Anyways, here's the method. I post only so that the physics and math buffs can point out errors (if some exist) and suggest a more optimistic model.

So, basically, it goes a little something like this:

The first things to assume, for very optimal conditions and ease of calculation are:

- Instead of balancing ambient pressure and taking into account the column of air after the projectile in the barrel, just assume the cannon is firing into a vacuum. Use the absolute pressure for these calculations, (i.e. if the operating pressure is 202 kpa, then the absolute pressure is 101 kpa).
- Assume there is no friction in the barrel.
- Assume you are using a 100% efficient valve, even better than a burst disk. Also, assume your propulsive gas has no limits to how fast it can expand. Basically, it will expand infinitely fast unless something gets in its way.
- Assume that all work done by the gases is transferred directly into kinetic energy of the object. No energy is lost.
- External ballistics also occur in a vacuum.

These parameters are very optimistic, but if you prove that even under these conditions, the cannon cannot make the range claimed, then the claim is disproved.

First, you need to figure out the work done by the projectile.

Using a composite function of Boyle's Law and simple geometric formulas for volume, you can derive an equation that yields pressure as a function of distance. Multiplication of the barrel area divided by ten will give the force as a function of distance if the pressure is in terms of kPa (1 kilopascal = 1 newton per 10 cm<sup>2</sup>). For ease of calculation, this is left undistributed into the function. The function is then simplified as much as possible.

Finally, you integrate the function from 0 to d to determine the work done on the projectile. Since all work is transferred into kinetic energy of the projectile, divide this energy by half the mass, and then take the square root to determine the muzzle velocity.

Now, it is a simple range calculation in vacuum to determine whether their claim is completely bogus or maybe has some merit to it...

So, here's a sample problem:

Initial Pressure: 800 kPa

Chamber volume: 1,000 mL

Barrel diameter: [4/sqrt(pi)] cm ≈ 2.26 cm

Barrel length: 100 cm

Projectile mass: 1 kg

Boyle's Law

Boyle's Law Rearranged

Composition into P as function of d

Simplified...

Multiplication by surface area divided by ten (simplified to 2/5)

Integral of force with respect to d (work)

Factoring out of constant to get integrand in the form of u'/u

Evaluated and simplified integral

Transcendentals approximated to rationals...

Equation for kinetic energy...

Posulated Approximate Muzzle Velocity

And, from here, you can calculate the range in a vacuum easily through standard high school physics.