Felt Recoil Equation

[rebliss]

I was talking with a buddy yesterday about the difference in felt recoil between bullets of differing weights. We know that typically, a lighter bullet will give less felt recoil. I wanted to know why.

:killingpc WARNING! I'm going to use some physics here at the risk of boring some people. The reason is that I know some of you fellows will be able to explain what I'm curious about.

Given: Two otherwise identical bullets fired from the same rifle under identical conditions.
Bullet 1: 151gr.
Bullet 2: 200gr.
Bullet 2 Velocity = 1700 fps

First, determining Mass of each bullet:

Bullet 1: 151gr/(7000gr x 31.174ft/sec/sec) = 6.705 e-4
Bullet 2: 200gr/(7000gr x 31.174ft/sec/sec) = 8.880 e-4

Next, determining what velocity the lighter bullet will need to give equal FORCE:

F= M x V(squared)

6.705 e-4 x V(2) = 8.880 e-4 x 2890000
6.705 e-4 x V(2) = 2566.40233
V(2) =3827814.57
V=1956 fps

So, a 200gr bullet traveling at 1700fps. gives equal FORCE to a 151gr bullet at 1956 fps. In my mind, equal force should equate to equal felt recoil. But what surprised me is that momentum is different for each bullet. If MOMENTUM equates to felt recoil, then I can understand the reason for the difference.

MOMENTUM = MASS x VELOCITY

SO

MOMENTUM (151) = 6.705 e-4 X 1956 = 1.311 lb-sec
and
MOMENTUM (200) = 8.880 e-4 x 1700 = 1.510 lb-sec

Therefore, even for cartridges providing the same Force, you've got more recoil in the heavier bullet according to Newton's Third Law.

Now my question. Recoil has been classified as "Pushy," vs. "Snappy," depending on the rate-of-burn for various powders. I.e. faster burn gives a more 'snappy' recoil. Is there a way to determine the ACCELERATION of a projectile of a given weight using various powders?

Do my calculations hold water as far as felt recoil goes, or is there something I'm missing?

To you folks who have made it this far through my post, I say thank you. I certainly look forward to hearing your thoughts on my theory.

[Uncle Grinch]

uh.... OK. I have a math deficiency....

(my wife won't let me keep the checkbook)

[Newboy]

The felt recoil has more to do with the force acting on the time duration.

A very fast pressure curve may, indeed have identical energies to a slow curve, but will be percieved as harsher, "snappier", simply because it is acting over a shorter time.

Don't confuse momentum and energy, this invariably starts the never-ending argument of big slow bullets versus small fast ones.

[MTWeatherman]

The easiest way to determine recoil is with an online recoil calculator. I don't know why anyone would want to go into all the math when its not necessary. You need the velocity and weight of the bullet, the powder charge, and the rifle weight to do it.

I'm not going to go into all the necessary calculations in your example, but if you want to do it yourself, I located the following link which does a good job of explaining it.

http://www.loadammo.com/Topics/August01.htm

Note: Momentum = MV, Force (F) =MA, and Kinetic Energy (KE)= 1/2 M(V squared).
What you want is the kinetic or recoil energy of the rifle...that's what felt recoil is...not force (or momentum).

In a nutshell: Newton's law states for every action...there is an equal and opposite one. The Force of the rifle must equal the force of the bullet and expanding powder. F=MA, however, momentum is directly related to F so we can substitue momentum in the formula....the momentum of the rifle has to be equal to the momentum of the bullet + the momentum of the powder. Calculate that (MV) for both, add them, and use that to solve for the velocity of the rifle (since we can calculate the M in the MV). Once you have the velocity of the rifle, you plug that into 1/2 M(V squared) to get kinetic energy (recoil energy) of the rifle. Do that with the two different weight bullets and you have your answer.

The reason recoil energy is noticably increased with heavier bullets is because typically the velocity of the heavier bullet doesn't fall enough in the MV equation to offset the mass increase...the result is an increase in the rifle velocity. Since that velocity is squared to get the recoil energy...it doesn't take much increase in rifle velocity for the recoil energy to become noticeable.

