Knockdown, Taylor Knockout, and Matunas' Optimal Range for Big Game are all based on momentum, not kinetic energy. The difference becomes important if the game is large enough or dangerous enough that the hunter wants to be able to break bones.
The often quoted comparison is the KE of a hot-loaded 22-250 is equal or greater than an over-the-counter Rem Core-Lokt 45-70. Both will kill a medium sized whitetail deer, but only one is effective on black bear and bigger game, including bison. KE= 1/2 m times velocity squared; Momentum is m times velocity.
To compare KE vs M dig out a couple boxes of factory ammo. Divide the published KE by the velocity and multiply that answer times 2 to get the momentum of that load.
The diameter of the bullet adds another important factor as does expansion. Now the formulas get complicated.
Yes, recoil is also calculated via momentum. Newton's law of action/reaction is momentum based, not KE based.