For those of you that don't, I'm (currently, at least) the UKSGC's 3rd top poster with over 1900 posts, I've built many cannons (mostly pneumatic), I've been reading here for a long time and I've got some plans for some other pretty neat stuff. ... so I am pretty experienced,don't take my post count here at face value.

I fancied a bit of more theoretical discussion today, and although UKSGC is a great community, they don't take to scientific stuff that much.

I've acquired a pretty heavy lump of steel rod that fits my usual barrel better than I could have hoped for, and naturally, I developed some concerns about the safety of firing it. Would the recoil generated be dangerous?

As the student engineer I am, I set up a model of the situation to get an idea of what should be expected.

Some assumptions were made, the main one being that the system is adiabatic. Over a matter of 50 milliseconds or so, the effects of heat transfer are low, so no major errors should arise by assuming the heat transfer to be negligible.

In a pneumatic, on firing, the potential energy in the chamber will be converted to:

- Recoil kinetic energy

Piston kinetic energy

Projectile kinetic energy/Sabot kinetic energy

Heat energy lost in the cooling from the rapid decompression

Potential energy losses from blow-by, either past the projectile or the pistion

The kinetic energy of the muzzle blast.

With the aid of GGDT, both the projectile's energy and losses to the decompression cooling are known.

To actually obtain a result, there is a need to cut back heavily on variables. So, Piston KE and blowby losses are taken to be zero. Patently not true, but the numbers are many times smaller than the other energies.

<sup>1</sup>/<sub>2</sub>m<sub>Projectile</sub>*v<sub>Projectile</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup> + nRT<sub>final</sub> - nRT<sub>initial</sub> = pV<sub>initial</sub>

The initial chamber energy is known, as is KE<sub>Projectile</sub> and (nRT<sub>final</sub> - nRT<sub>initial</sub>). This can therefore be taken as a single constant, E, for which the value is calculated as:

pV<sub>initial</sub> - nR(T<sub>final</sub> - T<sub>initial</sub>) - <sup>1</sup>/<sub>2</sub>m<sub>Projectile</sub>*v<sub>Projectile</sub><sup>2</sup>

So: E = <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>

It is also known that the recoil momentum is equal, but opposite, to the sum of the projectile and gas momentum.

m<sub>Launcher</sub>*v<sub>Launcher</sub> = m<sub>Projectile</sub>*v<sub>Projectile</sub> + m<sub>Gas</sub>*v<sub>Gas</sub>

Bear in mind that the masses of both the gases, and the launcher are known constants, which leaves us with only the velocities of the launcher and gas unknown.

To put some numbers in the situation, I'll use the data I used myself.

m<sub>Launcher</sub> = 4.5 kg (10 lbs)

m<sub>Projectile</sub> = 0.3 kg (300 grams, or 0.66 lbs)

m<sub>Gas</sub> = .018 kg (18 grams or .63 oz )

v<sub>Projectile</sub> = 58.7 ms<sup>-1</sup> (193 fps)

pV<sub>initial</sub> = 1359 J (1002 ft lb)

T<sub>initial</sub> = 293 K (68 <sup>o</sup>F)

T<sub>final</sub> = 206 K (-89 <sup>o</sup>F)

Which makes nR = 4.64 J K<sup>-1</sup>

And the constant E is equal to 488.5 J (331 ft. lb.)

So from: E= <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>

we get: <sup>1</sup>/<sub>2</sub>* 0.018 kg*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub> * 4.5 kg*v<sub>Launcher</sub><sup>2</sup> = 488.5 J

And: m<sub>Launcher</sub>*v<sub>Launcher</sub> = m<sub>Projectile</sub>*v<sub>Projectile</sub> + m<sub>Gas</sub>*v<sub>Gas</sub>

becomes: 4.5 kg *v<sub>Launcher</sub> = .3 kg * 58.7 ms<sup>-1</sup> + 0.018 kg *v<sub>Gas</sub>

rearranged to: 17.61 Ns = 0.018 kg *v<sub>Gas</sub> - 5 kg *v<sub>Launcher</sub>

So, this leaves a pair of simultaneous equations. Solving them to find v<sub>Gas</sub> and v<sub>Launcher</sub>:

v<sub>Launcher</sub> = 4.62 ms<sup>-1</sup>

v<sub>Gas</sub> = 177 ms<sup>-1</sup>

v<sub>Gas</sub> is of course, an average value. But, looking briefly at it, you can see that 177 m/s for 18 grams of gas is not an insignificant figure. With a lighter projectile, and noting the trend that lighter projectiles are accompanied by higher gas velocities, it might in fact turn out that the momentum of the gases is actually far greater than the momentum of the projectile.

The number we needed was v<sub>Launcher</sub>, which we then convert to recoil energy:

<sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>

For an energy of 48.1 J (35.5 ft lb.)

To get an idea of what this meant, I also calculated values for a few other projectiles I've used before:

Paintball recoil energy: 4.1 J (As I said before, in this case, most of the recoil energy is generated by the gases, not the projectile.)

25g spud slug recoil energy: 7.9 J

50g steel dart: 12.1 J

80g steel dart: 16.9J

Judging by these numbers, I need to do some more thinking about what 48.1 J actually means.

So, I went and found some typical firearm recoil energies. Seems that 48.1J is in the realm of a mid range game hunting rifle.

Personally, I've only ever fired a couple of different firearms cartridges, the .22 LR, and the 5.56x45 L2A1 Ball NATO, both well known for their low recoils, of about 1.8 and 5 J respectively, in typical weight rifles.

Ignoring the fact that I can get nearly as much recoil from firing a paintball as from the most common military round, I think that 48 J should be a manageable, but very solid kick.

However, I have a strong warning:

We have all heard tales of water filled cannons breaking. Launchers, particularly PVC ones, won't respond well to high recoil energies.

The 300g projectile I am considering is not much under the mass I would be firing with a full barrel of water, but I have a very carefully constructed metal cannon, which was specifically designed to handle recoil as best as possible, and even then, I will be taking extreme care.

Above certain limits, recoil may harm you or your launcher - please consider the potential consequences of heavy projectiles before you fire them.

I know this model only takes into account launcher weight, and not the user's hands, shoulder or anything else, but in that split second, the rigid launcher itself will primarily absorb the recoil energy over the more elastic user, so it's not too inaccurate.

Also, I apologize, I haven't got any information on combustions or hybrids. The heat losses complicate the matter significantly, and the increased dependence on C:B ratio makes it much harder to create a good algorithm, so I can't currently offer much advice to combustion spudgun users other than just being very careful with heavy rounds.

To get an idea of recoil distance, I took a definite integral of the area under a trapezium using a simplified model for the human system, treating it as just a preloaded spring.

This is an edgy model, but at a guess I can expect between 2 to 4 inches of recoil.

And that about wraps it up. I hope I didn't bore you to death, send your brain dribbling from your ears, or confuse you too heavily.

If you got this far, thanks for reading. I hope this gets you thinking a bit about recoil energies, and also helps you stay safe or avoid damaging your launcher.