As far as acceleration with various powders...only thing I can think of that might help there would be "Load from a Disk". I don't have it so can't address it directly. Go to "features" at the bottom of the above linked website and you'll find it.

FWIW...Using your example....from a recoil calculator...50 grains of powder and 1700 fps for both loads, rifle weight 7.5 lbs:

151 gr bullet...rifle recoil velocity = 9.37 fps = 10.22 ft lbs recoil energy
200 gr bullet................................=10.95 fps =13.97 ft lbs recoil energy

I used a ballistics calculator to determine that. Here's a good online version for recoil calculation...save yourself some work.

http://www.handloads.com/calc/recoil.asp

Hope this helps.

[Blackwater]

Stock dimensions also vary, which changes the manner in which the recoil is APPLIED to our bodies. Recoil pads lengthen the time in which the energy is applied also, which "softens" it to our sense of feel on the nerves that determine the sensation. The simple figure of the energy created doesn't tell the whole story. How and over what time period it's applied have a lot to do with how recoil's perceived.

I read somewhere where momentum is a better way to compare recoil than energy, but haven't ever given it any consideration, because it just hasn't been a fruitful pursuit for me. I've spent my time trying to adapt myself and my attitude and body to the recoil of whatever I've been shooting, but it is interesting how recoil sensation can vary.

Oddest, or maybe I should have said most unexpected thing I've encountered re recoil is when firing some long strings with my BPCR and 533 gr. bullets with black powder. Recoil didn't SEEM bad, but after 40 some shots, my shoulder was a bit tender later that day, and I had a little yellow bruise on my shoulder the next day. Didn't SEEM like enough to cause that. The steel butt probably had something to do with it, probably.

[Trailblazer]

John Barsness did an article in Rifle magazine several years ago about calculating recoil. In addition to using the mass and acceleration of the bullet and powder he added terms to the equation to account for what he called the rocket effect of the powder gases exiting the muzzle. Rocket effect is determined by weight of the powder and bore size. More powder in a smaller bore increases the rocket effect. Barsness felt his equation accounted for felt recoil better than the traditional equation. He used the example of the 300 magnum compared to a 338 magnum. He was trying to account for the fact that with equal weight bullets, velocities and rifles, the recoil of the 300 feels worse than the 338. Makes sense to me!

[felix]

Yeah, the rocket effect makes a lot of sense to me, too. Otherwise, the muzzle breaks would not work as well as they do. But, the noise criteria has even more damaging effects to my tendancy to flinch with the boomers. So, in retrospect, I'd rather have the physical recoil over that of mental. ... felix

[Scrounger]

Yeah, the rocket effect makes a lot of sense to me, too. Otherwise, the muzzle breaks would not work as well as they do. But, the noise criteria has even more damaging effects to my tendancy to flinch with the boomers. So, in retrospect, I'd rather have the physical recoil over that of mental. ... felix

Felix, ever hear of the Hydrocoil? The buttstock was in effect a piston tube the rest of the gun went into for an inch or so. Resistance to absorb the recoil came from hydraulics, pneumatics, or spring, I don't remember which. With all the ingenuity we have available here, we ought to be able to make a cheap one of those. Better yet, how about a small unit we could place at our shoulders and use any rifle with it for shooting off the bench. Friend of mine in CA had a good arrangement for shooting heavy calibers from the bench. He hooked a nylon towing strap at the front corner of the bench, ran it behind his buttplate, then to the other front corner of the bench. The strap took all the recoil and transferred it into trying to move the bench, whicj was quite heavy. Being lazier and less inventive, I just put a sandbag between my shoulder and the gun. Helps a lot.

[MTWeatherman]

The rocket effect is very real and is indeed the principle of a muzzle brake or a ported barrel. Seems to me I have heard about 10 to 20% added recoil to some firearms (won't swear to the value though)...but very dependent on caliber, etc.. Not insignificant . However, it doesn't lend itself to an easy calculation by hand. In a vacuum, the formula I gave would give the true answer (and that's the one usually used)....no air for the escaping gas to push on. However, in the real world, the escaping gas pushes on that air and creates the rocket effect. It is complicated even more in that there should be more rocket effect at sea level than in the mountains...more air to push on. The effect would also be dependent on the amount of powder gas, its velocity, and its duration. That would definately make its effect more pronounced with in the large cased magnums due to the heavy charges of slow powder.

