## "Knex War?" (The Math Bit)

Hey guy! Sharir1701 here and I just want to start off by saying that I'm not back into Knex warfare, but I have something to show you.

About a year and a half ago, I posted this forum topic. There I explained why, in my opinion, just adding more rubber bands to a gun will not actually produce an overall better gun (past a certain, low point). I'm not getting back into that discussion, but I simply wanted to share something regarding that.

Don't ask me how or why (I don't know, myself), but a few days ago I suddenly remembered that old assumption I made. Being a perfectionist, I hate leaving things unfinished or unexplained (particularly math and physics related inquiries). Back when I posted that topic, I had little to no formal Physics knowledge, and the beginnings of an understanding in mathematics. Now, I have a much broader understanding, much more knowledge, and an ability to explain and evaluate what I once couldn't. Anyway, what I'm trying to say is, back then, I didn't have the tools to prove my claims. I firmly believed in them, but I couldn't confirm them. It's nothing complicated, but, like I said, just something I'd like to share. I also want to point out that, although I personally don't enjoy spending days upon days perfecting a little plastic mechanism for firing (mostly) non-aerodynamic plastic projectiles, anymore, there still is a warm spot in my heart for the craft I once loved. More to the point - this is a small article just showing something I did in a few minutes the other day, that helps me to better understand how a Knex gun works. I hope that in writing this, more people that are still building guns, will think about more accurately calculating certain things about their guns to help improve their performance and hopefully produce more efficient guns. The final note I have is that I'm about to show you equations, all of which can be plugged with real, measurable numbers, to calculate to a high degree of accuracy, the forces at play. This means you can actually calculate the most efficient layout for a gun, and also that in designing your next, you will be able to use these equations, and many others, to find optimal solutions to your problems.

So, what's all this fuss about? Well, basically, I just proved with a few, painfully easy equations that my conjecture about the forces in a gun, working on the pin, is true. I'll just get to it:

First, Hooke's Law states that the force necessary to change the length of a spring or a (tense) rubber band is F=K*dX, where F is the force, dX is the distance you want to change, and K is a constant number, that each rubber band (or spring) has. You can quite easily measure both of these. For rubber bands connected parallel to each other (assuming they are the same type of rubber band, which ever is your chosen standard), this equation becomes F=K*dX*N, where N is the number of rubber bands used. dX and K are both constant in the regards of the pull of a pin on a standard Knex pin gun. Therefor, the amount of force required to cock a pin (pull it back to it's full length) is linearly correlated to the number of bands you put on your gun.

Next, if we examine Newton's equation of work and energy, W=dE=F*dX, where W is the work, dE is the change in energy in your system (input from an external force, i.e. your hand), F is the force applied along a length of movement, and dX is that length. Let us define the base position of the pin (not cocked, minimum tension on the rubber bands, fully in the barrel, etc.) as having 0 energy. This then means that the work applied to the pin by cocking it is equal to all the potential energy it has. From this, plugging in the force, we get Ep=K*(dX)^2*N. Let us assume a perfect world, where we neglect the effects of friction and air resistance, and assume all the momentum of the pin is transferred into the bullet as it fires (I will briefly mention in the end, why everything we're neglecting here just strengthens my claim in reality, but let's continue for now). After being released (in other words, shot), the maximum velocity the pin reaches right before the end of it's journey can be found using the equation for kinetic energy Ek=1/2*M*(Vmax)^2, using the fact that (again, neglecting energy wasted as heat due to friction) the energy is conserved, as no external force is working on the system, which then means that Ep,start=Ek,end => K*(dX)^2*N=1/2*M*(Vmax)^2 => Vmax = sqrt(2*K*(dX)^2/M) * sqrt(N). The first sqrt term in the final equation is all one big constant (again, K is the ratio associated with the rubber band, dX is the distance the pin travels, and M is the mass of the pin), meaning we can conclude that (C for constant) Vmax=C*sqrt(N). Finally, force applied by a moving, massive object can be calculated using Newton's second law, F=dP/dT (P is the pin's momentum, T is the time it takes for the pin to go from velocity Vmax to 0, transferring all it's energy into the gun and the bullet, but as I said, let us assume all of it goes into the bullet), or F=M*dV/dT (M, mass of the pin, dV is the difference in velocity, Vmax-0, which is simply Vmax. This is because P=M*V, which means dP=M*dV, ignoring relativity). So, F=M*C/dT*sqrt(N). The time varies slightly, but insignificantly, so let us assume it is a constant. So that's it. The force exerted by the pin on the bullet is some constant (calculatable, as mentioned and as shown), times the sqrt of N, the number of rubber bands on the gun.

