Free recoil has a square term but it the velocity of the firearm and has little to do with the energy of the projectile.
What is felt recoil? It's too subjective to put a number on "felt recoil". Recoil is a really nasty nut to talk about. There are many ways to put a number on it depending on the context we want to examine. The three most common are recoil impulse, free recoil energy and peak recoil force.
Recoil Impulse is the simplest. It is simply the change in momentum. Mass x Velocity of everything spit out the end of the barrel. It unfortunately leaves a lot of details out of it as it lumps all of it together. It also does not take into account what it is being fired from. (ie 357 mag from a 12 oz J-frame hurts, the same cartridge from a 40 oz N-frame, not too bad).
Free recoil energy is a theoretic metric, that does take into account the weight of the firearm. Free Recoil Energy is what Kinetic Energy would the firearm have if left to go the opposite direction from the bullet when fired if attached to nothing (ie zero-G). It is calculated off the above Recoil Impulse. A law of physics is that momentum is always conserved. If ,in our theoretic unsupported firearm setting, the bullet and propellant go one way with X amount of momentum the rifle must go the other way with the same amount of momentum. So the following has to be true in that setting. Mb x Vb + Mp x Vp = Mf x Vf. [ M = mass, V = velocity, b = bullet, p = propellant gases, f = firearm] Momentum does not sit well in most peoples heads so its common to take the resulting velocity of the firearm from the conservation of momentum equation and calculate the resulting kinetic energy the firearm would have. Free recoil energy is then 1/2 Mf x Vf^2. Notice that if you calculate the kinetic energy of the bullet and propellant it will NOT be the same as the firearm (good thing too). But in reality a firearm rarely achieve its free recoil energy since you are holding (hopefully firmly) onto the firearm and your body mass and muscle tension/elasticity effect the recoil greatly. But it is a good number to compare that takes into account the cartridge and the weight of the firearm it firing.
Another metric some measure is peak recoil force and impulse time. This take some good measurement equipment that can measure forces very fast. It is very subjective because the force of recoil will change a lot depending on how the firearm is supported in the recoil force measurement system. It is really only valid for measurements used comparatively when taken in the same rid setup the same way. Recoil pad manufactures love this number as a soft recoil pad does nothing for to recoil impulse or free recoil energy but it does reduce the peak force and spread the recoil out over time making it feel softer. This method is also the only one that will capture the felt recoil reduction most shooter experience when shooting a semi-auto compared to a similar weight and caliber weapon of a manual action. If this measurement is done correctly then if you integrate the force vs time curve you should arrive at the same number as the Recoil Impulse calculation.