Physics and Bullets

I'm trying to add weight to my rifle so that I can increase it's accuracy and precision without destroying my right shoulder. So I started to think about how much weight I should add. And then I remembered my physics.

My rifle weighs about 8 lbs (3.6 kg) and generates a bullet muzzle velocity of 3000 feet per second. I'm not absolutely concerned with the actual force that the gun exerts on me, inasmuch as I am about the final velocity of the gun (since acceleration equals the change in velocity with respect to time, and initial velocity is zero). It's this backwards movement that makes it hard to aim and hard to fire accurately.
So, if momentum is conserved in that the bullet's forward momentum is equal to the rifles backward momentum, calculating the momentum of the bullet will have the same absolute value of the rifle's momentum. 3000 feet per second equates to 914 meters per second, and a 130 grain bullet is equal to 8.43 grams. Multiplying those two together and using kilograms, I get the bullet's momentum to be 7.699 kgm/s, which means that a rifle with a weight of 3.6 kilograms moves backwards at 2.139 m/s. Increasing the weight of the rifle will decrease the actual velocity of the rifle, although the momentum will still be equal. I want the rifle to move at 1.5 m/s, so I should increase the weight of the rifle so that it weighs 5.13 kg or 11 pounds.

For right now I think I'm going to remove the recoil pad, put lead shot in a sock or sturdy cloth bag, and see if the rifle performs better. It can't hurt, but it's crushingly difficult to hold the rifle steady out past 100 yards. I can shoot from a sitting position groups that are .3 inches at 25 yards. But that's like saying you can hold your breath for 15 seconds; it's not very impressive. But, that kind of accuracy should theoretically hold out so that the groups I shoot at 100 yards would be about 1 minute of arc, or a 1" group. But it's just not happening. The rifle is too light for me to hold steady; my idea of aiming is to wait for the crosshairs to flit across the bullring upon which I squeeze the trigger quickly lest the rifle's line of position moves down 4 or 5 inches from where I want the bullet to impact. Even if I hold steady, the bullet either strays, or goes off target completely.

It's unreasonable to demand a lower end hunting rifle that can shoot bullets all day long into a .5" ring at 200 yard, but if a rifle can't shoot at least a 3" group with good ammunition at 100 yards(and admittedly a group like that isn't considered great at 100 yards), then you have a serious problem on your hands. And unfortunately, I managed to hit a cardboard box that was 16"x24" twice at 100 yards. Tomorrow I'll get the lead out and see if it improves my performance.