So there you go. Just a little something I did out of the blue the other day and thought I would share a proof of my conjecture from what feels like eons ago. I hope you enjoyed.

Finally, I would like to tell you guys, perhaps as a little tease, since I'm not sure if I will ever upload it, but I have made 1 more gun after I stopped posting. I have already slightly teased about it in my user info. I guess I'll tell you guys what it is if I'm already posting something here again :) Possibly my most enjoyable, most well received, and quite innovative gun of all time? The REMPAR-2. I built the REMPAR-3 (I was also going to call it S5 when I thought about posting it). In a brief summary, it's a pump action, chambering (or bolt action as I and many others falsely used to call it), magazine fed rifle, that's only 5 layers thick all over, except one tiny area where it's 8 layers thick (1.5 extra on either side), as a reinforcement, not necessary if you use less rubber bands. Oh, actually, there's another small necessary area where it's 7 layers thick, but it's tiny and doesn't make the gun look bulky at all, and who cares. Plus, the one is around the pump, which looks quite natural, and the other is "disguised" as a detachable sight (not really detachable) that also looks fine. It also looked fairly good for a gun that I made, it used (if I remember correctly) a grand total of 0 broken pieces, which is a big accomplishment for me (the mag has some broken white rods, but fuck off), and it worked flawlessly, reaching ranges over 80ft and being able to go at 2-4 rounds per second, depending on your skill level with it. In other words, I could shoot 4rps; My girlfriend, with no experience or practice, could do 2rps, which says something about the gun's comfortability and ease of use in my opinion. It shot blue rods up to an accuracy of about 5x5cm (2x2in) over 30ft, which is amazing for a Knex gun, and it was truly super comfortable. It wasn't even long at all, which is saying a lot, looking at some other people's attempts at 5-wide pump actions (I may also be guilty of an attempt several years ago). Being quite short and only 5 wide, it is also quite light. I'm sorry, I just really liked this gun, I think it was truly my best creation ever (of course, the S3 is by far the most innovative gun I've made, but it's mostly a concept gun, not meant for real effectiveness). I may post it in the future, but I'll make no promises. I suppose I won't leave you guys completely hanging and take a picture of it. Oh yes, there's also a neat, fun little thing I did, originally because I saw no other option, but then I actually really loved it. I'm talking about the mag-lock. It locks automatically (there's a band on it, but it ripped a long time ago, so...) and the mag cannot fall out. Then you pull on it with your middle finger, similar to a trigger, and the mag just drops right out. Love it :)

About a year and a half ago, I posted this forum topic. There I explained why, in my opinion, just adding more rubber bands to a gun will not actually produce an overall better gun (past a certain, low point). I'm not getting back into that discussion, but I simply wanted to share something regarding that.

Don't ask me how or why (I don't know, myself), but a few days ago I suddenly remembered that old assumption I made. Being a perfectionist, I hate leaving things unfinished or unexplained (particularly math and physics related inquiries). Back when I posted that topic, I had little to no formal Physics knowledge, and the beginnings of an understanding in mathematics. Now, I have a much broader understanding, much more knowledge, and an ability to explain and evaluate what I once couldn't. Anyway, what I'm trying to say is, back then, I didn't have the tools to prove my claims. I firmly believed in them, but I couldn't confirm them. It's nothing complicated, but, like I said, just something I'd like to share. I also want to point out that, although I personally don't enjoy spending days upon days perfecting a little plastic mechanism for firing (mostly) non-aerodynamic plastic projectiles, anymore, there still is a warm spot in my heart for the craft I once loved. More to the point - this is a small article just showing something I did in a few minutes the other day, that helps me to better understand how a Knex gun works. I hope that in writing this, more people that are still building guns, will think about more accurately calculating certain things about their guns to help improve their performance and hopefully produce more efficient guns. The final note I have is that I'm about to show you equations, all of which can be plugged with real, measurable numbers, to calculate to a high degree of accuracy, the forces at play. This means you can actually calculate the most efficient layout for a gun, and also that in designing your next, you will be able to use these equations, and many others, to find optimal solutions to your problems.

