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Engineering Nerf guns

ideal barrel length, range prediction, dart safety, etc.

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#1 Doom

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Posted 23 April 2013 - 07:12 PM

Many Nerfers want answers to engineering questions. Here are some examples:

  • What barrel length maximizes muzzle velocity?
  • What muzzle velocity will my Nerf gun have if it has X characteristics?
  • What will the range of my Nerf gun be if it has X characteristics?
  • How can I maximize the range of my Nerf gun?
  • How safe are my darts? Can I shoot an eye out or bruise someone with my Nerf gun?
  • How tight do darts have to fit to get good performance?
  • How can I make my Nerf gun more accurate?
  • How many shots can I get from a tank?
Unfortunately, a lot of the information available about these topics in the NIC and at other places like SpudFiles and the PunkWorks forum is wrong. There is a great deal of confusion about these topics even among actual engineers.

I've been informally distributing some fairly mathematical notes I wrote about engineering Nerf guns for several months now. These notes are a compilation of many things I've written since 2009 and are in no way complete or entirely accurate. But they are much better than what passes for Nerf physics currently. All the questions above are addressed in some form.

2014-03-29 version of my notes

The point of this thread is to discuss these notes and other theoretical aspects of Nerf gun design. Generally, "theory" threads are disallowed at NH, but that's because they are usually have no basis in reality. This thread hopes to remedy that.

Post if you have any questions about or suggestions for my notes, the details about how Nerf guns work and how to improve their function, or new ideas to investigate theoretically or experimentally. My hope is that engineers in the hobby can congregate in this thread and help each other, rather than have to wade through the muck elsewhere.

PM me about typos rather than posting them here.

Edited by Doom, 29 March 2014 - 05:18 PM.

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#2 storm98

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Posted 23 April 2013 - 08:46 PM

Wow, I am in Geometry (I am only in 8th grade) and this is way more than I expected, This will definitely help anyone who is serious, props!
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#3 Doom

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Posted 23 April 2013 - 09:18 PM

Yes, much of the theoretical parts of my notes require a background in calculus and differential equations. Anyone with a college-level background in engineering, math, computer science, or physics should be comfortable with the math required.

However, there's still a lot in my notes that is accessible to someone with only basic algebra knowledge. You can take equations from there and use them to understand or design a Nerf gun. You don't need to know how to derive it, just what the assumptions are behind its derivation. There's also a lot of experimental data in there you can just understand as-is.

Also, all of these topics can be learned by a dedicated high school student. I'm hoping I can inspire a few of you all to learn a thing or two about applied math and engineering.
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#4 LockeWatts

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Posted 23 April 2013 - 10:35 PM

Damn man, I'm seriously impressed with this. This reads like a senior thesis. I'm just starting now, I'll chime in from the CS prospective here as soon as I get through it.

EDIT: Though a table of contents would be nice.

Edited by LockeWatts, 23 April 2013 - 10:42 PM.

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#5 Guitarzan

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Posted 24 April 2013 - 06:57 AM

Wow man this is extremely cool. I'll weigh in my thoughts after reading through the rest of it. All I can say now is that I'm extremely impressed at the way you account for all aspects of a nerf gun, such as deadspace and the true initial and final volumes of a plunger tube. For what it's worth I'm a junior Chem major so the physics and thermodynamics stuff is right up my alley.
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#6 Doom

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Posted 24 April 2013 - 07:25 AM

LockeWatts, Guitarzan, thanks for the positive comments.

LockeWatts, a table of contents is a good idea. Not sure why I didn't have one before. I've updated the file to have a table of contents. Also, these notes lack a big part that you might be interested in: computer simulations. I've spent a lot of time developing computer simulations of compressed gas guns (not just Nerf guns), but I haven't got around to writing about them yet. I'll fix that this year. Computers simulations are much more accurate that than simple theory I develop in my notes. The theory exists mainly to help understand what controls what (it is qualitative) but the computer simulations are quantitative.
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#7 Daniel Beaver

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Posted 24 April 2013 - 09:39 AM

Thanks for updating these, I'll give it a couple read-throughs this week (most of this is familiar to me, but I think it was a couple years ago last time I read it).

I'm finishing some of my major projects from earlier this spring at work, so we'll get together sometime and discuss getting a new round experimental data (and fill out sections 5 and 6 some more).
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#8 KaneTheMediocre

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Posted 24 April 2013 - 06:33 PM

LockeWatts, Guitarzan, thanks for the positive comments.

