, a digital one could be used; this would most likely take the form of a digital tire pressure meter. However, the inherent risk of such a meter is that in order to measure pressure, some air leaks out as the meter makes use of the tire stem. The tradeoff between air released and accuracy in measurement of air pressure can be circumvented through the use of a built-in digital sensor, though.

Another source of error is the ratio between the diameter of the ping pong ball and the air cannon barrel. This ratio is 1.6”:2”, or 0.8. Thus, not all of the air released through the cannon imparts work onto the ball; a certain percentage of the air is ‘lost’ (about 35%, based on the areas of the ball and barrel, when these are viewed as circular cross-sections). This came about due to materials available; PVC pipe is only sold in specific diameters, and these diameters do not coincide with the diameters of any common small sports balls. One of these two diameters would have to be changed in order to mitigate the error that results from the loss of air pressure stemming from the disparity in diameters. The easier of the two to mitigate would be the diameter of the ball, as there exist many non-standard balls (not used for official sports), while it would be significantly more difficult to procure pipe of a nonstandard diameter.

Another instance of error in this experiment is the shape of the ball, and the manner in which it interacts with the photogates. Since the ball is spherical, it is very difficult for it to pass through both photogates with the same point of its surface triggering the sensor, at least with the influence of gravity and air drag acting upon it. Thus, the time data gathered is not wholly exact, as it does not reflect the precise moments in which the ball was within the sensor array.

A potentially significant source of error is the surrounding air pressure encountered by the ping pong ball as it is launched from the air cannon. While the photogates were set up 10cm from one another in order to minimize the effect of air pressure, the very light ping pong ball still is effected by air drag. In order to truly minimize this, the test could be conducted in a zero-gravity environment; however, since this is practically infeasible, the test could instead be conducted with projectiles of higher mass. This is such because the inertia of an object depends on its mass, and thus the ping pong ball is more affected by the air resistance than a projectile of higher mass would be (as an object of higher mass has greater inertia). However, the diameter of the higher-mass object would have to be around equal (less than or equal to) the diameter of the barrel in order for the error to be truly mitigated.

A further source of error is the length of the barrel, as the distance the ball travels before it encounters the photogates affects its velocity. The barrel is not frictionless, and therefore the ball is slowed by the friction of the sides of the barrel in conjunction with the air resistance. The effect of a longer barrel on a ping pong ball would be detrimental to the experiment, as it would noticeably alter the data gathered. A shorter barrel would be beneficial in the case of the ping pong ball, as it would minimize friction between the ball and the barrel and thus minimize the effect on the velocity of the ball as it goes through the photogates. However, in the case of a more massive projectile (either in size or in mass), a shorter barrel may negatively affect the data gathered as the ball may not have enough time to speed up (since heavier objects have a greater inertia). Thus, the optimal barrel length is dependent on the diameter and mass of the projectile selected. For the experiment conducted, a shorter barrel might prove beneficial so as to attempt to mitigate the effect of barrel friction on the data gathered. Likewise, the volume of the air tank is a variable that, if changed, could potentially mitigate error. PSI is a unit that measures the pressure of a container independent of volume; thus, an increase of volume could have a meaningful effect on the outcome of the data. The volume of the air tank would have to be tailored to that of the barrel (as it is recommended that the volume of the two be equivalent), which is further dependent upon the projectile used. In the case of this experiment, the volume of the air tank was roughly equivalent to that of the barrel; thus, an increase or decrease in the volume of the air tank without a proportional change to the volume of the barrel could actually end up being a detriment to the experiment.

Another potential (minor) source of error is the natural fluctuations in ambient temperature and air pressure: in order to mitigate whatever minute bearing this factor has on the data gathered, the experiment could be conducted in a temperature and pressure controlled-environment.

A final source of error is the condition of the ball used through the experiment. Ping pong balls, being hollow and made of soft plastic, are prone to external damage. After a certain number of trials, (of being launched from an air cannon) the ball ceases to be spherical. Even though the he deformities gathered by the ball over the course of the experiment are all rather minor, the use of several balls so as to mitigate any deformities required could alleviate some error inherent in the design of the experiment. However, with this solution, the masses of the balls would have to be similar so that a new source of error is not introduced. Overall, while there are only several major sources of error present due to the design of the experiment (ratio between ball and pipe diameters, ball mass, uncertainty introduced by analogue pressure gauge) there are many smaller sources of error that could be accounted for in order to produce a truly accurate set of data, that controlled for all but two variable so as to isolate the relationship between air tank pressure and muzzle velocity.

[1]"US Air Force Measures Potato Cannon Muzzle Velocities." MIT Technology Review. N.p., 8 May 2013. Web. 29 Nov. 2015. <http://www.technologyreview.com/us-air-force-measures-potato-cannon-muzzle-velocities>

[2]"ABO-Home-page." ABO-Home-page. N.p., n.d. Web. 30 Nov. 2015. <http://www.airboostiso.com/airblaster.htm>.

[3]Wise, Daniel G., and Mark A. Mitchell. Universal Air Gun Launch System (1992): n. pag. Defense Technical Information Center. United States Government. Web. 29 Nov. 2015. <http://www.dtic.mil/dtic/>

[4]"Theory/physics behind the Spudgun." The Spudgun Technology Center. N.p., n.d. Web. 29 Nov. 2015. <https://www.spudtech.com/content.asp?id=6>.

[5]Carl E. Mungan, “Internal Ballistics of a Pneumatic Potato Cannon”, European Journal of Physics Vol. 30 (2009) 453-457

Not How To Build An Air Cannon – The very first web page about air cannons, made circa 1993. Details Bill Mills’ air cannon design.

Spudgunner Air Cannon Designs – Compilation of several practical air cannon designs.

Lu Laboratories Air Cannons – The air cannon development of Chris Lu, featuring three iterations (Marks I – III).

Internal Ballistics of a Pneumatic Potato Cannon – A short research paper detailing the mathematics of a pneumatic potato cannon.

Pneumatic Cannon – A series of posts detailing the construction of an air cannon, designed to be mounted onto a 1/5 scale Sherman tank.

Pneumatic Antenna Launching Systems – Web page dedicated to pneumatic antenna launching systems. Very similar to air cannon in construction, has many related links of its own.