[Unclenick]

Originally Posted by Jimro

]Yes, forward energy imparted to the bullet at the muzzle, plus energy lost to muzzle blast should equal the rearward energy of the rifle in a perfect system.

No, it looks like I caused this confusion. When I said, "Peak energy is the same as final energy for the gun", I meant the peak energy of the recoiling gun will equal the final energy of the recoiling gun at the moment the bullet and gas have cleared the muzzle; not that the gun energy will equal that of the bullet and powder mass. It won't come close. A typical hunting bullet has about 2000 ft-lbs of muzzle energy, but a typical hunting rifle firing it will have only about 15 ft-lbs of free recoil energy. It is momentum that is equal and opposite, not energy.

Originally Posted by Jimro

But you missed the point, VELOCITY is dependent on TIME. The longer the time, SPEED = DISTANCE/TIME the slower the velocity. Acceleration is not the same as barrel time which is a velocity. A bullet that leaves the bore at 2200 FPS will have a LONGER barrel time than the same bullet leaving the bore at 3200 FPS. The faster bullet from the same barrel had a greater acceleration to get to that final velocity, and therefore less TIME in the barrel.

But you missed the point. The object of the discussion is not either velocity or speed, but what causes felt recoil to change with bullet weight. That is not dependent on elapsed time alone, but rather depends on the recoil energy which varies with the combinations of gun mass, time and force applied to the breech of the gun. For an extreme example of there being no unique dependence on time, having a small grandchild sit on your lap for five seconds would not have the same felt effect as having morbidly obese aunt Bertha sit on your lap for five seconds. Even though the amount of time is the same, you will feel the difference in force imposed by gravity on these two affectionate creatures.

Similarly, in the gun, there are two ways to increase the barrel time. One is to reduce powder charge, which lengthens barrel time by reducing the force accelerating the bullet, simultaneously reducing recoil. The other is to increase the bullet mass so the exact same increase in barrel time is needed to get it out of the muzzle, but that does not reduce the accelerating force. So now you have the same original force applied for a longer time, accelerating the gun mass more, increasing recoil.

In the real world its not that simple because progressive powder will burn faster and make more pressure with a heavier bullet because it can't push the greater weight to increase the powder burning space (expansion) as quickly. So powder charge has to be reduced or the load has to be switched to a slower powder to keep the pressure profiles the same.

Originally Posted by Jimro

Acceleration is given in distance per time per time, such as feet per second per second. Newtonian physics again, the greater the acceleration on the bullet, the greater the acceleration on the rifle.

For a given gun mass, that's only true if you keep the bullet mass the same. The premise of the OP, however, is that the bullet mass will be changed.

Originally Posted by Jimro

Follow me on a thought experiment. Two bullets of different mass have the same barrel time. The barrel time being the same means that the recoil should be the same? Obviously not. It takes more energy to push the heavier bullet at the same velocity.

Yes, we agree that more energy in proportion to mass has to transfer to the heavier bullet to give it the same velocity, though that's not what is normally done. We normally limit the force on the bullet in both cases by limiting pressure to something the gun can withstand. That means accepting lower velocity with longer barrel time for heavier bullets. Indeed, typically, if guns are optimally loaded for each round by adjusting powder burn rate and quantity, the heavy and slow bullets exit with similar levels of muzzle energy, but the recoil energy is not the same. Again, energy is not equal and opposite, only force and its resulting change in momentum are, and accelerating the gun rearward for a longer time with the same level of force will give it more velocity and more momentum and with that comes more recoil energy.

Originally Posted by Jimro

Second, the "jet blast" effect of all that hot gas shooting out the muzzle at around 5,200 fps. This is where the bulk of felt recoil comes into play. That heavier bullet with the same barrel time as the lighter bullet is going to have more pressure behind it when it leaves the muzzle, and this is where you are going to get the biggest portion of recoil.

That would happen if the heavier bullet were driven to the same velocity as the lighter one, but, again, that's not normally the case because that would require a pressure increase.

I will add that while it is true with some overbore guns that muzzle blast can be the majority contributor to recoil energy, with most common calibers, like a .308, it's responsible for about a third of the recoil energy. That's because it gets about a third of the momentum and only momentum, not energy, is equally and oppositely created in the gun.