LockeWatts, a table of contents is a good idea. Not sure why I didn't have one before. I've updated the file to have a table of contents. Also, these notes lack a big part that you might be interested in: computer simulations. I've spent a lot of time developing computer simulations of compressed gas guns (not just Nerf guns), but I haven't got around to writing about them yet. I'll fix that this year. Computers simulations are much more accurate that than simple theory I develop in my notes. The theory exists mainly to help understand what controls what (it is qualitative) but the computer simulations are quantitative.


Writing about them is good, and you will probably do that at some point, but lacking that graphs of value A vs value B under simulated conditions SDKJHAODSALKASJD would be very helpful to gain a practical understanding.
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#9 Doom

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Posted 24 April 2013 - 07:05 PM

Writing about them is good, and you will probably do that at some point, but lacking that graphs of value A vs value B under simulated conditions SDKJHAODSALKASJD would be very helpful to gain a practical understanding.


Yeah, I should have done that already. I'll add it to my to-do list. I can fire up my existing simulator, but I need to clean it up a bit before then and I don't have the time at the moment.
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#10 Lucian

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Posted 24 April 2013 - 07:16 PM

This is a great read and very interesting, I applaud your efforts Ben, I aspire to comprehend more of this.

Also here!
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#11 azrael

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Posted 24 April 2013 - 10:42 PM

Oooh. What's been updated? I think I first found your Nerf Ballistics Physics awesomeness about two months ago?

Really enjoyed it, mostly because a lot of it either confirmed some of my theories or gave me a new way of thinking about things.
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#12 Draconis

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Posted 26 April 2013 - 08:11 PM

Thanks for the update, Doom! I'll be reading this again tonight Instead of doing laundry. Hope I don't get in to trouble for that.
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#13 Doom

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Posted 27 April 2013 - 09:25 AM

Lucian, thanks for spotting the typo. The notes are probably full of them.

In the future, PM me about typos to avoid cluttering up the thread. I added this to the first post.

As for what's changed, I've listed the revision history of the document below. If I gave you an older version, check the revision number on the top left of the first page and look at the table below. Note that a lot of the things listed below showed up in todo lists that do not appear in the actual document. I'll change that. I also have a lot of simple programs to calculate things and produce plots. I'll look into releasing these programs. Finally, a lot of little typos get fixed without being noted in the revision log.

------------------------------------------------------------------------
r43 | ben | 2013-04-25 17:07:46 -0400 (Thu, 25 Apr 2013) | 1 line

Cleaned up code a bit.
------------------------------------------------------------------------
r42 | ben | 2013-04-24 08:08:25 -0400 (Wed, 24 Apr 2013) | 1 line

Added a TOC.
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r41 | ben | 2013-04-14 22:58:00 -0400 (Sun, 14 Apr 2013) | 1 line

Fixed a typo in the LaTeX code and added a few more checks to checkdoc.sh.
------------------------------------------------------------------------
r40 | ben | 2013-04-02 22:32:51 -0400 (Tue, 02 Apr 2013) | 1 line

Added a text note about contact mechanics books.
------------------------------------------------------------------------
r39 | ben | 2013-03-26 23:08:24 -0400 (Tue, 26 Mar 2013) | 1 line

Added script to calculate energy stored in a gas chamber.
------------------------------------------------------------------------
r38 | ben | 2013-03-17 21:21:15 -0400 (Sun, 17 Mar 2013) | 1 line

Added a script to convert from scfm to g/s.
------------------------------------------------------------------------
r37 | ben | 2013-02-17 12:21:28 -0500 (Sun, 17 Feb 2013) | 1 line

Added a few notes to the TeX file.
------------------------------------------------------------------------
r36 | ben | 2013-02-10 22:47:52 -0500 (Sun, 10 Feb 2013) | 1 line

Added a few TODOs to the tex file.
------------------------------------------------------------------------
r35 | ben | 2013-02-06 13:15:28 -0500 (Wed, 06 Feb 2013) | 1 line

Added a few TODO items.
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r34 | ben | 2013-02-06 09:40:06 -0500 (Wed, 06 Feb 2013) | 1 line

Fixed a few minor errors.
------------------------------------------------------------------------
r33 | ben | 2013-02-06 09:29:49 -0500 (Wed, 06 Feb 2013) | 3 lines

Fixed a mistake in the explicit range efficiency equation.

