Years ago I asked a fellow shooter that happened to be a physicist to calculate the recoil force of a gun based on bullet mass, bullet energy and weight of the gun. Well what I didn't tell him was that I would be working on the same problem from a engineering perspective.
He used an entirely different approach than I did. I nor he at no time knew how the other was approaching the problem. When he emailed me his solution I compared it to mine - the formula was exactly the same! Exactly!
1- Neither of us took the reciprocation of the slide into account; unquestionably valid for revolvers since they have no slides. And, I'm not sure the slide reciprocation would make much difference, but even if it did, if we are comparing semi-to semi, i.e. a G22 to a G17 the slide effects would be a wash anyway.
2- Resistance between the bullet and the barrel was not included. Could make some difference but how would you ever determine what the friction was?
The formula for recoil force we both came up with using different approaches is:
Er = Mb * Eb / Mg
Er = recoil energy
Mb = bullet mass
Eb = bullet muzzle energy
Mg = mass of the gun
Realizing that mass is proportional to weight and that in the formula the mass of the bullet and the mass of the gun is expressed as a ratio, the formula can be simplified to:
Fr = Wb * Eb / Wg
Wb is the weight, instead of mass, of the bullet in pounds, i.e. bullet grains divided by 7000
Wg is the weight of the gun in lbs.
Here's an example
A Federal .40 cal 180 gn vs a Federal 9mm 115 gn bullet. One fired from a G22, the other from a G17
The weight of the guns:
G22 = 22.9 oz or 1.43 lbs
G17 = 22 oz or 1.357 lbs
muzzle energy = 400 ft-lbs
bullet weight = 180 / 7000 = 0.0257 lbs
muzzle energy = This is a bit problematic, the specific spec says 356 ft-lbs, the chart says 400 ft-lbs, let's use 400.
bullet weight = 115 / 7000 = 0.0164 lbs
Fr(40) = 0.0257 lbs * 400 ft-lbs / 1.43 lbs = 7.18 lbs
Er(9mm) = 0.0164 lbs * 400 ft-lbs / 1.357 lbs = 4.83 lbs
% difference = ( (7.18 - 4.83) / 4.83) * 100% = 48.7%
Interestingly, simply assuming the two bullets to have the same energy, dividing 180 by 115 gives 1.56 which would represent a 56% increase in recoil. However there is a significant difference in the weight of the G22 and G17 and that's why the formula gave a lower percentage.
This is theoretical, but the bullet weights happened to be the very ones I was shooting when I perceived felt recoil to be about 50% more and I had no idea about the math at the time, I was simply going by feel.
And, the M1V1 = M2V2 relationship suggests that a 20 oz slide could move 0.2" by the time the bullet exits the barrel. That assumes a motionless frame and that the slide unlocks immediately.