Brakes work by diverting the gas energy away from the bore axis. The muzzle rises because the force vector is high above the pivot point (your hand). By diverting much of the gas up at the muzzle the resulting gas force pushes down on the muzzle and reduces muzzle rise. It also reduces the total recoil straight back - it has to, simple physics. The amount of recoil reduction depends on the amount of gas generated and the energy (pressure) of that gas at the ports versus the bullet weight and velocity.
Using the .375 JDJ as an example, with the brake efficency of 50% (probably average):
Relative bullet momentum* = 270 grains x 2000 fps = 540,000
Relative gas momentum = 48 grains x 4500 fps x 50% = 108,000
Total recoil momentum = 108,000 + 540,000 = 648,000
Percent reduction = 108,000/648,000 = 17%
So the total reduction is about 17%. Designed correctly that means no muzzle rise and 17% less momentum into your hand....but straight back instead of angular. I understand GB's pain.
* You can calculate momentum, recoil energy, etc. but relative momentum is simple and reasonably comparable - you can successfully argue to use another method. The 4500 fps gas velocity is the average from several sources - but it really depends on the muzzle pressure. The brake on a 14" .270 WCF will work better than one on a 14" .375JDJ for that reason.
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