Originally Posted by Jimro

E = 0.5mass*velocity squared, so that bullet with a 150 grain mass going 2800 fps pushed against the rifle less than that 45 grains of powder going 5200 fps.

The root of the problem is expressed there. It is that you are calculating energy on the assumption it will be equal and opposite in the gun and ejected mass. If you apply the same force to the bullet and gun breech in opposite directions, you'll find the acceleration of the bullet and the gun mass are inversely proportional to their masses. That makes the resulting velocities at the end of the barrel time inversely proportional to their masses. If you multiply these oppositely directed masses by their oppositely directed velocities you end up with two equal and oppositely directed quantities. Velocity times mass is momentum, so momentum is what those equal and oppositely directed quantities are called.

Energy, on the other hand, is proportional to half the square of velocity times the mass, so square the velocity ratio and divide by the mass ratio (same magnitude) and also by 2. If the mass ratio is about 250:1, rearranging you have: 250²/(2×250)=250/2= or 125 times more energy in the bullet and gas mass than in the gun. The bottom line is the bullet gets more energy out of the powder than the gun does, basically as a result of having less inertia and therefore being easier to move.

So, now we see momentum rather than energy is equal and opposite, let's run a calculation based on the momentum.

150 gr = 0.0214 lbs.
45 gr = 0.0064 lbs.

0.0214 lbs + (0.0064 lb/2) = 0.0246 lbs × 2800 fps = 68.9 lb-ft/s momentum added to the bullet and half the powder mass during their time in the barrel.

0.0064 lb × 5200 fps = 33.3 lb-ft/s momentum added to gas expelled at muzzle.

68.9 lb-ft/s + 33.3 lb-ft/s = 102 lb-ft/s total momentum.

The gun weighs 8 lbs with scope. Momentum will be equal and opposite, so:

102 lb-ft/s / 8 lb = 12.8 ft/s gun recoil velocity.

For kinetic energy, the lbs in ft-lbs is pounds force (commonly also written as lbf), the amount of force needed to accelerate 1 lb of mass (1/g slugs) at 1 ft/s². The gun weighs 8 lbs on a scale because gravity imposes enough enough force on it to do that. But gravity isn't involved in the force from gas pressure in the gun. That force acts on the mass of the gun, not its weight, so we have to get that mass from the weight by dividing by gravity.

8 lb/32.2 ft/s² = 0.248 slugs.

mV²/2 = 0.5 × 0.248 slugs × (12.8 ft/s)² = 0.124 slugs × 163 ft²/s² = 20.3 ft-lbs recoil energy.

If you rework this example for a 9 lb rifle, you wind up with 16.0 ft-lbs of recoil, so you can see how gun weight is a significant factor.

That bullet had 0.5 × 0.0214 lb/32.2 ft/s² × (2800 ft/s)² or 2610 ft-lbs of energy. Lets switch to a 180 grain bullet also propelled to 2610 ft-lbs of energy by 45 grains of a slightly slower burning powder (so the pressure doesn't increase). It will be traveling at about 2560 fps.

We now have:

180 grains =0.0257 lbs
45 gr = 0.0064 lbs.

0.0257 lbs + (0.0064lb/2) = 0.0289 lbs × 2560 fps = 74.0 lb-ft/s momentum added to the bullet and half the powder mass during their time in the barrel.

0.0064 lb × 5200 fps = 33.3 lb-ft/s momentum added to gas expelled at muzzle.

74.0 lb-ft/s + 33.3 lb-ft/s = 107 lb-ft/s total momentum.

The gun is still 8 lbs, so:

107 lb-ft/s / 8 lb = 13.4 ft/s gun recoil velocity.

8 lb/32.2 ft/s² = 0.248 slugs.

0.5 mV² = 0.5 × 0.248 slugs × (13.4 ft/s)² = 0.124 slugs × 180 ft²/s² = 22.3 ft-lbs recoil energy.

Reworking for a 9 lb gun it comes out to 17.6 ft-lbs.

Bottom line, firing the heavier bullet to the same energy level, as is common since the case only can hold so much stored energy, the heavier bullet recoils more.