So, what's all this fuss about? Well, basically, I just proved with a few, painfully easy equations that my conjecture about the forces in a gun, working on the pin, is true. I'll just get to it:

First, Hooke's Law states that the force necessary to change the length of a spring or a (tense) rubber band is F=K*dX, where F is the force, dX is the distance you want to change, and K is a constant number, that each rubber band (or spring) has. You can quite easily measure both of these. For rubber bands connected parallel to each other (assuming they are the same type of rubber band, which ever is your chosen standard), this equation becomes F=K*dX*N, where N is the number of rubber bands used. dX and K are both constant in the regards of the pull of a pin on a standard Knex pin gun. Therefor, the amount of force required to cock a pin (pull it back to it's full length) is linearly correlated to the number of bands you put on your gun.

Next, if we examine Newton's equation of work and energy, W=dE=F*dX, where W is the work, dE is the change in energy in your system (input from an external force, i.e. your hand), F is the force applied along a length of movement, and dX is that length. Let us define the base position of the pin (not cocked, minimum tension on the rubber bands, fully in the barrel, etc.) as having 0 energy. This then means that the work applied to the pin by cocking it is equal to all the potential energy it has. From this, plugging in the force, we get Ep=K*(dX)^2*N. Let us assume a perfect world, where we neglect the effects of friction and air resistance, and assume all the momentum of the pin is transferred into the bullet as it fires (I will briefly mention in the end, why everything we're neglecting here just strengthens my claim in reality, but let's continue for now). After being released (in other words, shot), the maximum velocity the pin reaches right before the end of it's journey can be found using the equation for kinetic energy Ek=1/2*M*(Vmax)^2, using the fact that (again, neglecting energy wasted as heat due to friction) the energy is conserved, as no external force is working on the system, which then means that Ep,start=Ek,end => K*(dX)^2*N=1/2*M*(Vmax)^2 => Vmax = sqrt(2*K*(dX)^2/M) * sqrt(N). The first sqrt term in the final equation is all one big constant (again, K is the ratio associated with the rubber band, dX is the distance the pin travels, and M is the mass of the pin), meaning we can conclude that (C for constant) Vmax=C*sqrt(N). Finally, force applied by a moving, massive object can be calculated using Newton's second law, F=dP/dT (P is the pin's momentum, T is the time it takes for the pin to go from velocity Vmax to 0, transferring all it's energy into the gun and the bullet, but as I said, let us assume all of it goes into the bullet), or F=M*dV/dT (M, mass of the pin, dV is the difference in velocity, Vmax-0, which is simply Vmax. This is because P=M*V, which means dP=M*dV, ignoring relativity). So, F=M*C/dT*sqrt(N). The time varies slightly, but insignificantly, so let us assume it is a constant. So that's it. The force exerted by the pin on the bullet is some constant (calculatable, as mentioned and as shown), times the sqrt of N, the number of rubber bands on the gun.

So there you go. Just a little something I did out of the blue the other day and thought I would share a proof of my conjecture from what feels like eons ago. I hope you enjoyed.

Finally, I would like to tell you guys, perhaps as a little tease, since I'm not sure if I will ever upload it, but I have made 1 more gun after I stopped posting. I have already slightly teased about it in my user info. I guess I'll tell you guys what it is if I'm already posting something here again :) Possibly my most enjoyable, most well received, and quite innovative gun of all time? The REMPAR-2. I built the REMPAR-3 (I was also going to call it S5 when I thought about posting it). In a brief summary, it's a pump action, chambering (or bolt action as I and many others falsely used to call it), magazine fed rifle, that's only 5 layers thick all over, except one tiny area where it's 8 layers thick (1.5 extra on either side), as a reinforcement, not necessary if you use less rubber bands. Oh, actually, there's another small necessary area where it's 7 layers thick, but it's tiny and doesn't make the gun look bulky at all, and who cares. Plus, the one is around the pump, which looks quite natural, and the other is "disguised" as a detachable sight (not really detachable) that also looks fine. It also looked fairly good for a gun that I made, it used (if I remember correctly) a grand total of 0 broken pieces, which is a big accomplishment for me (the mag has some broken white rods, but fuck off), and it worked flawlessly, reaching ranges over 80ft and being able to go at 2-4 rounds per second, depending on your skill level with it. In other words, I could shoot 4rps; My girlfriend, with no experience or practice, could do 2rps, which says something about the gun's comfortability and ease of use in my opinion. It shot blue rods up to an accuracy of about 5x5cm (2x2in) over 30ft, which is amazing for a Knex gun, and it was truly super comfortable. It wasn't even long at all, which is saying a lot, looking at some other people's attempts at 5-wide pump actions (I may also be guilty of an attempt several years ago). Being quite short and only 5 wide, it is also quite light. I'm sorry, I just really liked this gun, I think it was truly my best creation ever (of course, the S3 is by far the most innovative gun I've made, but it's mostly a concept gun, not meant for real effectiveness). I may post it in the future, but I'll make no promises. I suppose I won't leave you guys completely hanging and take a picture of it. Oh yes, there's also a neat, fun little thing I did, originally because I saw no other option, but then I actually really loved it. I'm talking about the mag-lock. It locks automatically (there's a band on it, but it ripped a long time ago, so...) and the mag cannot fall out. Then you pull on it with your middle finger, similar to a trigger, and the mag just drops right out. Love it :)

