If wearing body armor that could effectively stop a 12 Gauge Slug from penetrating, would the ridiculous momentum from the slug Knock the wearer over, if not send them flying back? A 1.25 Oz Slug @ 1551 FPS delivers 2920 Ft/Lbs of kinetic energy and 121 Ft/Lbs of momentum. What would happen?
Depends.
Ever shot a 12 gauge with magnum rounds? Either 3" magnum or 3 1/2" super magnums? You need to snug that butt into your shoulder and lean into it. The same amount of force felt at your shoulder would be felt upon impact with the armor. You can easily bruise your shoulders, especially if you insist on shooting a lot of them.
If you're not already braced or otherwise unbalanced in your stance, yes it could topple you or otherwise knock you for a loop. "Stagger" would probably be a good word to use. But that's because you're a biological, bipedal machine. That means your balance, especially in motion, is less stable than, say, an inanimate object like an equivalent mass in the form of a block. You are also more flexible.
Send you flying back? No.
It's not energy which pushes something over. It's force. Force is equal to mass times acceleration. Therefore, acceleration is equal to force divided by mass.
For a given impact force, the larger the target compared to the slug, the less it will accelerate.
SO...let's look at it one way:
Assume the mass of the slug is one ounce. A 180 pound person after being shot has a mass of 2,881 ounces. (180 pounds times 16 ounces per pound, plus the slug).
The force the bullet delivers (mass times acceleration) has to be calculated from momentum. How the bullet decelerates determines its force. And this depends on the slug's terminal performance.
Acceleration would be calculated by the change slug in velocity upon impact divided by the change in time. You need to use the inelastic equation for this, since the slug does not pass through the person.
These calculations require you to make some assumptions with respect to the change in velocity of the slug and the time it takes for that to happen. And that's difficult.
BUT...using momentum equations straight up, you can get a better feel. Momentum is mass times velocity.
Momentum is always conserved.
So momentum of the bullet is the mass of the bullet times the velocity of the bullet.
Momentum of the target is the (mass of the person + mass of the bullet) times the velocity the combined masses.
If you look at the relative difference in mass (1 ounce vs 2881 ounces), you can see that whatever the initial velocity of the slug is will be radically reduced for the combined mass of the person and the slug together.