I do believe that some of the recoil calculators attempt to compensate for the "rocket effect" in their equations. Another good reason to use them. However, not many describe the way they do their actual calculations.

[woody1]

I just put a sandbag between my shoulder and the gun. Helps a lot.
About 1/3 of a bag of shot between stock and shoulder doubles the weight of most rifles. Regards, Woody

[fourarmed]

As a physics teacher, I'm always a sucker for this discussion. The basis for what happens is explained by Newton's laws of motion (naturally). The 3rd one says that if the rifle exerts a force on the bullet, then the bullet exerts an equal and opposite force on the rifle. The 2nd one says that the acceleration of each is therefore in inverse proportion to their mass. So the rifle accelerates much less than the bullet, and the bigger the disparity in their masses, the bigger the difference in their accelerations, and the slower the rifle will be going when it hits your shoulder. That's what's called the "hand-waving" description of what happens.

A consequence of the second law is that the total momentum (MV, as has been pointed out) must be the same after a third law event as it was before. Thus the free recoil velocity (backwards) of the rifle, times its mass, must equal the sum of the MVs (forward) of the ejecta - the bullet and the powder gases. The bullet velocity is easily measurable, the velocity of the gases is not. However, the usual empirical formulas estimate this with accuracy that is good enough. So you take the bullet mass times its velocity, plus the powder mass times the estimated velocity of the escaping gases (this is the "rocket" part). You divide that by the rifle mass, and you get the recoil velocity of the rifle. You square that, divide by 2, and multiply by the rifle mass to get the rifle energy, which contrary to what many people think, is NOT equal to the ejecta energy.

The reason the muzzle brakes work is that they redirect part of the gas sideways, taking it out of the equation. Another common misconception is the part about rockets not working in a vacuum because there is nothing for the escaping gases to push against. In fact, there is...they push against the rocket.

So a lighter bullet could give more recoil IF IT WERE GOING FAST ENOUGH.

[felix]

The spring effect (in the stock, et. al.) by any means (actual spring, hydro, etc.) would help the shoulder. Just like those cannons that allow the barrel assembly to move rearward. Whatever method is chosen, it would have to be a completely overdamped system in itself, which means absolutely no foward recoil from the system should be allowed to occur. BB gun scopes are made to cancel out (as much as possible) the foward recoil after the rearward recoil, because it is damn near impossible to contain the energy in the "recoil" absorbing spring without some kickback. I see a real engineering job here. ... felix

[felix]

Like forarmed says, some sideward motion "of something" within the stock spring, would be indicated to eliminate any foward recoil. Or, at least, some kind of energy diversion that would not be transferred into the foward direction. ... felix

[felix]

Well, just another thought. Magnetic damping! Electric motor in reverse, charging an external battery. Instead of a bag of shot in or outside of the stock, why not an electric motor of sorts within the stock? Forearmed, you need to put your young engineers to work! ... felix

[BABore]

I like to use the felt recoil meter I had installed at the factory. Those nerve endings running along your rotator cuff.

Level 1 Recoil: Huh
Level 2 Recoil: Uh Huh
Level 3 Recoil: Hempt
Level 4 Recoil: Ooh
Level 5 Recoil: Ooof
Level 6 Recoil: Holy Sh!!!!!t

:veryconfu :shock:

[MTWeatherman]

Forearmed:
The whole issue is exceeding complex isn't it?

As defined here, the "rocket effect" is the presumed extra force supplied in recoil by gas pressure against the air...not the force actually supplied by a true rocket or the bullet and gas momentum. In the a true rocket, the force is largely provided by the momentum of the escaping gas, but also by gas pressure against the atmosphere if within it.

