Ouch - I really screwed that up. I was in the middle of two divergent methods, was late for a meeting and hit submit when I should have deleted it.
First, I was trying to simplify the problem down to the differences between the two events (Smokeless powder vs Black powder). And I got one part very right: It is the total time the event takes that makes the recoil difference. And I had another basic idea right, that you can cancel out many differences: Same basic rifle weight, same shoulder to push, etc.
Measuring perceived impact is neither a pure Force event not a pure Energy event. It is a (fast) Work event. And measuring Work can be tricky.
In a lab, we would set up two rifles and two identical resistance weights to model the shoulders. Then we would film them both at very high speed and plot two key things: (1) How long did the total event (recoil absorption) take; (2) What did the event curve look like.
The event with the highest peak curve (instantaneous Work per second) would have the highest perceived impact.
Neither Force nor Energy can quite get you there.
And I was in such a hurry, I switched from Force to Energy in mid calculation. (I do better math when drinking than in a hurry.)
F = M*A - my attempt at force was decent on the "A" Acceleration = feet/second/second. But the M is dubious. I was trying to calculate based on the bullet Mass, but there is no telling how much of the actual blast goes into moving the bullet and how much goes into noise, heat and lost gas. Probably almost impossible to measure. (The lab equipment to measure this kind off event is called a "bomb" for a reason.)
KE = 1/2 M*V^2 has the same challenges. What "M" to measure. So none of my shorthands was going to get you there, and I apologize to those who wasted time trying to make sense of my incorrect methods.
Sorry about that ...