active| newest | oldestMath is fun. It's pretty natural to understand that more bands doesn't linearly make a gun shoot further. Graphing or unrolling the equation makes it pretty easy to see.

There's still an increase in power with each rubberband, though. In a war, it's important to maximize range any way you can. If it means adding just enough rubberbands so that your gun doesn't explode, then that's what you need to do.

Yes, I do remember your previous comment on the aforementioned topic. I will have to say I disagree from my Knex war experiences, but that's getting into opinions and personal preference. As I said, I'm not here to start up this discussion again, but merely to show mathematically why this happens, in the hopes to aspire the current generation of Knex gun builders to perhaps use more accurate measurements and more mathematics in their design process.

Just out of curiosity, you seem to still be active on these forums. Do you still build guns and have wars with your group from KI?

I'm not completely active. We still have wars every year, though. We usually build a few new guns in the weeks before the war. Knex wars are always a lot of fun, even though I'm barely active.

You talked about your knex war experiences. Were they indoors or outdoors? Against friends or other designers? Were people using TRs or replicas?

Dude I am so hyped to see you posting again. LOL, in all honesty, I didn't read the whole thing (its quite a long epistle), I might though...

I'd agree to that. While mathematically this proves your point I

think you may be leaving out that the constant K should change to equal

the new rubberband force. When adding more bands it changes the elastic

constant, as to how, no idea...

I do believe you're wrong. The constant K does not change, assuming you're using identical rubber bands. It may vary slightly because of production imperfections, but it generally stays the same. That's why it's called a constant. You may claim that it changes linearly, depending on the number of rubber bands, but that's the same as multiplying the equation by N, the number of rubber bands, as I did.

This is highly generalized but basically you get less and less return on each band you add. There you go, kiddies.

I wish we could see tests in slow mo. Someone did that with Killerk's take on my Oodassault pistol and, though he set it up wrong, it was interesting watching it. It'd be nice to see some side by side shots adding more and more bands just to see how much faster the pin actually travels.

Something else I was thinking: The traditional method to get very high returns is using a longer weapon built for maximizing acceleration, such as a slingshot or rail pin, but this is cumbersome in my opinion. You have to resist more and more force over a long distance. So, another thing I wanted to test was focusing on acceleration via numerous, highly stretched bands over a short distance and then utilizing torque to charge the pin, like a lever. Obviously there'd be far less return per band overall, but I was wondering, ease of use wise, whether it'd get any sort of practical range on top of a decent RoF.

If you could get that working, sure, it would be very easy to use. Unfortunately though, I had a similar idea (Allow me to refer you to my "failed" closed barrel slingshot concept), of using torque to more easily charge the pin (or in my case, slingshot, but the concept is all the same). I learned that the plastic pieces we use here simply aren't strong enough in their joints to handle the pressure of that kind of torque. Basically, using this torque idea, you're moving away some of the force your hand exerts traditionally towards the "leverage" acting on the hinge, but Knex simply can't handle this force, even at low rubber band usage. Had this been possible, I'm sure we'd be seeing a lot more Gatling-style guns that are based on this. Unfortunately, I don't believe it's possible with a reasonable amount of force, even over a short distance, as you're suggesting.

I also share your dislike for the traditional methods of getting higher returns such as slingshots. That's why I never messed with them. The only time I did was when I thought I could bring it to a point where you have no interaction with the slingshot itself. But that failed, as you know.

I'd like too see more of the gun too, but I can't help but say I beat you too the punch on this concept. I built Zip3 and it's five layers, seven on the pump. Enough of me ranting though, I'd love to build your s3 I've heard your pump actions are legendary and strong.

You beat me to posting it, but I finished mine round about at January of last year, so I beat you to building it by almost a year :) None-the-less, objectively speaking, mine's smaller, doesn't use an external rail (more stable), and only uses 2 broken white rods for the mag, and none for the gun. Also, I think mine looks better :P Arrogance aside (>:D), your Zip3 certainly looks like a cool weapon indeed.