As defined here, in order for the gas to exert a "rocket force" against the gun...it would have to be pushing on something besides the gun...the exerted force on the rifle would have to be equal to the exerted force on something (air) or we've got a reaction without an equal and opposite one. Without air,There still would be acceleration of the firearm due to the ejecting gas and its associated momentum, however, that's not what I would consider a "rocket" effect as discussed here. Rockets in space (a vacuum) work...we know that. However, once they leave the atmosphere that is completely due to the ejection of mass and the associated momentum....they've just lost the component of gas pressure on the atmosphere. Air is not required for a rocket to work.

How much effect that "rocket effect" has in recoil is an open question. First off, the effects of redirecting of gas in a muzzle brake or ported barrel affects recoil not just due to the "rocket" effect. The direction of the mass acceleration has changed so even if there were no "rocket" effect there would still be a force acting against the direction of recoil due to the momentum of the gas.

It makes it real tough to calculate any "rocket" effect on recoil. With sophisticated equipment, one could usea good chronograph and a recoil calculator to calculate the theoretical recoil...then measure the actual recoil velocity of the rifle. Any difference would have to be due to the "rocket effect". All this would assume no porting which would greatly complicate the issue.

I quoted the 10 to 20% increase in energy due to "rocket" effect but I'm skeptcial its effect could be could be that high on actual felt recoil. Since recoil due to the bullets momentum ends when the bullet exits the barrel, at that point we have a short period of recoil contribution due to gas momentum and "rocket" effect. Seems to me, that its main effect would be lengthening of the recoil pulse...more of an increase in duration rather than severity. However, don't really know...or even if the effect has been rigorously tested. The 10 to 20% could be just a number pulled from you know where.

[BruceB]

I have placed that "Handloads" recoil calculator on my Favorites list.

I ran the numbers for my two hunting loads which I took to Alberta last fall, namely the .338 Winchester Magnum(225 TSX @ 2800), and the .416 Rigby (365 RCBS, 2100 fps).

It was quite a coincidence, because the 'paper' recoil energy was almost precisely the same for the two loads, at around 38 pounds.

What is REALLY interesting is how different the two feel to the shooter, with the .338 being much sharper, due of course to higher velocity both forward and back. The .338 has a muzzle brake, and it really does tame it to a large degree.

Good thread here, very educational and interesting.

[fourarmed]

MTW, when I teach these types of problems, I draw a box around the object of interest (mostly so I can tell the students to "think inside the box") In the case - or rather box - of the rocket, there is the rocket and there is the hot gases it is ejecting. The rocket pushes on the gases, the gases push on the rocket. If you want to draw the box around the gases and the air mass behind it, then the gases push on the air mass, and the air mass pushes back on the gases. The air mass doesn't push on the rocket - at least not on the rear of it. The air does push against the front of the rocket, slowing it down. Which is why the rocket works BETTER in vacuum than it does in air, not worse.

Maybe it would help to think of it this way: A propeller plane pushes back on the air, the air pushes forward on the propeller. A rocket works in the absence of air because it carries its own air to push against.

[fourarmed]

Felix, I've been trying to put them to work for over 30 years. You'd be amazed what a tough job that is. Maybe if I set a better example...

MTW, the 10-20% reduction with a MB you quote is conservative if anything. The bigger the proportion of the reaction momentum is due to the gases, the better they work. A big case using lots of powder to fire a small bullet is where the brake really shines. On something like a .30-.378 I have read figures over 30%.

Bruce B, you are correct regarding the difference in recoil effect when it happens fast. Momentum divided by time is force. The shorter the elapsed time of the event, the greater the perceived force.

[MTWeatherman]

Fourarmed:
I agree with your first paragraph...and the first sentence in paragraph 2...perhaps we're looking at or trying to say the same thing in different ways.

I hope we can agree to disagree on one point however. I would disagree with:
"A rocket works in the absence of air because it carries its own air to push against. "

Maybe I'm misunderstanding your wording here. However, I would prefer to say a rocket works in the absence of air because it doesn't need it for combustion...and the momentum of the gas being expelled provides an equal amount of momentum to the rocket to accelerate it.

MV (expelled gas) = MV (rocket) The old equal and opposite reaction again.