Upgraded to a more recent LaTeX compilation script.
------------------------------------------------------------------------
r32 | ben | 2013-01-21 10:18:03 -0500 (Mon, 21 Jan 2013) | 1 line

Moved a few files around.
------------------------------------------------------------------------
r31 | ben | 2013-01-19 18:25:41 -0500 (Sat, 19 Jan 2013) | 1 line

Made a few changes to the text.
------------------------------------------------------------------------
r30 | ben | 2013-01-13 22:40:06 -0500 (Sun, 13 Jan 2013) | 1 line

Made a few fixes.
------------------------------------------------------------------------
r29 | ben | 2013-01-13 15:35:15 -0500 (Sun, 13 Jan 2013) | 1 line

Added small angle approximation to external ballistics section.
------------------------------------------------------------------------
r28 | ben | 2013-01-12 22:44:57 -0500 (Sat, 12 Jan 2013) | 1 line

Added some springer notes from 2011.
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r27 | ben | 2013-01-09 16:28:12 -0500 (Wed, 09 Jan 2013) | 1 line

Added some scanned notes.
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r26 | ben | 2013-01-09 10:06:45 -0500 (Wed, 09 Jan 2013) | 1 line

Added some external ballistics experimental data.
------------------------------------------------------------------------
r25 | ben | 2013-01-04 23:17:59 -0500 (Fri, 04 Jan 2013) | 1 line

Cleaned up a few things.
------------------------------------------------------------------------
r24 | ben | 2013-01-04 20:46:39 -0500 (Fri, 04 Jan 2013) | 1 line

Added optimal firing angle section.
------------------------------------------------------------------------
r23 | ben | 2013-01-03 12:34:57 -0500 (Thu, 03 Jan 2013) | 1 line

Started added a lumped parameter model of heat transfer in a gas reservoir. Fixed the compilation script so that references work on MightyMouse.
------------------------------------------------------------------------
r22 | ben | 2012-12-30 20:47:56 -0500 (Sun, 30 Dec 2012) | 3 lines

Updated scripts to not make plots visible by default.

Completed section on effective pressure. Added a section on compensating chronometer readings to better represent the true muzzle velocity.
------------------------------------------------------------------------
r21 | ben | 2012-12-29 11:29:11 -0500 (Sat, 29 Dec 2012) | 1 line

Changed octave directory to be titled scripts. Added some recent (but still a little old) internal ballistic models.
------------------------------------------------------------------------
r20 | ben | 2012-12-28 20:05:44 -0500 (Fri, 28 Dec 2012) | 1 line

Started adding sections on heat transfer losses and effective pressure for the alpha valve model.
------------------------------------------------------------------------
r19 | ben | 2012-12-26 23:04:10 -0500 (Wed, 26 Dec 2012) | 1 line

Added a few scripts from other ballistics work. Cleaned up some other scripts. Added a few notes.
------------------------------------------------------------------------
r18 | ben | 2012-11-09 23:08:25 -0500 (Fri, 09 Nov 2012) | 1 line

Updated list of ideas.
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r17 | ben | 2012-08-26 22:28:03 -0400 (Sun, 26 Aug 2012) | 1 line

Updated to-do lists.
------------------------------------------------------------------------
r16 | ben | 2012-07-12 22:30:57 -0400 (Thu, 12 Jul 2012) | 1 line

Forgot to add the range test data Octave script in the last commit.
------------------------------------------------------------------------
r15 | ben | 2012-07-12 22:30:32 -0400 (Thu, 12 Jul 2012) | 1 line

A few small corrections.
------------------------------------------------------------------------
r14 | ben | 2012-07-11 10:12:52 -0400 (Wed, 11 Jul 2012) | 1 line

Made a few small changes, including adding an estimate for the final plunger volume.
------------------------------------------------------------------------
r13 | ben | 2012-07-08 20:39:16 -0400 (Sun, 08 Jul 2012) | 15 lines

Added code to remove howpublished part of bibliography, which made the URL of a page print twice.

Added two frames atomatron's high speed video of a translucent springer shooting to illustrate that the final plunger volume is not necessarily zero.

Fixed subscripts in figures so that the non-variable/index subscripts are all \mathrm.

Started adding a glossary.

Expanded the adiabatic process ideal barrel length section.

Added a derivation of the range energy efficiency equation with the Lambert W function.

Added barrels section.

Added outlines of a few other sections.
------------------------------------------------------------------------
r12 | ben | 2012-07-08 00:41:05 -0400 (Sun, 08 Jul 2012) | 1 line

Added springer model and gas reservoirs section.
------------------------------------------------------------------------
r11 | ben | 2012-07-05 00:50:48 -0400 (Thu, 05 Jul 2012) | 1 line

Made many correction as suggested by suggest.sh.
------------------------------------------------------------------------
r10 | ben | 2012-07-04 18:28:14 -0400 (Wed, 04 Jul 2012) | 1 line

Added analytical pneumatic gun model section, among other things.
------------------------------------------------------------------------
r9 | ben | 2012-07-01 23:16:43 -0400 (Sun, 01 Jul 2012) | 1 line