One of the long range rockets under consideration for deep space exploration involves an ion generation propulsion system. No combustion gas, and no air...just ions being expelled to the rear. Its not designed for initial launch...just to be used after the effects of earths gravity are negligble. Once underway, the actual amount of force generated by those ions is quite small due to the very small mass involved...but Newtons law still applies. The resulting force applied to the rocket will accelerate the rocket to very high velocities (theoretically) due to the long period of time involved (many years) and the lack of a decelerating force (other than weak gravitational fields).

Also, the 10 to 20% I quoted is not related to reduced recoil from a muzzle brake...it is the amount of incease in recoil energy supposedly related to the "rocket" effect alone. As mentioned previously, a muzzle brake is going to reduce recoil due the effect of the gas momentum being redirected to move against the recoil as well as the gas being redirected to cause the associated rocket effect. I would have no reason to argue with that 30%...just still don't know what the actual recoil contribution of the "rocket" effect is or how that increase would be perceived to the shooter.

[rebliss]

MTW said, "I would prefer to say a rocket works in the absence of air because it doesn't need it for combustion...and the momentum of the gas being expelled provides an equal amount of momentum to the rocket to accelerate it."

I ask both MTW and Fourarmed, would it be too simplistic to state that a rocket operates due to difference in pressure within the combustion chamber compared to that outside (Bernoulli's law, if memory serves)? If that were the case, wouldn't a rocket be more efficient at higher altitudes (ignoring decreased friction due to less air density)?

[felix]

Rebliss, you are correct. It is the pressure difference between that WITHIN the motor versus that OUTSIDE of the motor. A baloon with a perfect hole in it will move (in outer space) in the direction which is exactly 180 degrees on the other side (of the baloon) of the hole. This is how a rocket engine, or jet engine for that matter, really works. It has nothing to do with the exhaust itself of the motor (engine). Part of the efficiency of the motor is how well that 180 degrees is maintained in actual practice. ... felix

[felix]

OK, for those of you who cannot see the baloon principle, consider the baloon with a bunch of arrows in it, all pointing towards the outside of the baloon from the center. Those arrows represent force vectors. More pressure, the longer the vector pushing against the baloon surface. Punch a hole, albeit a tiny, tiny one. That force vector at that hole now looses some length, meaning less force in that outward direction. Now, add up all the vectors algebrically and they will all cancel out, except the two opposite each other, one pointing to the hole (actually through the hole by now), and the other pointing to the opposite direction, 180 degrees from the one having the hole. The net movement of the baloon will be where the sum of the force vectors is still the longest, and that will be the only one pointing to the opposite direction. The idea of an engine is to control that hole size (if you will) and to make sure all the other force vectors remain all cancelled out. In practice, this can never happen perfectly. Always room for improvement. ... felix

[Underclocked]

Okay then, why does a 94 Winchester in 30-30 kick the crap outta me? ;-)

[MTWeatherman]

I ask both MTW and Fourarmed, would it be too simplistic to state that a rocket operates due to difference in pressure within the combustion chamber compared to that outside (Bernoulli's law, if memory serves)? If that were the case, wouldn't a rocket be more efficient at higher altitudes (ignoring decreased friction due to less air density)?

I make no claim to any knowledge of rocket design, however, since you asked (and against my better judgement), I'll wade in and try to confine the issue to physics.

Yes the rocket does indeed operate due to the difference in pressure between the combustion chamber and the outside. However, that's not the whole story. It wouldn't work if it were a closed system. You need to allow mass to pass from the combustion chamber to the outside...in other words a hole, tailpipe, exhaust, port, etc. Seal that hole up and it stops working.

The key is the acceleration of mass between the combustion chamber and the outside...inducing a force on the rocket, balloon, etc. directly opposite the hole (equal and opposite reaction issue again). The higher the difference in pressure, the more mass could be expect to be transferred (assuming equal sized hole, etc.)so the greater the acceleration of the rocket, etc. Its the old third law of Newton again...already well discussed. If we are accelerating mass through that hole, the very definition of force (F=MA), we are inducing an equal and opposite force on the rocket. We have to continue accelerating mass through that hole to maintain the force on the rocket...when its gone, the rockets done. Thats also called running out of fuel.

As to whether or not a rocket is more efficient (ignoring air friction) at higher altitudes, my guess is yes due to the greater difference in pressure. However, It strikes me as possible that it might just create the potential for higher performance using more fuel in the process to offset efficiency. Hopefully someone with more expertise can give you a true and more detailed answer.

[fourarmed]

************************************************** *
Okay then, why does a 94 Winchester in 30-30 kick the crap outta me?
************************************************** **

You mean why does it say "Winchester" on your shoulder after shooting one in a tee shirt? Force divided by area is pressure. The smaller the buttplate area, the more pressure it exerts for a given force.

MTW, you are absolutely correct about the momentum of the ejecta. So now I don't understand why you think the hot gases need something else to push against other than the rocket (or the rifle muzzle). My comment about the rocket carrying it's own "air" instead of "reaction mass" was sort of a parable - a small lie told to illustrate an important truth.

The pressure differential is just another way of saying the gases are pushing against the rocket. Gases exert force via pressure. Same with rifle recoil. Instead of thinking about the momentum of the hot gases, you could say that when the bullet uncorks the barrel, the pressure of the gases against the breech of the gun is no longer balanced by pressure on the base of the bullet.

[lovedogs]

Man, I'm staying out of this one. Just listening to all this makes my head and shoulder hurt! Oh yeah, Underclocked... your '94 kicks 'cuz of poor stock design... poor for handling the kick. It hasn't got many foot pounds, but it's how they are transferred to you. There are so many variables involved in this topic it boggles the imagination. All I can say is if it kicks use some sort of recoil management. Manage it down to your comfort zone. Pad it, weight it, restock it... whatever it takes. Recoil is interesting to ponder but in the end all we can do is learn to manage it.

[carpetman]

I think I am a rocket---I just expelled some gas. Better not say that too loud,Africa will be chopping off part of my colon. I sure dont want to be a semi colon.

[MTWeatherman]

[QUOTE=fourarmed

MTW, you are absolutely correct about the momentum of the ejecta. So now I don't understand why you think the hot gases need something else to push against other than the rocket (or the rifle muzzle). [/QUOTE]

I think we may have a case of miscommunication here. Maybe we've been agreeing all along but not realizing it.

The issue is one of the "rocket effect"...real or otherwise. Maybe I'm misunderstanding here but many people attribute an increase in recoil due to the exhausting gases from the barrel pushing on the atmosphere...over and above the effect of the bullet and powder gas momentum. If that is its real definition, then you do need air for that "rocket" effect.

However, that effect(if real) is not normally calculated for when calulating for recoil. The simple fact is that whether or not there is that defined "rocket effect" (believe a case can be made for it), most recoil is related to the issue of momentum. Its the case of the momentum of the rifle equaling the momentum of the bullet + the powder as in my initial post. Pure momentum delivers the recoil and pushes the rocket regardless. I don't think the hot gases need something else to push against other than the rocket (or rifle muzzle)...unless one is talking about "rocket effect" as defined above. I didn't intend to leave the impression I did.

[fourarmed]

There are really two stages of recoil. The first is while the bullet is still in the barrel. There the momentum of the rifle backward at any instant equals the momentum of the bullet forward plus the momentum of the powder products forward. The speed of the powder products during this stage is usually taken to be half that of the bullet, since the powder stuff right behind the bullet is going at its speed, but the powder stuff in the chamber isn't going anywhere. During this stage, the bullet momentum generally dominates by a large factor

Things change considerably when the bullet exits the barrel. At this point the powder gases speed up considerably. If you look at the formula given in the link posted earlier, they use 4000 fps as the speed of the powder stuff. That is conservative, because it has to include the momentum of the powder during the first stage, and the escaping gases don't all travel in a straight line behind the bullet. This momentum of the unconfined gases is what is called the "rocket effect." That effect will be the same whether the rifle is fired in the atmosphere or in vacuum.

[Newboy]

If this type of discussion intrigues you, read Harold Vaughn's book "Rifle Accuracy Facts", from precision rifle magazine.

He is a retired rocket scientist who looks at these things in depth.