Added time-to-target section and moved files around to automatically generate plots with Octave.
------------------------------------------------------------------------
r8 | ben | 2012-07-01 20:16:50 -0400 (Sun, 01 Jul 2012) | 1 line

Added many previously written sections.
------------------------------------------------------------------------
r7 | ben | 2012-03-11 13:17:31 -0400 (Sun, 11 Mar 2012) | 1 line

Added simplified analytical lumped-parameter internal ballistics perturbation equations.
------------------------------------------------------------------------
r6 | ben | 2012-03-11 10:59:05 -0400 (Sun, 11 Mar 2012) | 1 line

Started adding perturbation expansion for analytical lumped parameter internal ballistics section.
------------------------------------------------------------------------
r5 | ben | 2012-03-10 17:23:12 -0500 (Sat, 10 Mar 2012) | 1 line

Added a change of variables for the analytical internal ballistics case.
------------------------------------------------------------------------
r4 | ben | 2012-03-10 16:29:43 -0500 (Sat, 10 Mar 2012) | 1 line

Derived dimensionless lumped-parameter internal ballistics model that will be solved or approximated analytically.
------------------------------------------------------------------------
r3 | ben | 2012-03-10 11:40:45 -0500 (Sat, 10 Mar 2012) | 1 line

Fixed svn keywords.
------------------------------------------------------------------------
r2 | ben | 2012-03-10 11:39:19 -0500 (Sat, 10 Mar 2012) | 1 line

Added governing equations for the analytical lumped-parameter case.
------------------------------------------------------------------------
r1 | ben | 2012-03-10 10:04:11 -0500 (Sat, 10 Mar 2012) | 1 line

Added initial files.
------------------------------------------------------------------------


As you can see in the revision log above, I've been working on this particular document for over a year now. Some components were written before then and copied from other places. So I've been working on all of this for a while, but intermittently. Slow and steady probably won't win the race, but at least you'll finish.

Also, I'm aware of the problem with some square roots in the valves section. This is a bug in the typesetting program I am using, LuaTeX.

Edited by Doom, 27 April 2013 - 01:04 PM.

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#14 Samurai kidd

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Posted 28 April 2013 - 01:39 PM

Do I see statistics in there? I do!

Great read.

Something I was surprised not to find was a discussion on faux barrels. You touched on extra length and dart stability, but not necessarily faux barrels like the ones on the recon, retaliator, etc. Hopefully, anyone on this forum understands the physics behind extra plastic on the front of your blaster, but elsewhere I have seen several people claim that faux barrels improve accuracy, and in some cases even claims of increased velocity. I'd like to hear your thoughts.
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#15 Doom

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Posted 28 April 2013 - 01:51 PM

Something I was surprised not to find was a discussion on faux barrels. You touched on extra length and dart stability, but not necessarily faux barrels like the ones on the recon, retaliator, etc. Hopefully, anyone on this forum understands the physics behind extra plastic on the front of your blaster, but elsewhere I have seen several people claim that faux barrels improve accuracy, and in some cases even claims of increased velocity. I'd like to hear your thoughts.


Those are interesting claims. Could you provide a few links to where people claim faux barrels improve accuracy or muzzle velocity?

I did discuss porting (but not suppressors or muzzle brakes, see section 3.10.1), which might act similar to faux barrels in terms of accuracy. A faux barrel could reduce muzzle blast, decreasing the chance the dart will be knocked around when exiting the barrel.

Edit: Here's an interesting page about the faux barrels with some data. And here's the claim:

These results are clear visually: the "rifled" Retaliator barrel is more accurate than the unrifled Recon barrel, which is in turn more accurate than the absense of a barrel attachment.


I just skimmed the page and it seems that the author realizes that the barrel isn't really rifled and that the dart doesn't actually rotate. I think the difference between the faux barrel and not is explained by the muzzle blast effect. I'm not sure how the pattern inside of the barrel changes things.

Edit #2: To make my notes more popular, I've decided I'm going to revamp them one section at a time and post each section as an article here on NH. I'll start with the entire terminal ballistics section. I need to add a part about soft tips, KED limits for bruising (and welts?), and do a more thorough literature search to see if I missed any articles on projectile injury or other injuries (especially of the eyes). I'll make all of the hard to find references available as PDFs.

Edit #3: More data about barrel extensions.

Edited by Doom, 01 May 2013 - 08:25 PM.

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#16 PVC Arsenal 17

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Posted 13 May 2013 - 08:45 PM

Interesting work, Doom. I look forward to reading it when I have some time.

Also I'm glad to see that you and a few others are still active as I get back into this hobby.
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